theory of holonomy groups for supermanifolds · these holonomy groups have all properties of the...

MASARYKOVA UNIVERZITA

PRIRODOVEDECKA FAKULTA

Anton Galaev, Dr. rer. nat.

THEORY OF HOLONOMY GROUPS

FOR SUPERMANIFOLDS

habilitacnı prace

Brno 2013

Abstract

This thesis is dedicated to the study of the holonomy groups of pseudo-Riemannian ma-

nifolds and supermanifolds. First we review the classical classification of Berger of the

connected holonomy groups of Riemannian manifolds. This result has many applications

in geometry and theoretical physics and provides the motivation to our research. Next, we

discuss the classification of connected holonomy groups of Lorentzian manifolds and some

applications. After that we turn to supermanifolds. We introduce the theory of holonomy

groups for superconnections on supermanifolds. Due to the speciality of supermanifolds,

the standard approach to the holonomy groups does not work, never the less, we find the

way to define the holonomy group of a superconnection on a supermanifold. We show that

these holonomy groups have all properties of the usual holonomy groups. Then we provide

a classification of some classes of holonomy groups of Riemannian supermanifolds, in par-

ticular, we generalize the Berger result about the classification of the connected irreducible

holonomy groups of pseudo-Riemannian manifolds.

Obsah

1 Introduction 4

2 Holonomy groups of connections on vector bundles 6

3 Holonomy groups of pseudo-Riemannian manifolds 8

3.1 Holonomy groups of Riemannian manifolds . . . . . . . . . . . . . . . . . . 8

3.2 Irreducible holonomy groups of pseudo-Riemannian manifolds . . . . . . . . 10

3.3 Holonomy groups of Lorentzian manifolds . . . . . . . . . . . . . . . . . . . 10

4 Holonomy groups of connections on locally free sheaves over superma-

nifolds 13

4.1 Supermanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.2 Parallel sections and definition of the holonomy . . . . . . . . . . . . . . . . 15

4.3 Holonomy groups of linear superconnections on supermanifolds . . . . . . . 18

5 Holonomy groups of Riemannian supermanifolds 20

5.1 Berger superalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

5.2 Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

References 26

Paper A: A. S. Galaev, Metrics that realize all Lorentzian holonomy algebras. Int. J. Geom.

Methods Mod. Phys. 3 (2006), no. 5–6, 1025–1045.

Paper B: A. S. Galaev, Holonomy of supermanifolds. Abh. Math. Semin. Univ. Hambg. 79

(2009), no. 1, 47–78.

Paper C: A. S. Galaev, Irreducible complex skew-Berger algebras. Differential Geom. Appl. 27

(2009), no. 6, 743–754.

Paper D: A. S. Galaev, Irreducible holonomy algebras of odd Riemannian supermanifolds. Lo-

bachevskii Journal of Mathematics 32 (2011), no. 2, 163–173.

Paper E: A. S. Galaev, Irreducible holonomy algebras of Riemannian supermanifolds. Annals

of Global Analysis and Geometry 42 (2012), no. 1, 1–27.

1 Introduction

The holonomy group of a pseudo-Riemannian manifold is an important invariant of the

Levi-Civita connection; it gives information about the curvature tensor of the manifold

and about parallel sections of the vector bundles associated with the manifold, e.g. the

tangent bundle or the spinor bundle.

The most important result in the holonomy theory is the classification of the connected

holonomy groups of Riemannian manifolds obtained in 1955 by Berger [12]. It is important

to note that Berger obtained only candidates to the holonomy groups and it took three

decades before it was shown that all these groups can really appear as the holonomy

groups [19]. This classification has many applications both in geometry and theoretical

physics, in particular, in string theory compactifications, supersymmetry and M-theory,

see [14, 25, 56, 59, 60, 67]. We review this classification in Section 3.1.

In Section 3.3 we discuss the case of Lorentzian manifolds. In that case the classification

was achieved recently. The works [11, 51, 63] provide the list of possible connected ho-

lonomy groups. In [49] we give construction of metrics showing that all these groups are

holonomy groups. Recently we found some applications of the obtained classification. We

classified connected holonomy groups of Einstein Lorentzian manifolds [44] and obtained

a way to simplify the Einstein equation [42]. This work was motivated by the physical

paper [54]. Next, we classified Lorentzian manifolds with zero second covariant derivative

of the curvature tensor [39]. Remark that afterwords this result was proven in another

much more complicated way that does not use the holonomy groups [16]. In [52] we find

the local form of all conformally flat Lorentzian manifolds with special holonomy groups.

Note that before this problem was considered in the case of dimension four [58, 53] and

no solution was found. Further applications we discuss in [48, 35]. Lorentzian manifolds

with special holonomy groups and their relations to the relativity and supersymmetry are

considered in physical literature [22, 23, 24, 29, 55].

Since theoretical physicists discovered supersymmetry, supermanifolds began to play an

important role both in mathematics and physics [30, 64, 66, 76]. There is also an interest to

Riemannian supermanifolds, see e.g. [5, 3, 15, 26, 27, 28, 57, 65]. In particular, Calabi-Yau

supermanifolds (i.e. Riemannian supermanifolds with the holonomy algebras contained in

su(p0, q0|p1, q1)) were considered recently in several physical papers, e.g. [1, 72, 79].

We introduce the holonomy groups and holonomy algebras for superconnections on su-

4

permanifolds. Supermanifolds are generalizations of manifolds. On supermanifolds one

considers usual (even) coordinates and additional anticommuting (odd) coordinates. Even

coordinates parametrize the points of the supermanifold, while the odd coordinates do

not correspond to points. The functions on supermanifolds (called superfunctions) depend

both on even and odd coordinates. One may consider the value of such function at a point

of the supermanifold, but such values do not define the function: the language of points is

not appropriate for supermanifolds. Instead of vector bundles over supermanifolds one con-

sider locally free sheaves of supermodules over the structure sheaf of superfunctions. One

may define a connection and its curvature on such sheaf. The main examples corresponds

to the tangent sheaf, whose sections are vector fields. It is natural to ask for the definition

of the holonomy group of a connection over a locally free sheaf on a supermanifold. The

approach using parallel displacements along loops at a point does not work: considering

the points we loose some information, moreover, it is natural to expect that the holonomy

group should be a Lie supergroup, which is in particular a supermanifold and it is not

defined by the points. The main property of the usual holonomy group of a connection on

a vector bundle is that it gives the information about parallel sections of this bundle. First

we study the parallel sections of locally free sheaves on supermanifolds, we show that such

section is defined by its value at a single point (in the same time, an arbitrary section is

not defined by its values at all points). Then using the properties of parallel sections, we

define the holonomy algebra that is a Lie superalgebra that allows to control local parallel

sections. After that we define the holonomy group as a Lie supergroup.

The defined by us holonomy groups carry a lot of properties of the usual holonomy groups.

We consider examples of possible parallel tensor structures on a supermanifold with a

linear connection and give the corresponding holonomy groups, see Section 4.3. Some of

these structures have no classical analogues. In [46] we introduce the notion of the Berger

superalgebras, these algebras are candidates to the holonomy algebras of linear torsion-free

connections of supermanifolds. We give many examples of such algebras, again, some of

them have no classical analogues.

Then we consider the case of Riemannian supermanifolds. In this case the holonomy

algebra g is contained in the orthosymplectic Lie superalgebra osp(p, q|2m). We generalize

the result of Berger for the case of Riemannian supermanifolds. We assume that g is of

the form

g = (⊕igi)⊕ z, (1)

5

where gi are simple Lie superalgebras of classical type and z is a trivial or one-dimensional

center. This is natural assumption, since in the case of pseudo-Riemannian manifolds with

irreducible g this property follows automatically. In [36] we classify all possible holonomy

algebras of that form, see Section 5.2. The obtained list generalizes the list of Berger of

irreducible holonomy algebras of pseudo-Riemannian manifolds. We describe geometric

structures corresponding to each possible holonomy algebra. The classification in the case

of purely odd supermanifolds is obtained in [37, 45]. The proof in the general case is based

on this classification.

Now the natural problem is to construct examples of supermanifolds with each of the

obtained possible holonomy algebras. Note that examples of special Kahlerian manifolds

(i.e. Calabi-Yau manifolds) delivers the Calabi-Yau Theorem. In [72] it is shown that the

Calabi-Yau Theorem does not hold for Kahlerian supermanifolds of real odd dimension

two. In [79] it is shown that the arguments of [72] work only for the odd dimension two and

it is conjectured that Calabi-Yau Theorem is true for manifolds of odd dimensions bigger

then two. In [1] some examples of Calabi-Yau supermanifolds are constructed. Examples

of quaternionic-Kahlerian supermanifolds are constructed in [27]. The constructions in the

case of Riemannian and Lorentzian manifolds suggest that all the obtained examples of

possible holonomy algebras may be realized.

Thus we have introduced the theory of holonomy groups of the connections on superma-

nifolds that generalizes the theory of the usual holonomy groups. These results can be

used first of all in supergeometry. Since, on one hand, the holonomy groups of Riemannain

manifolds has applications in theoretical physics (in particular, in supersymmetry), on the

other hand, the concept of supermanifolds appeared as the mathematical formulation of

supersymmatry, we believe that our results will found applications in supersymmetry as

well.

2 Holonomy groups of connections on vector bundles

In this section we recall some standard facts about holonomy of connections on vector

bundles over smooth manifolds.

Let E be a vector bundle over a connected smooth manifold M and ∇ a connection on

E. It is known that for any smooth curve γ : [a, b] ⊂ R → M and any X0 ∈ Eγ(a) there

exists a unique section X of E defined along the curve γ and satisfying the differential

6

equation ∇γ(s)X = 0 with the initial condition Xγ(a) = X0. Consequently, for any smooth

curve γ : [a, b] ⊂ R → M we obtain the isomorphism τγ : Eγ(a) → Eγ(b) defined by

τγ : X0 7→ Xγ(b). The isomorphism τγ is called the parallel displacement along the curve

γ. The parallel displacement can be defined in the obvious way also for piecewise smooth

curves. Let x ∈ M . The holonomy group Hol(∇)x of the connection ∇ at the point x is

the subgroup of GL(Ex) that consists of parallel displacements along all piecewise smooth

loops at the point x ∈ M . If we consider only null-homotopic loops, we get the restricted

holonomy group Hol(∇)0x. Obviously, Hol(∇)0

x ⊂ Hol(∇)x is a subgroup. If the manifold

M is simply connected, then Hol(∇)0x = Hol(∇)x. It can be proved that the group Hol(∇)x

is a Lie subgroup of the Lie group GL(Ex) and the group Hol0x is the connected identity

component of the Lie group Hol(∇)x. The Lie algebra hol(∇)x of the Lie group Hol(∇)x

(and of Hol(∇)0x) is called the holonomy algebra of the connection ∇ at the point x. Since

the manifold M is connected, the holonomy groups of ∇ at different points of M are

isomorphic.

Remark that by the holonomy group (resp. holonomy algebra) we understand not just

the Lie group Hol(∇)x (resp. Lie algebra hol(∇)x), but the Lie group Hol(∇)x with the

representation Hol(∇)x ↪→ GL(Ex) (resp. the Lie algebra hol(∇)x with the representation

hol ↪→ gl(Ex)). These representations are called the holonomy representations.

The theorem of Ambrose and Singer states that the holonomy algebra hol(∇)x coincides

with the vector subspace of gl(Ex) spanned by the elements of the form

τ−1γ ◦Ry(Y,Z) ◦ τγ ,

where R is the curvature tensor of the connection ∇, γ is any curve in M beginning at

the point x; y ∈M is the end-point of the curve γ and Y,Z ∈ TyM .

Note that if E = TM is the tangent bundle of M , then

τ−1γ ◦ ∇rYr,...,Y1

Ry(Y, Z) ◦ τγ ∈ hol(∇)x,

where Y, Z, Y1, ..., Yr ∈ TyM .

A section X ∈ Γ(E) is called parallel if ∇X = 0. This is equivalent to the condition that

X is parallel along all curves in M , i.e. for any curve γ : [a, b]→M holds τγXγ(a) = Xγ(b).

The following theorem is usually called The fundamental principle is the main charac-

terisation of the holonomy groups and it is one of the main applications of the holonomy

groups.

7

Theorem 2.1. Let M be a smooth manifold, x ∈ M , E a vector bundle over M , and ∇a connection on E. Then there is a one-to-one correspondence between parallel sections

X ∈ Γ(E) and vectors Xx ∈ Ex preserved by the holonomy group Hol(∇)x.

Proof. Having a parallel section X ∈ Γ(E) it is enough to take the value Xx ∈ Ex. Since X

is invariant under the parallel displacements, the vector Xx is invariant under the parallel

displacements along the loops at the point x, i.e. under the holonomy representation.

Conversely, for a given vector Xx ∈ Ex preserved by Hol(∇)x define the section X ∈ Γ(E)

such that at any point y ∈ M it holds Xy = τγXx, where γ is any curve beginning at x

and ending at y. It is clear that Xy does not depend on the choice of the curve γ. �

3 Holonomy groups of pseudo-Riemannian mani-

folds

Let (M, g) be a connected pseudo-Riemannian manifold of signature (r, s) (r is the number

of minuses in the signature of the metric g). We will be interested in the case of Riemannian

manifolds (r = 0, i.e. g is positive definite) and in the case of Lorentzian manifolds (r = 1).

Denote by ∇ the Levi-Civita connection on M defined by the metric g; ∇ is the unique

torsion-free linear connection on M that preserves the metric g: ∇g = 0. Denote by G the

holonomy group of the connection ∇, then G ⊂ O(TxM, gx) ' O(r, s). For the holonomy

algebra it holds g ⊂ so(TxM, gx) ' so(r, s).

The connection ∇ can be extended to a connection on the tensor bundle T pqM of tensors

of type (p, q), and its holonomy group coincides with the representation of G on the

corresponding tensor space. Theorem 2.1. Implies that there is a one to one correspondence

between parallel tensor fields, i.e. sections of T pqM and tensors in T pqxM preserved by G.

E.g. (M, g) is a pseudo-Kahlerian manifolds (i.e. it admits a parallel Hermitian structure)

if and only if G ⊂ U( r2 ,s2).

3.1 Holonomy groups of Riemannian manifolds

If we consider the product of two Riemannian manifolds, then the holonomy group of

the obtaining manifolds is the product of the holonomy groups of these two manifolds

acting on the direct sum of the corresponding tangent spaces. Conversely, according to

8

the de Rham decomposition theorem [14], if the holonomy group of a Riemannian manifold

is not irreducible, then the manifold is at least locally a product of a (possibly trivial) flat

Riemannian manifold and of Riemannian manifolds with irreducible holonomy groups. In

particular, the holonomy group is the direct product of irreducible holonomy groups, and

this allows to restrict the attention only to irreducible holonomy groups.

Possible connected irreducible holonomy groups (i.e. possible irreducible holonomy alge-

bras) of not locally symmetric Riemannian manifolds classified Berger in [12]. Later it was

proved that all these algebras can be realized as the holonomy algebras of Riemannian

manifolds [19].

The holonomy algebra of a locally symmetric Riemannian space (M, g) at a point x co-

incides with {Rx(X,Y )|X,Y ∈ TxM}. Locally (M, g) is isometric to a symmetric space

H/G, where H is the group of transvections of that space; the holonomy group of that

space coincides with the isotropy representation of the stabilizer H of a point. The list of

indecomposable Riemannian spaces can be found e.g. in [14].

Here we list connected irreducible holonomy groups of not locally symmetric Riemannian

manifolds and we give the description of the corresponding geometries including the Ein-

stein condition and parallel forms (we do not include trivial parallel forms i.e. constant

function and the volume form on an orientable manifold):

• SO(n): generic Riemannian manifolds, no parallel forms;

• U(m) ⊂ SO(2m): Kahlerian manifolds, parallel Kahlerian 2-form and its powers,

not Ricci-flat;

• SU(m) ⊂ SO(2m): special Kahlerian manifolds or Calabi-Yau manifolds, parallel

Kahlerian 2-form, its powers, parallel complex volume form and its conjugate, Ricci-

flat;

• Sp(k) ⊂ SO(4k): hyper-Kahlerian manifolds, 3 independent parallel Kahlerian 2-

forms and forms obtained from their combinations, Ricci-flat;

• Sp(k) ⊕ Sp(1) ⊂ SO(4k): quaternionic-Kahlerian manifolds, parallel 4-from and its

powers, Einstein and not Ricci-flat;

• Spin(7) ⊂ SO(8): Ricci-flat, parallel 4-form;

• G2 ⊂ SO(7): Ricci-flat, a parallel 3-form and its dual.

9

Compact Riemannian manifolds with the holonomy groups SU(2), SU(3), G2 and Spin(7)

are extremely useful in theoretical physics, see [25, 56, 60] and references therein.

3.2 Irreducible holonomy groups of pseudo-Riemannian ma-

nifolds

The Wu decomposition Theorem [78] generalizes the de Rham Theorem for the case of

pseudo-Riemannian manifolds. It states that a pseudo-Riemannian manifold (M, g) with

not weakly irreducible connected holonomy group can be decomposed at list locally into the

product of pseudo-Riemannian manifolds (M0, g0),...,(Mr, gr) such that (M0, g0) is flat and

the holonomy groups of (M1, g1),...,(Mr, gr) are weakly irreducible. Recall that a subgroup

of a pseudo-orthogonal group is weakly irreducible, if it does not preserve any proper non-

degenerate subspace of the pseudo-Euclidean space. Consequently, a pseudo-Riemannian

manifold (M, g) is locally indecomposable if and only if its connected holonomy group is

weakly irreducible.

The particular case of weakly irreducible groups are irreducible ones. Possible connected

irreducible holonomy groups of non-locally symmetric pseudo-Riemannian manifolds clas-

sified Berger [12]. The list is the following: SO(r, s), SO(p,C), U(r, s), SU(r, s), Sp(r, s),

Sp(r, s)·Sp(1), Sp(r,R)·SL(2,R), Sp(r,C)·sl(2,C), Spin(7) ⊂ SO(8), Spin(4, 3) ⊂ SO(4, 4),

Spin(7,C) ⊂ SO(8, 8), G2 ⊂ SO(7), G∗2(2) ⊂ SO(4, 3) and GC2 ⊂ SO(7, 7). This list is the

generalization of the list from the previous section and the corresponding special geomet-

ries can be described in the same way as above.

Connected weakly irreducible not irreducible holonomy groups of pseudo-Riemannian ma-

nifolds are classified only in the case of Lorentzian manifolds, and we deal with this case

in the next section.

3.3 Holonomy groups of Lorentzian manifolds

Let (M, g) be a connected locally indecomposable Lorentzian manifold of dimension (1, n+

1). Then the holonomy algebra g ⊂ so(1, n + 1) of (M, g) is weakly irreducible. If g ⊂so(1, n+1) is irreducible, then g = so(1, n+1). This follows from the Berger classification,

and recently direct proofs of this result were obtained in [17, 31]. Suppose that g ⊂so(1, n+ 1) is not irreducible, the g preserves an isotropic line in R1,n+1.

10

We fix a basis p, e1, ..., en, q of R1,n+1 such that p and q are light-like vectors, g(p, q) =

g(q, p) = 1 and g(ei, ei) = 1 and the subspace E ⊂ R1,n+1 spanned by e1, ..., en is an

Euclidean subspace orthogonal to p and q. We obtain the decomposition

TxM = R1,n+1 = Rp⊕ E ⊕ Rq. (2)

Denote by sim(n) the subalgebra of so(1, n+ 1) that preserves the isotropic line Rp. The

Lie algebra sim(n) can be identified with the following matrix algebra:

sim(n) =

a Xt 0

0 A −X0 0 −a

∣∣∣∣∣∣∣∣ a ∈ R, A ∈ so(n), X ∈ Rn

. (3)

The above matrix can be identified with the triple (a,A,X). We get the decomposition

sim(n) = (R⊕ so(n)) nRn,

which means that R⊕so(n) ⊂ sim(n) is a subalgebra and Rn ⊂ sim(n) is an ideal, and the

Lie brackets of R⊕ so(n) with Rn are given by the standard representation of R⊕ so(n) in

Rn. The Lie algebra sim(n) is isomorphic to the Lie algebra of the Lie group of similarity

transformations of Rn. We assume that g ⊂ sim(n). We identify Rn and E.

Let h ⊂ so(n) be a subalgebra. Recall that h is a compact Lie algebra and we have the

decomposition h = h′ ⊕ z(h), where h′ is the commutant of h, and z(h) is the center of h.

If h ⊂ so(n) is irreducible, then z(h) 6= 0 implies h ⊂ u(n2 ); in this case h′ ⊂ su(n2 ) and

z(h) = RJ , where J is the complex structure.

The next theorem gives the classification of weakly irreducible not irreducible holonomy

algebras of Lorentzian manifolds.

Theorem 3.1. A subalgebra g ⊂ sim(n) is the weakly irreducible holonomy algebra of a

Lorentzian manifold if and only if it is conjugated to one of the following subalgebras:

type 1. g1,h = (R⊕ h) nRn, where h ⊂ so(n),

type 2. g2,h = hnRn,

type 3. g3,h,ϕ = {(ϕ(A), A, 0)|A ∈ h}nRn,

type 4. g4,h,m,ψ = {(0, A,X + ψ(A))|A ∈ h, X ∈ Rm},

11

where h ⊂ so(n) is the holonomy algebra of a Riemannian manifold; for g3,h,ϕ it holds

z(h) 6= {0}, and ϕ : h → R is a non-zero linear map with ϕ|h′ = 0; for g4,h,m,ψ it holds

0 < m < n is an integer, h ⊂ so(m), dim z(h) ≥ n−m, a decomposition Rn = Rm⊕Rn−m

is fixed, and ψ : h→ Rn−m is a surjective linear map with ψ|h′ = 0.

The subalgebra h ⊂ so(n) associated to a weakly irreducible Lorentzian holonomy algebra

g ⊂ sim(n) is called the orthogonal part of g.

The proof of the above theorem is the result of the works [11, 63, 49, 48]. First Berard-

Bergery and Ikemakhen [11] classified weakly irreducible subalgebras of sim(n). These

algebras are exhausted by the algebras of the above theorem without any assumption on

the subalgebra h ⊂ so(n). In [50] we obtain this result in a geometric way: we consider the

the connected Lie subgroup G ⊂ SO(n) corresponding to g ⊂ sim(n), using the boundary

of the Lobachevskian space, it is possible to consider G as a subgroup of the group of

similarity transformations of the Euclidean space Rn; this action is transitive and all such

groups are known. Thus the algebras g ⊂ sim(n) of the above theorem with arbitrary

subalgebras h ⊂ so(n) are exactly the Lie algebras of connected transitive Lie groups of

similarity transformations of the Euclidean space. Next, the important result stating that

if g ⊂ sim(n) is weakly irreducible holonomy algebra of a Lorentzian manifold, then the

corresponding subalgebra h ⊂ so(n) is the holonomy algebra of a Riemannian manifold

proved Leistner in [63]. The statement can be reduced to the case of irreducible subalgebras

h ⊂ so(n). Then, Leistner used the classification of such irreducible representations and

obtained the statement using difficult algebraic computations. We give a short direct proof

of this result for the case when h ⊂ so(n) is an irreducible semi-simple and not simple

subalgebra.

The last step of the proof of the above theorem is the construction of examples of Lo-

rentzian manifolds with all possible holonomy algebras. We give such construction in [49].

Recall that the solution of the corresponding problem in the Riemannian case was obtained

only three decades after the list of possible holonomies was obtained.

Let (M, g) be a Lorentzian manifold with the holonomy algebra g ⊂ sim(n). Then (M, g)

admits a parallel distribution of isotropic lines `. According to [77], locally there exist so

called Walker coordinates v, x1, ..., xn, u such that the metric g has the form

g = 2dvdu+ h+ 2Adu+H(du)2, (4)

where h = hij(x1, ..., xn, u)dxidxj is an u-dependent family of Riemannian metrics,

12

A = Ai(x1, . . . , xn, u)dxi is an u-dependent family of one-forms, andH = H(v, x1, ..., xn, u)

is a local function on M . An important example of such spaces are pp-waves that are given

by h =∑

(dxi)2, A = 0, ∂vH = 0. Equivalently, pp-waves are Walker manifolds with the

holonomy algebras contained in Rn ⊂ sim(n).

In our construction we take h to be flat. Then we define the 1-form A in such a way that

the orthogonal part of the holonomy algebra g coincides with a given h ⊂ so(n). Finally we

use the function H in order to get the required type of g. There are examples of globally

hyperbolic Lorentzian manifolds with some special holonomy algebras [8, 10].

The curvature tensor of the Lorentzian manifolds with holonomy algebras g ⊂ sim(n)

is completely described by us in [51, 43]. In [44, 42, 39, 52, 35, 33] we obtain different

geometric applications of the above classification. Let us note that there is a recent review

[7]. Some result about not connected holonomy groups can be found in [9].

4 Holonomy groups of connections on locally free

sheaves over supermanifolds

Now we introduce the theory of the holonomy groups for supermanifolds.

4.1 Supermanifolds

In this section we give some necessary preliminaries on supermanifolds. An introduction

to linear superalgebra and to the theory of supermanifolds can be found in [30, 64, 66, 76].

A real smooth (analytic) supermanifold M of dimension n|m is a pair (M,OM), where

M is a Hausdorff topological space and OM is a sheaf of commutative superalgebras with

unity over R locally isomorphic to Rn|m = (Rn,ORn|m = ORn⊗Λη1,...,ηm), where ORn is the

sheaf of smooth (analytic) functions on Rn and Λη1,...,ηm is the Grassmann superalgebra of

m generators. The sections of the sheaf OM are called superfunctions (or just functions) on

M. The ideal (OM)1⊕ ((OM)1)2 consists of the nilpotent elements of OM, and the sheaf

OM defined as the quotient OM/((OM)1 ⊕ ((OM)1)2) furnish M with the structure of a

real smooth (analytic) manifold. We get the canonical projection ∼: OM → OM , f 7→ f .

The value of a superfunction f at a point x ∈ M is by definition f(x). If m = 0, then

M = M is a smooth (analytic) manifold. Any isomorphism as above OM(U) ' ORn|m(U1),

13

where U ⊂ M and U1 ⊂ Rn, defines on M a local coordinate system (U, x1, ..., xn+m)

(x1, ..., xn+m are the functions corresponding to the functions y1, ..., yn, η1, ..., ηm, where

y1, ..., yn are the standard coordinates on U1 ⊂ Rn). It is known [6] that there exists an

atlas onM consisting of local coordinate systems (U, x1, ..., xn+m) such that (U, x1, ..., xn)

are local coordinate systems on M . We will always use such coordinates. We will use the

following convention about the ranks of the indices i, j, k = 1, ..., n, α, β, γ = 1, ...,m and

a, b, c = 1, ..., n + m. We will use the Einstein rule for sums. Let (U, x1, ..., xn+m) be as

above and denote xn+α by ξα. For any f ∈ OM(U) we get

f =m∑r=0

∑α1<···<αr

fα1...αrξα1 · · · ξαr , (5)

where fα1...αr ∈ OM (U) and f∅ = f . For any α1 < · · · < αr and permutation σ :

{α1, ..., αr} → {α1, ..., αr} we assume that fσ(α1)...σ(αr) = signσ fα1...αr . If two of the num-

bers α1, ..., αr are equal, we assume that fα1...αr = 0. The value f(x) at a point x ∈ Uequals to f(x), and we see that superfunctions are not defined by their values at the points

of the underlying smooth manifold.

Denote by TM the tangent sheaf, i.e. the sheaf of superderivatives of the sheaf OM. If

(U, xi, ξα) is a system of local coordinates, then the vector fields ∂xi , ∂ξα form a basis of

the supermodule TM(U) over the superalgebra OM(U). The vector fields ∂xi and ∂ξα are

defined in the usual way.

LetM = (M,OM) be a supermanifold and E a locally free sheaf of OM-supermodules on

M, e.g. TM. Let p|q be the rank of E , then locally there exists a basis (eI , hΦ)I=1,...,p;Φ=1,...,q

of sections of E . We denote such basis also by eA, where ep+Φ = hΦ. We will always

assume that A,B,C = 1, ..., p + q. For a point x ∈ M consider the vector space Ex =

E(V )/(OM(V )xE(V )), where V ⊂ M is an open subset and OM(V )x is the ideal in

OM(V ) consisting of functions vanishing at the point x. The vector space Ex does not

depend on choice of V ; it is a real vector superspace of dimension p|q. For any open subset

V ⊂ M we have the projection map from E(V ) onto Ex. For example, if E = TM, then

(TM)x is the tangent space TxM to M at the point x.

A connection on E is an even morphism ∇ : TM⊗R E → E of sheaves of supermodules over

R such that

∇fYX = f∇YX and ∇Y fX = (Y f)X + (−1)|Y ||f |f∇YX

for all homogeneous functions f , vector fields Y on M and sections X of E , here | · | ∈Z2 = {0, 1} denotes the parity. In particular, |∇YX| = |Y | + |X|. Locally we get the

14

superfunctions ΓAaB such that ∇∂aeB = ΓAaBeA. Obviously, |ΓAaB| = |a| + |A| + |B|, where

|a| = |∂xa | and |A| = |eA|.

The curvature tensor of the connection ∇ is given by

R(Y,Z) = [∇Y ,∇Z ]−∇[Y,Z], (6)

where Y and Z are vector fields on M. Let ∇ be a connection on TM.

Let Π denote the parity change functor. For example, if V = V0⊕V1 is a vector superspace,

than Π is the vector superspace with (Π(V ))0 = V1 and (Π(V ))1 = V0.

4.2 Parallel sections and definition of the holonomy

Our main task is to define the holonomy group and the holonomy algebra of a connection

on a locally-free sheaf E over a supermanifoldM. The holonomy group should satisfy the

analogue of Theorem 2.1, i.e. it should control parallel sections of E . By that reason we

first study parallel sections.

Let (M,OM) be a supermanifold, E a locally free sheaf of OM-supermodules of rank

p|q on M and ∇ a connection on E . Consider the vector bundle E over M defined as

E = ∪x∈MEx. The rank of E is p + q. For each open subset U ⊂ M , we get the natural

projection map ∼: E(U) → Γ(E,U), X 7→ X. The section X consists of the values of

the section X and it does not define X, so X is not defined by its values. Define the

subbundles E0 = ∪x∈M (Ex)0 and E1 = ∪x∈M (Ex)1 of E. Obviously, the restriction ∇ =

(∇|Γ(TM)⊗Γ(E))∼ : Γ(TM) ⊗ Γ(E) → Γ(E) is a connection on E. Since ∇ is even, the

subbundles E0, E1 ⊂ E are parallel. Let γ : [a, b] ⊂ R → M be a curve and τγ : Eγ(a) →Eγ(b) the parallel displacement along γ. Since the subbundles E0, E1 ⊂ E are parallel, we

have τγ(E0)γ(a) = (E0)γ(b) and τγ(E1)γ(a) = (E1)γ(b). We get the even isomorphism

τγ : Eγ(a) → Eγ(b)

of vector superspaces. We call this isomorphism the parallel displacement in E along γ.

A section X ∈ E(M) is called parallel if ∇X = 0. For the section X ∈ Γ(M) we get

∇X = 0. Hence for any curve γ : [a, b] ⊂ R→M we have τγXγ(a) = Xγ(b). Consequently,

τγXγ(a) = Xγ(b), where τγ : Eγ(a) → Eγ(b), i.e. X is parallel along curves in M .

Consider a system of local coordinates (U, xa) and a basis eA of E(U). Let X ∈ E(U),

then X = XAeA. The condition ∇X = 0 is equivalent to the following condition in local

15

coordinates

∂iXA +XBΓAiB = 0, (7)

∂γXA + (−1)|X

B |XBΓAγB = 0. (8)

These equations can be written as

∂iXAγ1...γr + signγ1,...,γr

r∑l=0

∑{α1,...,αr}={γ1,...,γr}α1<···<αl,αl+1<···<αr

signα1,...,αr XBα1...αl

ΓAiBαl+1...αr= 0, (9)

XAγγ1...γr + signγ1,...,γr

r∑l=0

∑{α1,...,αr}={γ1,...,γr}α1<···<αl,αl+1<···<αr

signα1,...,αr(−1)lXBα1...αl

ΓAγBαl+1...αr= 0.

(10)

Using this, we can prove the following proposition.

Proposition 4.1. Let M = (M,OM) be a supermanifold, E a locally free sheaf of OM-

supermodules on M and ∇ a connection on E. Then a parallel section X ∈ E(M) is

uniquely defined by its value at any point x ∈M .

Proof. Let ∇X = 0, x ∈ M and Xx be the value of X at the point x. Since X is parallel

along curves in M , using Xx, we can find the values of X at all points of M . Consider the

local coordinates as above. As we know the values of X at all points, we know the functions

XA. Using (10) for r = 0, we can find the functions XAγ . Namely, XA

γ = −XBΓAγB. Using

(10) for r = 1, we get XAγγ1

= −XBΓAγBγ1+ XB

γ1ΓAγB. In the same way we can find all

functions XAγγ1...γr , i.e. we know the functions XA and we reconstruct the section X in any

coordinate system. The proposition is proved. �

Thus in spite of the fact that sections of E are not defined by their values at the points, a

parallel section is defined by its value at a single point. The above considerations suggest

us how to define the holonomy that will control parallel sections.

First we define the holonomy algebra generalizing the Ambrose-Singer Theorem.

Definition 4.1. Let (M,OM) be a supermanifold, E a locally free sheaf of OM-supermodules

on M and ∇ a connection on E. The holonomy algebra hol(∇)x of the connection ∇ at a

point x ∈M is the supersubalgebra of the Lie superalgebra gl(Ex) generated by the operators

of the form

τ−1γ ◦ ∇rYr,...,Y1

Ry(Y, Z) ◦ τγ : Ex → Ex,

16

where γ is any curve in M beginning at the point x; y ∈M is the end-point of the curve γ,

r ≥ 0, Y, Z, Y1, ..., Yr ∈ TyM and ∇ is a connection on TM|U for an open neighbourhood

U ⊂M of y.

Proposition 4.2. The definition of the holonomy algebra hol(∇)x does not depend on the

choice of the connection ∇.

The next proposition simplifies the expression for the holonomy algebra. In particular it

shows that it is not necessary to take the covariant derivatives of the curvature tensor in

the directions of the vectors tangent to M .

Proposition 4.3. The holonomy algebra hol(∇)x coincides with the supersubalgebra of

the Lie superalgebra gl(Ex) generated by the operators of the form

τ−1γ ◦Ry(∂i, ∂j) ◦ τγ and τ−1

γ ◦ ∇r∂γr ,...,∂γ1Ry(∂γ , ∂a) ◦ τγ ,

where γ is any curve starting at the point x; y is the end-point of the curve γ, r ≥ 0,

γr > · · · > γ1, xa are local coordinates on M over an open neighbourhood U of the point

y and ∇ is a connection on TM|U .

Let E be the vector bundle over M and ∇ the connection on E as above. Then the

holonomy algebra hol(∇)x is contained in (hol(∇)x)0, but these Lie algebras must not

coincide, this shows the following example.

Example 4.1. Consider the supermanifold R0|1 = ({0},Λξ). Define the connection ∇ on

TR0|1 by ∇∂ξ∂ξ = ξ∂ξ. Then, hol(∇)0 = {0} and (hol(∇)0)0 = hol(∇)0 = gl(0|1).

Now we define the holonomy group. Recall that a Lie supergroup G = (G,OG) is a group

object in the category of supermanifolds. The underlying smooth manifoldG is a Lie group.

The Lie superalgebra g of G can be identified with the tangent space to G at the identity

e ∈ G. The Lie algebra of the Lie group G is the even part g0 of the Lie superalgebra g.

Any Lie supergroup G is uniquely given by a pair (G, g) (Harish-Chandra pair), where

G is a Lie group, g = g0 ⊕ g1 is a Lie superalgebra such that g0 is the Lie algebra of

the Lie group G and there exists a representation Ad of G on g that extends the adjoint

representation of G on g0 and the differential of Ad coincides with the Lie superbracket

of g restricted to g0 × g1, see [30, 57].

Denote by Hol(∇)0x the connected Lie subgroup of the Lie group GL((Ex)0) × GL((Ex)1)

corresponding to the Lie subalgebra (hol(∇)x)0 ⊂ gl((Ex)0) ⊕ gl((Ex)1) ⊂ gl(Ex). Let

17

Hol(∇)x be the Lie subgroup of the Lie group GL((Ex)0) × GL((Ex)1) generated by the

Lie groups Hol(∇)0x and Hol(∇)x. Clearly, the Lie algebra of the Lie group Hol(∇)x is

(hol(∇)x)0. Let Ad′ be the representation of the connected Lie group Hol(∇)0x on hol(∇)x

such that the differential of Ad′ coincides with the Lie superbracket of hol(∇)x restricted

to (hol(∇)x)0 × (hol(∇)x)1. Define the representation Ad′′ of the Lie group Hol(∇)x on

hol(∇)x by the rule

Ad′′τµ(τ−1γ ◦ ∇rYr,...,Y1

Ry(Y,Z) ◦ τγ) = τµ ◦ τ−1γ ◦ ∇rYr,...,Y1

Ry(Y,Z) ◦ τγ ◦ τ−1µ .

Note that Ad′ |Hol(∇)0x∩Hol(∇)x

= Ad′′ |Hol(∇)0x∩Hol(∇)x

. Consequently, we get a represen-

tation Ad of the group Hol(∇)x on hol(∇)x. It is obvious that Hol(∇)0x ∩ Hol(∇)x =

Hol(∇)0x and if M is simply connected, then Hol(∇)x ⊂ Hol(∇)0

x and Hol(∇)x = Hol(∇)0x.

Definition 4.2. The Lie supergroup Hol(∇)x given by the Harish-Chandra pair

(Hol(∇)x, hol(∇)x) is called the holonomy group of the connection ∇ at the point x. The

Lie supergroup Hol(∇)0x given by the Harish-Chandra pair (Hol(∇)0

x, hol(∇)x) is called the

restricted holonomy group of the connection ∇ at the point x.

Now we generalize Theorem 2.1. This shows that the defined by us holonomy groups satisfy

the main property of the usual holonomy groups.

Theorem 4.1. Let M = (M,OM) be a supermanifold, x ∈ M , E a locally free sheaf

of OM-supermodules on M and ∇ a connection on E. Then there exists a one-to-one

correspondence between parallel sections X ∈ E(M) and vector Xx ∈ Ex annihilated by the

holonomy algebra hol(∇)x and preserved by the group Hol(∇)x.

Recall that the connection ∇ is called flat if E admits local bases of parallel sections. Here

is the generalization of another well-known fact.

Corollary 4.1. Let M = (M,OM) be a supermanifold, E a locally free sheaf of OM-

supermodules onM and ∇ a connection on E. Then the following conditions are equivalent:

(i) ∇ is flat; (ii) R = 0; (iii) hol(∇)x = 0.

4.3 Holonomy groups of linear superconnections on super-

manifolds

Let M = (M,OM) be a supermanifold of dimension n|m. In this section we consider a

connection ∇ on the tangent sheaf TM of M. Then in Definition 4.1 of the holonomy

18

algebra hol(∇)x we may choose ∇ = ∇. If we put E = TM, then in the above notation,

E = ∪y∈MTyM = TM. In particular, E0 = TM is the tangent bundle over M . We

get the connections ∇ and ∇|TM on the vector bundles TM and TM , respectively. We

identify the holonomy algebra hol(∇)x and the group Hol(∇)x with a supersubalgebra

hol(∇) ⊂ gl(n|m,R) and a Lie subgroup Hol(∇) ⊂ GL(n,R)×GL(m,R), respectively.

Consider the sheaf of tensor fields of type (r, s) over M,

T r,sM = ⊗rOMTM⊗OM

⊗sOMT∗M.

One defines the connection ∇r,s on this sheaf in the standard way. Thus we see that the

holonomy algebra hol(∇r,s)x of the connection ∇r,s at a point x ∈ M and the group

Hol(∇r,s)x coincide with the tensor extension of the representation of the holonomy al-

gebra hol(∇)x and with the tensor extension of the representation of the group Hol(∇)x,

respectively. From Theorem 4.1 we immediately read the following.

Theorem 4.2. Let M = (M,OM) be a supermanifold, x ∈ M , and ∇ a connection on

TM. Then there exists a one-to one correspondence between parallel tensors P ∈ T r,sM (M)

and tensor Px ∈ T r,sx M annihilated by the tensor extension of the representation of the

holonomy algebra hol(∇)x and preserved by tensor extension of the representation of the

group Hol(∇)x.

Example 4.2. In Table 4.1 we give equivalent conditions for existence of some parallel

tensors on supermanifolds and inclusions of the holonomy. The definitions of the conside-

red Lie superalgebras can be found in [46].

Table 4.1. Examples of parallel structures and the corresponding holonomy

parallel structure on M hol(∇) is Hol(∇) is restriction

contained in contained in

Riemannian supermetric, osp(p, q|2k) O(p, q)× Sp(2k,R) n = p+ q,m = 2k

i.e. even non-degenerate

supersymmetric metric

even non-degenerate ospsk(2k|p, q) Sp(2k,R)×O(p, q) n = 2k,m = p+ q

super skew-symmetric metric

odd non-degenerate pe(n,R){(A 0

0 (At)−1

)∣∣A ∈ GL(n,R)}

m = n


19

odd non-degenerate super pesk(n,R){(A 0

0 (At)−1

)∣∣A ∈ GL(n,R)}

m = n

skew-symmetric metric

complex structure gl(k|l,C) GL(k,C)×GL(l,C) n = 2k, l = 2m

odd complex structure, q(n,R){(

A 00 A

)∣∣A ∈ GL(n,R)}

m = n

i.e. odd automorphism

J of TM with J2 = − id

5 Holonomy groups of Riemannian supermani-

folds

A Riemannian supermanifold (M, g) is a supermanifold M of dimension n|2m endowed

with an even non-degenerate supersymmetric metric

g : TM ⊗OM TM → OM,

see e.g. [27]. In particular, the value gx of g at a point x ∈M satisfies: gx((TxM)0, (TxM)1) =

0, gx|(TxM)0×(TxM)0is non-degenerate, symmetric and gx|(TxM)1×(TxM)1

is non-degenerate,

skew-symmetric. The metric g defines a pseudo-Riemannian metric g on the manifold M .

Note that g is not assumed to be positively defined. The supermanifold (M, g) has a

unique linear connection ∇ such that ∇ is torsion-free and ∇g = 0. This connection is

called the Levi-Civita connection. We denote the holonomy algebra of the connection ∇ by

hol(M, g). Since g is parallel, hol(M, g) ⊂ osp(p, q|2m) and Hol(∇) ⊂ O(p, q)×Sp(2m,R),

where (p, q) is the signature of the pseudo-Riemannian metric g.

We call a supersubalgebra g ⊂ osp(p, q|2m) weakly-irreducible if it does not preserve any

non-degenerate vector supersubspace of Rp,q ⊕ ΠR2m. The following theorem generalizes

the Wu theorem [78].

Theorem 5.1. [46] Let (M, g) be a Riemannian supermanifold such that the pseudo-

Riemannian manifold (M, g) is simply connected and geodesically complete. Then there

exist Riemannian supermanifolds (M0, g0), (M1, g1), ..., (Mr, gr) such that

(M, g) = (M0 ×M1 × · · · ×Mr, g0 + g1 + · · ·+ gr), (11)

the supermanifold (M0, g0) is flat and the holonomy algebras of the supermanifolds

20

(M1, g1),...,(Mr, gr) are weakly-irreducible. In particular,

hol(M, g) = hol(M1, g1)⊕ · · · ⊕ hol(Mr, gr). (12)

For general (M, g) decomposition (11) holds locally.

5.1 Berger superalgebras

Let V = V0⊕V1 be a real or complex vector superspace and g ⊂ gl(V ) a supersubalgebra.

The space of algebraic curvature tensors of type g is the vector superspace R(g) = R(g)0⊕R(g)1, where

R(g) =

R ∈ Λ2V ∗ ⊗ g

∣∣∣∣∣∣∣∣R(X,Y )Z + (−1)|X|(|Y |+|Z|)R(Y,Z)X

+(−1)|Z|(|X|+|Y |)R(Z,X)Y = 0

for all homogeneous X,Y, Z ∈ V

.

Here | · | ∈ Z2 denotes the parity. The identity that satisfy the elements R ∈ R(g) is called

the first Bianchi super identity. Obviously, R(g) is a g-module with respect to the action

A ·R = RA,

RA(X,Y ) = [A,R(X,Y )]− (−1)|A||R|R(AX,Y )− (−1)|A|(|R|+|X|)R(X,AY ), (13)

where A ∈ g, R ∈ R(g) and X,Y ∈ V are homogeneous.

If M is a supermanifold and ∇ is a linear torsion-free connection on the tangent sheaf

TM with the holonomy algebra hol(∇)x at some point x, then for the covariant derivatives

of the curvature tensor we have (∇rYr,...,Y1R)x ∈ R(hol(∇)x) for all r ≥ 0 and tangent

vectors Y1, ..., Yr ∈ TxM . Moreover, |(∇rYr,...,Y1R)x| = |Y1|+ · · ·+ |Yr|, whenever Y1, ..., Yr

are homogeneous.

Define the vector supersubspace

L(R(g)) = span{R(X,Y )|R ∈ R(g), X, Y ∈ V } ⊂ g.

From (13) it follows that L(R(g)) is an ideal in g. We call a supersubalgebra g ⊂ gl(V ) a

Berger superalgebra if L(R(g)) = g.

If V is a vector space, which can be considered as a vector superspace with the trivial odd

part, then g ⊂ gl(V ) is a usual Lie algebra, which can be considered as a Lie superalgebra

with the trivial odd part. Berger superalgebras in this case are the same as the usual

Berger algebras.

21

Proposition 5.1. [46] LetM be a supermanifold of dimension n|m with a linear torsion-

free connection ∇. Then its holonomy algebra hol(∇) ⊂ gl(n|m,R) is a Berger superalge-

bra.

Consider the vector superspace

R∇(g) =

S ∈ V ∗ ⊗R(g)

∣∣∣∣∣∣∣∣SX(Y,Z) + (−1)|X|(|Y |+|Z|)SY (Z,X)

+(−1)|Z|(|X|+|Y |)SZ(X,Y ) = 0

for all homogeneous X,Y, Z ∈ V

.

IfM is a supermanifold and∇ is a linear torsion-free connection on TM, then (∇rYr,...,Y2,·R)x ∈R∇(hol(∇)x) for all r ≥ 1 and Y2, ..., Yr ∈ TxM . Moreover, |(∇rYr,...,Y2,·R)x| = |Y2|+ · · ·+|Yr|, whenever Y2, ..., Yr are homogeneous.

A Berger superalgebra g is called symmetric if R∇(g) = 0. This is a generalization of the

usual symmetric Berger algebras, see e.g. [73], and the following is a generalization of the

well-known fact about smooth manifolds.

Proposition 5.2. [46] Let M be a supermanifold with a torsion free connection ∇. If

hol(∇) is a symmetric Berger superalgebra, then (M,∇) is locally symmetric (i.e. ∇R =

0). If (M,∇) is a locally symmetric superspace, then its curvature tensor at any point is

annihilated by the holonomy algebra at this point and its image coincides with the holonomy

algebra.

A geometric theory of Riemannian symmetric superspaces is developed recently in [57].

The proof of the following proposition is as in [73].

Proposition 5.3. Let g ⊂ gl(V ) be an irreducible Berger superalgebra of the form (1). If

g annihilates the module R(g), then g is a symmetric Berger superalgebra.

In [75] simply connected symmetric superspaces of simple Lie supergroups of isometries

are classified. In particular this implies the classification of the holonomy algebras of Rie-

mannian symmetric superspaces and of irreducible Berger superalgebras g ⊂ osp(p, q|2m)

of the form (1). Hence we assume that the Riemannian supermanifolds under the consi-

deration are not locally symmetric.

22

5.2 Classification

First let us consider irreducible holonomy algebras of odd Riemannian supermanifolds, in

this case

g ⊂ osp(0|2m) ' sp(2m,R)

is a usual Lie algebra. The following result from [37] is the mirror analogue of the Berger

classification:

Theorem 5.2. Possible irreducible holonomy algebras g ⊂ sp(2m,R) of not symmetric

odd Riemannian supermanifolds are listed in Table 5.1.

Table 5.1. Possible irreducible holonomy algebras g ⊂ sp(2m,R) = sp(V ) of not symmetric odd

Riemannian supermanifolds.

g V restriction

sp(2m,R) R2m m ≥ 1

u(p, q) Cp,q p+ q ≥ 2

su(p, q) Cp,q p+ q ≥ 2

so(n,H) Hn n ≥ 2

sp(1)⊕ so(n,H) Hn n ≥ 2

sl(2,R)⊕ so(p, q) R2 ⊗ Rp,q p+ q ≥ 3

spin(2, 10) ∆+2,10 = R32

spin(6, 6) ∆+6,6 = R32

so(6,H) ∆H6 = H8

sl(6,R) Λ3R6 = R20

su(1, 5) {ω ∈ Λ3C6| ∗ w = w}su(3, 3) {ω ∈ Λ3C6| ∗ w = w}sp(6,R) R14 ⊂ Λ3R6

sp(2m,C) C2m m ≥ 1

sl(2,C)⊕ so(m,C) C2 ⊗ Cm m ≥ 3

spin(12,C) ∆+12 = C32

sl(6,C) Λ3C6 = C20

sp(6,C) Vπ3 = C14

In order to get this results, we first classify in [45] irreducible Berger supersubalgebras g ⊂gl(0|m,C). Note that gl(0|m,C) ' gl(m,C) and in fact we deal with the usual Lie algebras

satisfying some algebraic conditions. We use classification of irreducible representations

of semi-simple complex Lie algebras. We use results from [73], in the same time, there

are many cases that require additional considerations and some new ideas. Then for the

23

proof of the above theorem we use the complexification procedure, similar to one from [73].

For that we need a classification of irreducible subalgeras g of gl(m,R) and gl(m,C) with

non-trivial first skew-symmetric prolongations. We get such classifications using results of

Kac [61]. This classification is of independent interest [4, 69].

The main result form [46] is the following theorem.

Theorem 5.3. Let (M, g) be a not locally symmetric Riemannian supermanifold of di-

mension p+q|2m (p+q > 0) with an irreducible holonomy algebra hol(M, g) ⊂ osp(p, q|2m)

that is of the form (1), then hol(M, g) ⊂ osp(p, q|2m) coincides with one of the Lie super-

algebras from Table 5.2.

Table 5.2. Irreducible non-symmetric Berger supersubalgebras g ⊂ osp(p, q|2m) (p+ q >

0) of the from (1) and the connected Lie subgroups G ⊂ SO(p, q)×Sp(2m,R) corresponding

to g0 ⊂ so(p, q)⊕ sp(2m,R).

g G (p, q|2m)

osp(p, q|2m) SO(p, q)× Sp(2m,R) (p, q|2m)

osp(p|2k,C) SO(p,C)× Sp(2k,R) (p, p|4k)

u(p0, q0|p1, q1) U(p0, q0)×U(p1, q1) (2p0, 2q0|2p1 + 2q1)

su(p0, q0|p1, q1) U(1)(SU(p0, q0)× SU(p1, q1)) (2p0, 2q0|2p1 + 2q1)

hosp(r, s|k) Sp(p0, q0)× SO∗(k) (4r, 4s|4k)

hosp(r, s|k)⊕ sp(1) Sp(1)(Sp(p0, q0)× SO∗(k)) (4r, 4s|4k)

ospsk(2k|r, s)⊕ sl(2,R) Sp(2k,R)× SO(r, s)× SL(2,R) (2k, 2k|2r + 2s)

ospsk(2k|r,C)⊕ sl(2,C) Sp(2k,C)× SO(r,C)× SL(2,C) (4k, 4k|4r)

The Ricci tensor of a supermanifold (M, g) is defined by the formula

Ric(Y,Z) = str(X 7→ (−1)|X||Z|R(Y,X)Z), (14)

where U ⊂M is open and X,Y, Z ∈ TM(U) are homogeneous.

The definitions of the supermanifolds considered below are similar to the usual ones,

see e.g. [27, 46]. Foe example, a Riemannian supermanifold (M, g) is called a Kahlerian

supermanifold if it admits an even parallel g-orthogonal complex structure. Since the

holonomy algebra annihilates the values of the parallel tensors [46], in this case it must be

contained in u(p0, q0|p1, q1). By definition, a special Kahlerian supermanifold or a Calabi-

Yau supermanifold is a Ricci-flat Kahlerian supermanifold.

24

Proposition 5.4. [46] Let (M, g) be a Kahlerian supermanifold, then Ric = 0 if and only

if hol(M, g) ⊂ su(p0, q0|p1, q1).

Riemannian supermanifolds with the holonomy algebras osp(p, q|2m) are generic. Here we

give the geometric characterization of simply connected supermanifolds with the holonomy

algebras g different from osp(p, q|2m) (the conditions on the holonomy algebra and the

corresponding geometries are equivalent for simply connected supermanifolds):

g ⊂ osp(p|2k,C): holomorphic Riemannian supermanifolds;

g ⊂ u(p0, q0|p1, q1): Kahlerian supermanifolds;

g ⊂ su(p0, q0|p1, q1): special Kahlerian supermanifolds or Calabi-Yau supermani-

folds;

g ⊂ hosp(r, s|k): hyper-Kahlerian supermanifolds;

g ⊂ hosp(r, s|k)⊕ sp(1): quaternionic-Kahlerian supermanifolds;

g ⊂ ospsk(2k|r, s)⊕ sl(2,R): para-Kahlerian supermanifolds;

g ⊂ ospsk(2k|r,C)⊕ sl(2,C): holomorphic para-Kahlerian supermanifolds.

Proposition 5.5. [46] Let (M, g) be a quaternionic-Kahlerian supermanifold, then Ric =

0 if and only if hol(M, g) ⊂ hosp(p0, q0|p1, q1). In particular, if (M, g) is hyper-Kahlerian,

then Ric = 0; if M is simply connected, (M, g) is quaternionic-Kahlerian and Ric = 0,

then (M, g) is hyper-Kahlerian.

For the proof of Theorem 5.3 we consider the space R(g) for an irreducible supersubalge-

bra g ⊂ osp(p, q|2m) of the form (1). We restrict the attention to the representation of the

even part g0 ⊂ g. Roughly speaking, we show that if g ⊂ osp(p, q|2m) is a Berger super-

subalgebra, then prso(p,q) g0 ⊂ so(p, q) is the holonomy algebra of a pseudo-Riemannian

manifold and prsp(2m,R) g0 ⊂ sp(2m,R) is in the list of possible reductive holonomy alge-

bras of Riemannian odd supermanifolds. This is a strong condition on g0. Using the theory

of representation of simple Lie superalgebras [62, 74], we analyse which representations of

g0 can be extended to irreducible representations of g.

25

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31

October 10, 2006 9:10 WSPC/IJGMMP-J043 00157

International Journal of Geometric Methods in Modern PhysicsVol. 3, Nos. 5 & 6 (2006) 1025–1045c© World Scientific Publishing Company

METRICS THAT REALIZE ALL LORENTZIANHOLONOMY ALGEBRAS

ANTON S. GALAEV

Institut fur Mathematik, Humboldt-Universitat zu BerlinSitz: Rudower, Chaussee 25, 12489

Berlin, [email protected]

Received 27 September 2005Accepted 25 January 2006

Dedicated to Dmitri Vladimirovich Alekseevsky on the occasion of his 65th birthday

All candidates to the weakly-irreducible not irreducible holonomy algebras of Lorentzianmanifolds are known. In the present paper metrics that realize all these candidates asholonomy algebras are given. This completes the classification of the Lorentzian holon-omy algebras. Also new examples of metrics with the holonomy algebras g2 � R7 ⊂so(1, 8) and spin(7) � R

8 ⊂ so(1, 9) are constructed.

Keywords: Lorentzian manifold; holonomy algebra; local metric.

Mathematics Subject Classification: 53C29, 53C50, 53B30

0. Introduction

The classification of the holonomy algebras for Riemannian manifolds is a well-known classical result. By the Borel–Lichnerowicz theorem a Riemannian man-ifold is locally a product of Riemannian manifolds with irreducible holonomyalgebras [8]. In 1955, Berger [7] gave a list of possible irreducible holonomy algebrasof Riemannian manifolds. Later, in 1987, Bryant [12] constructed metrics for theexceptional algebras of this list. For more details see [1, 6, 20].

The classification problem for the holonomy algebras of pseudo-Riemannianmanifolds is still open. The main difficulty is that the holonomy algebra can pre-serve an isotropic subspace of the tangent space. A subalgebra g ⊂ so(r, s) is calledweakly-irreducible if it does not preserve any nondegenerate proper subspace of R

r,s.The Wu theorem states that a pseudo-Riemannian manifold is locally a product ofpseudo-Riemannian manifolds with weakly-irreducible holonomy algebras, see [27].So, it is enough to consider only weakly-irreducible holonomy algebras. If a holon-omy algebra is irreducible, then it is weakly-irreducible. In [7], Berger also gave aclassification of possible irreducible holonomy algebras for pseudo-Riemannian man-ifolds, but there is no classification for weakly-irreducible not irreducible holonomy

1025

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1026 A. S. Galaev

algebras for general pseudo-Riemannian manifolds. About the Lorentzian case seelater. There are some partial results for holonomy algebras of pseudo-Riemannianmanifolds of signature (2, N), see [19, 16, 17], and (N, N), see [5].

We consider the holonomy algebras of Lorentzian manifolds. From Berger’slist it follows that the only irreducible holonomy algebra of Lorentzian mani-folds is so(1, n + 1), see [13, 11] for direct proofs of this fact. In 1993, BerardBergery and Ikemakhen [4] classified weakly-irreducible not irreducible subalge-bras of so(1, n + 1). More precisely, they divided these subalgebras into 4 types,and associated to each such subalgebra g ⊂ so(1, n + 1) a subalgebra h ⊂ so(n),which is called the orthogonal part of g. In [15] more geometrical proof of this resultwas given. The Lie algebras of types 1 and 2 have the forms (R⊕h)�R

n and h�Rn

respectively. The Lie algebras of types 3 and 4 can be obtained from the first twoby some twistings. Recently Leistner [21–24] proved that if g ⊂ so(1, n + 1) is theholonomy algebra of a Lorentzian manifold, then its orthogonal part h ⊂ so(n) is theholonomy algebra of a Riemannian manifold. This gives a list of possible holonomyalgebras for Lorentzian manifolds (Berger algebras). To complete the classificationof holonomy algebras one must prove that all Berger algebras can be realized asholonomy algebras. In [4] were given metrics that realize all Berger algebras of types1 and 2. In [9, 10], Boubel studied possible shapes of local metrics for Lorentzianmanifolds with weakly-irreducible not irreducible holonomy algebras. In particular,he gave equivalent conditions for such manifolds to have the holonomy of type 1,2, 3 or 4 and parameterized the set of germs of metrics giving a holonomy algebraof each type. In [25], Sfetsos and Zoakos constructed metrics with the holonomyalgebras su(2) � R

4 ⊂ so(1, 5), su(3) � R6 ⊂ so(1, 7) and g2 � R

7 ⊂ so(1, 8).In the present paper we construct metrics that realize all Berger algebras as

holonomy algebras. The method of the construction generalizes an example ofIkemakhen given in [18]. The coefficients of the constructed metrics are polyno-mial functions, hence the holonomy algebra at a point is generated by the values ofthe curvature tensor and of its derivatives at this point, and it can be computed.This completes the classification of the holonomy algebras of Lorentzian manifolds.As application we construct new examples of metrics with the holonomy algebrasg2 � R

7 ⊂ so(1, 8) and spin(7) � R8 ⊂ so(1, 9).

1. Preliminaries

Let (R1,n+1, η) be a Minkowski space of dimension n + 2, where η is a metric onR

n+2 of signature (1, n + 1). We fix a basis p, e1, . . . , en, q of R1,n+1 such that the

Gram matrix of η has the form 0 0 1

0 En 01 0 0

,

where En is the n-dimensional identity matrix. We will denote by Rn ⊂ R

1,n+1 theEuclidean subspace spanned by the vectors e1, . . . , en.

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Metrics that Realize all Lorentzian Holonomy Algebras 1027

Definition 1. A subalgebra g ⊂ so(1, n + 1) is called irreducible if it does notpreserve any proper subspace of R

1,n+1; g is called weakly-irreducible if it does notpreserve any nondegenerate proper subspace of R

1,n+1.

Obviously, if g ⊂ so(1, n + 1) is irreducible, then it is weakly-irreducible.Denote by so(1, n+1)Rp the subalgebra of so(1, n+1) that preserves the isotropic

line Rp. The Lie algebra so(1, n + 1)Rp can be identified with the following matrixalgebra

so(1, n + 1)Rp =

a X 0

0 A −Xt

0 0 −a

∣∣∣∣∣∣ a ∈ R, X ∈ R

n, A ∈ so(n)

.

This matrix can be identified with the triple (a, A, X). Define the following sub-algebras of so(1, n + 1)Rp, A = {(a, 0, 0)|a ∈ R}, K = {(0, A, 0)|A ∈ so(n)} andN = {(0, 0, X)|X ∈ R

n}. We see that A commutes with K, and N is a commuta-tive ideal. We also see that

[(a, A, 0), (0, 0, X)] = (0, 0, aX + AX).

We have the decomposition

so(1, n + 1)Rp = (A⊕K) � N = (R ⊕ so(n)) � Rn.

If a weakly-irreducible subalgebra g ⊂ so(1, n + 1) preserves a degenerate propersubspace U ⊂ R

1,n+1, then it preserves the isotropic line U∩U⊥, and g is conjugatedto a weakly-irreducible subalgebra of so(1, n + 1)Rp.

Let h ⊂ so(n) be a subalgebra. Recall that h is a compact Lie algebra and wehave the decomposition h = h′ ⊕ z(h), where h′ is the commutant of h and z(h) isthe center of h [26].

The following result is due to Berard Bergery and Ikemakhen.

Theorem 2 [4]. A subalgebra g ⊂ so(1, n + 1)Rp is weakly-irreducible if and onlyif g belongs to one of the following types

Type 1.

g1,h = (R ⊕ h) � Rn =

a X 0

0 A −Xt

0 0 −a

∣∣∣∣∣∣ a ∈ R, A ∈ h, X ∈ R

n

,

where h ⊂ so(n) is a subalgebra;

Type 2.

g2,h = h � Rn =

0 X 0

0 A −Xt

0 0 0

∣∣∣∣∣∣ A ∈ h, X ∈ R

n

;

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1028 A. S. Galaev

Type 3.

g3,h,ϕ = {(ϕ(A), A, 0)|A ∈ h} � Rn =

ϕ(A) X 0

0 A −Xt

0 0 −ϕ(A)

∣∣∣∣∣∣ A ∈ h, X ∈ R

n

,

where h ⊂ so(n) is a subalgebra with z(h) �= {0}, and ϕ : h → R is a non-zero linearmap with ϕ|h′ = 0;

Type 4.

g4,h,m,ψ = {(0, A, X + ψ(A))|A ∈ h, X ∈ Rm}

=

0 X ψ(A) 00 A 0 −Xt

0 0 0 −ψ(A)t

0 0 0 0

∣∣∣∣∣∣∣∣ A ∈ h, X ∈ R

m

,

where 0 < m < n is an integer, h ⊂ so(m) is a subalgebra with dim z(h) ≥ n − m,

and ψ : h → Rn−m is a surjective linear map with ψ|h′ = 0.

Definition 3. The subalgebra h ⊂ so(n) associated with a weakly-irreducible sub-algebra g ⊂ so(1, n + 1)Rp in Theorem 2 is called the orthogonal part of g.

Let (M, g) be a Lorentzian manifold of dimension n+2 and g the holonomy algebra(that is the Lie algebra of the holonomy group) at a point x ∈ M . By Wu’s theorem(see [27]) (M, g) is locally indecomposable, i.e. it is not locally a product of twopseudo-Riemannian manifolds if and only if the holonomy algebra g is weakly-irreducible. If the holonomy algebra g is irreducible, then g = so(TxM, gx) [7]. Sowe may assume that it is weakly-irreducible and not irreducible. Then g preservesan isotropic line � ⊂ TxM . We can identify the tangent space TxM with R

1,n+1

such that gx corresponds to η and � corresponds to the line Rp. Then g is identifiedwith a weakly-irreducible subalgebra of so(1, n + 1)Rp.

Let W be a vector space and f ⊂ gl(W ) a subalgebra.

Definition 4. The vector space

R(f) = {R ∈ Hom(W ∧ W, f)|R(u ∧ v)w

+ R(v ∧ w)u + R(w ∧ u)v = 0 for all u, v, w ∈ W}is called the space of curvature tensors of type f. Denote by L(R(f)) the vectorsubspace of f spanned by R(u ∧ v) for all R ∈ R(f), u, v ∈ W,

L(R(f)) = span{R(u ∧ v)|R ∈ R(f), u, v ∈ W}.A subalgebra f ⊂ gl(W ) is called a Berger algebra if L(R(f)) = f.

From the Ambrose–Singer theorem [2] it follows that if g ⊂ so(1, n + 1)Rp is theholonomy algebra of a Lorentzian manifold, then g is a Berger algebra.

Let h ⊂ so(n) be a subalgebra.

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Definition 5. The vector space

P(h) = {P ∈ Hom(Rn, h)| η(P (u)v, w) + η(P (v)w, u)

+ η(P (w)u, v) = 0 for all u, v, w ∈ Rn}

is called the space of weak-curvature tensors of type h. A subalgebra h ⊂ so(n) iscalled a weak-Berger algebra if L(P(h)) = h, where

L(P(h)) = span{P (u)|P ∈ P(h), u ∈ Rn}

is the vector subspace of h spanned by P (u) for all P ∈ P(h) and u ∈ Rn.

The following theorem was proved in [14].

Theorem 6 [14]. A weakly-irreducible subalgebra g ⊂ so(1, n + 1)Rp is a Bergeralgebra if and only if its orthogonal part h ⊂ so(n) is a weak-Berger algebra.

Recently Leistner proved the following theorem.

Theorem 7 [21–24]. A subalgebra h ⊂ so(n) is a weak-Berger algebra if and onlyif h is a Berger algebra.

Recall that from the classification of Riemannian holonomy algebras it follows thata subalgebra h ⊂ so(n) is a Berger algebra if and only if h is the holonomy algebra ofa Riemannian manifold. Thus a subalgebra g ⊂ so(1, n + 1) is a weakly-irreduciblenot irreducible Berger algebra if and only if g is conjugated to one of the subalgebrasg1,h, g2,h, g3,h,ϕ, g4,h,m,ψ ⊂ so(1, n + 1)Rp, where h ⊂ so(n) is the holonomy algebraof a Riemannian manifold.

To complete the classification of Lorentzian holonomy algebras we must realizeall weakly-irreducible Berger subalgebra of so(1, n+1)Rp as the holonomy algebras.There are some examples.

Example 8 [4]. In 1993, Berard Bergery and Ikemakhen realized the weakly-irreducible Berger subalgebra of so(1, n + 1)Rp of types 1 and 2 as the holonomyalgebras. They constructed the following metrics. Let h ⊂ so(n) be the holon-omy algebra of a Riemannian manifold. Let x0, x1, . . . , xn, xn+1 be the standardcoordinates on R

n+2, h be a metric on Rn with the holonomy algebra h, and

f(x0, . . . , xn+1) be a function with ∂f∂x1 �= 0, . . . , ∂f

∂xn �= 0. If ∂f∂x0 �= 0, then the

holonomy algebra of the metric

g = 2dx0dxn+1 + h + f · (dxn+1)2

is g1,h. If ∂f∂x0 = 0, then the holonomy algebra of the metric g is g2,h.

In the next section we will construct metrics that realize all weakly-irreducibleBerger algebras. We will use the space P(h) and the fact that h = L(P(h)). Theidea of the constructions is given by the following example of Ikemakhen.

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1030 A. S. Galaev

Example 9 [18]. Let x0, x1, . . . , x5, x6 be the standard coordinates on R7. Con-

sider the following metric

g = 2dx0dx6 +5∑

i=1

(dxi)2 + 25∑

i=1

uidxidx6,

where

u1 = −(x3)2 − 4(x4)2 − (x5)2, u2 = u4 = 0,

u3 = −2√

3x2x3 − 2x4x5, u5 = 2√

3x2x5 + 2x3x4.

The holonomy algebra of this metric at the point 0 is g2,ρ(so(3)) ⊂ so(1, 6), whereρ : so(3) → so(5) is the representation given by the highest irreducible component ofthe representation ⊗2 id : so(3) → ⊗2so(3). The image ρ(so(3)) ⊂ so(5) is spannedby the matrices

A1 =

0 0 −1 0 00 0

√3 0 0

1 −√3 0 0 0

0 0 0 0 −10 0 0 1 0

, A2 =

0 0 0 −4 00 0 0 0 00 0 0 0 −24 0 0 0 00 0 2 0 0

,

A3 =

0 0 0 0 −10 0 0 0 −√

30 0 0 −1 −10 0 1 0 01

√3 1 0 0

.

We have prso(n)

(R(

∂∂x3 , ∂

∂x6

)0

)= A1, prso(n)

(R(

∂∂x4 , ∂

∂x6

)0

)= A2, prso(n)

(R(

∂∂x5 ,

∂∂x6

)0

)= A3 and prso(n)

(R(

∂∂x1 , ∂

∂x6

)0

)= prso(n)

(R(

∂∂x2 , ∂

∂x6

)0

)= 0.

Note the following. Let P ∈ Hom(Rn, h) be a linear map defined as followsP (e1) = P (e2) = 0, P (e3) = A1, P (e4) = A2 and P (e5) = A3. We have P ∈ P(h),P (Rn) = h and prso(n)

(R(

∂∂xi ,

∂∂x6

)0

)= P (ei) for all 1 ≤ i ≤ 5.

2. Main Results

In this section we will construct metrics that for any Riemannian holonomy algebrah ⊂ so(n) realize the Lie algebras g1,h, g2,h, g3,h,ϕ and g4,h,m,ψ (if g3,h,ϕ and g4,h,m,ψ

exist) as holonomy algebras.Recall that the Lie algebra g3,h,ϕ exists only for h ⊂ so(n) with z(h) �= {0} and

the Lie algebra g4,h,m,ψ exists only for h ⊂ so(m) with dim z(h) ≥ n − m.

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2.1. Constructions of the metrics

Let h ⊂ so(n) be the holonomy algebra of a Riemannian manifold. The Borel–Lichnerowicz theorem [8] states that we have an orthogonal decomposition

Rn = R

n1 ⊕ · · · ⊕ Rns ⊕ R

ns+1 (1)

and the corresponding decomposition into the direct sum of ideals

h = h1 ⊕ · · · ⊕ hs ⊕ {0} (2)

such that h annulates Rns+1 , hi(Rnj ) = 0 for i �= j, and hi ⊂ so(ni) is an irreducible

subalgebra for 1 ≤ i ≤ s. Moreover, the Lie algebras hi are the holonomy algebrasof Riemannian manifolds. Note that we have [21, 14]

P(h) = P(h1) ⊕ · · · ⊕ P(hs). (3)

We will assume that the basis e1, . . . , en of Rn is compatible with this decomposition

of Rn.

Let n0 = n1 + · · · + ns = n − ns+1. We see that h ⊂ so(n0) and h does notannulate any proper subspace of R

n0 . Note that in the case of the Lie algebrag4,h,m,ψ we have 0 < n0 ≤ m.

We will always assume that the indices b, c, d, f run from 0 to n + 1, the indicesi, j, k, l run from 1 to n, the indices i, j, k, l run from 1 to n0, the indices ˆi, ˆj, ˆk,

ˆl

run from n0 + 1 to n, and the indices α, β, γ run from 1 to N . In case of the Liealgebra g4,h,m,ψ we will also assume that the indices i, j, k, l run from n0 + 1 to m

and the indices ˜i, ˜j, ˜k,˜l run from m+1 to n. We will use the Einstein rule for sums.

Let (Pα)Nα=1 be linearly independent elements of P(h) such that the subset

{Pα(u)|1 ≤ α ≤ N, u ∈ Rn0} ⊂ h generates the Lie algebra h. For example, it

can be any basis of the vector space P(h). We have Pα|Rns+1 = 0 and Pα can beconsidered as linear maps Pα : R

n0 → h ⊂ so(n0). For each Pα define the numbersP k

αjisuch that Pα(ei)ej = P k

αjiek. Since Pα ∈ P(h), we have

P j

αki= −P k

αjiand P k

αji+ P i

αkj+ P j

αik= 0. (4)

Define the following numbers

akαji

=1

3 · (α − 1)!(P k

αji+ P k

αij

). (5)

We have

akαji

= akαij

. (6)

From (4) it follows that

P kαji

= (α − 1)!(ak

αji− aj

αki

)and ak

αji+ ai

αkj+ aj

αik= 0. (7)

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1032 A. S. Galaev

Let x0, . . . , xn+1 be the standard coordinates on Rn+2. Consider the following

metric

g = 2dx0dxn+1 +n∑

i=1

(dxi)2 + 2n0∑i=1

uidxidxn+1 + f · (dxn+1)2, (8)

where

ui = aiαjk

xjxk(xn+1)α−1 (9)

and f is a function that will depend on the type of the holonomy algebra that wewish to obtain.

For the Lie algebra g3,h,ϕ (if it exists) define the numbers

ϕαi =1

(α − 1)!ϕ(Pα(ei)). (10)

For the Lie algebra g4,h,m,ψ (if it exists) define the numbers ψαii

such that

1(α − 1)!

ψ(Pα(ei)) =n∑

˜i=m+1

ψαii

e˜i . (11)

Suppose that f(0) = 0, then g0 = η and we can identify the tangent space to Rn+2

at 0 with the vector space R1,n+1.

Theorem 10. The holonomy algebra hol0 of the metric g at the point 0 ∈ Rn+2

depends on the function f in the following way

f hol0

(x0)2 +∑n

î=n0+1(x

î)2 g1,h

∑nî=n0+1

(xî)2 g2,h

2x0ϕαixi(xn+1)α−1 +

∑nî=n0+1

(xî)2 g3,h,ϕ(if z(h) �= {0})

2ψαii

xix˜i(xn+1)α−1 +

∑mi=n0+1(x

i)2 g4,h,m,ψ(if dim z(h) ≥ n − m)

As the corollary we get the classification of the weakly-irreducible not irreducibleLorentzian holonomy algebras.

Theorem 11. A weakly-irreducible not irreducible subalgebra g ⊂ so(1, n + 1) isthe holonomy algebra of a Lorentzian manifold if and only if g is conjugated to oneof the subalgebras g1,h, g2,h, g3,h,ϕ, g4,h,m,ψ ⊂ so(1, n + 1)Rp, where h ⊂ so(n) is theholonomy algebra of a Riemannian manifold.

From Theorem 11, Wu’s theorem and Berger’s list it follows that the holonomyalgebra hol ⊂ so(1, N + 1) of any Lorentzian manifold of dimension N + 2 has the

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form hol = g⊕h1⊕· · ·⊕hr, where either g = so(1, n+1) or g is a Lie algebra fromTheorem 11, and hi ⊂ so(ni) are the irreducible holonomy algebras of Riemannianmanifolds (N = n + n1 + · · · + nr).

2.2. Explanation of the idea of the constructions

Now we compare our method of constructions with the example of Ikemakhen.Let us construct the metric for g2,ρ(so(3)) ⊂ so(1, 6) by our method. Take P ∈P(ρ(so(3))) defined as P (e1) = P (e2) = 0, P (e3) = A1, P (e4) = A2 andP (e5) = A3. By our constructions, we have

g = 2dx0dx6 +5∑

i=1

(dxi)2 + 25∑

i=1

uidxidx6,

where

u1 = −23((x3)2 + 4(x4)2 + (x5)2), u2 = −2

√3

3((x3)2 − (x5)2),

u3 =23(x1x3 −

√3x2x3 − 3x4x5 − (x5)2), u4 = −8

3x1x4,

u5 =23(x1x5 +

√3x2x5 + 3x3x4 + x3x5).

We still have prso(n)

(R(

∂∂x3 , ∂

∂x6

)0

)= A1, prso(n)

(R(

∂∂x4 , ∂

∂x6

)0

)= A2, prso(n)(

R(

∂∂x5 , ∂

∂x6

)0

)= A3 and prso(n)

(R(

∂∂x1 , ∂

∂x6

)0

)= prso(n)

(R(

∂∂x2 , ∂

∂x6

)0

)= 0.

The reason why we obtain another metric is the following. The idea of ourconstructions is to find the constants ak

αjisuch that

prso(n)

(R

(∂

∂xi,

∂

∂xn+1

)0

)= P1(ei),

...

prso(n)

(∇N−1R

(∂

∂xi,

∂

∂xn+1;

∂

∂xn+1; · · · ; ∂

∂xn+1

)0

)= PN (ei).

These conditions give us the system of equations (α − 1)!

(ak

αji− aj

αki

)= P k

αji,

akαji

− akαij

= 0.

One of the solutions of this system is given by (5), but this system canhave other solutions. In the example we use the solution given by (5), takinganother solution of the above system, we can obtain the metric constructed byIkemakhen.

Thus the choice of the functions ui given by (9) guarantees us that the orthogonalpart of the holonomy algebra hol0 coincides with the given Riemannian holonomyalgebra h ⊂ so(n) (the other values of prso(n)(∇rR) does not give us any thing new).

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1034 A. S. Galaev

This also guarantees us the inclusion Rn0 ⊂ hol0. The reason why we choose the

function f as in Theorem 10 can be easily understood from the following formulas

prR

(R

(∂

∂x0,

∂

∂xn+1

)0

)=

12

∂2f

(∂x0)2

(we use this for g1,h),

prR

(∇α−1R

(∂

∂xi,

∂

∂xn+1;

∂

∂xn+1; · · · ; ∂

∂xn+1

)0

)=

12

∂α+1f

∂x0∂xi(∂xn+1)α−1

(we use this for g3,h,ϕ),

prRn

(∇α−1R

(∂

∂xˆi,

∂

∂xn+1;

∂

∂xn+1; · · · ; ∂

∂xn+1

)0

)=

12

n∑ˆj=n0+1

∂α+1f

∂xˆi∂x

ˆj(∂xn+1)α−1eˆj

(for α = 0 we use this for all algebras, for α ≥ 0 we use this for g4,h,m,ψ).

As application we construct metrics for the Lie algebras g2,g2 ⊂ so(1, 8) andg2,spin(7) ⊂ so(1, 9).

Example 12 (Metric with the holonomy algebra g2,g2 ⊂ so(1, 8)). Con-sider the Lie subalgebra g2 ⊂ so(7). The vector subspace g2 ⊂ so(7) is spanned bythe following matrices [3]

A1 = E12 − E34, A2 = E12 − E56, A3 = E13 + E24, A4 = E13 − E67,

A5 = E14 − E23, A6 = E14 − E57, A7 = E15 + E26, A8 = E15 + E47,

A9 = E16 − E25, A10 = E16 + E37, A11 = E17 − E36, A12 = E17 − E45,

A13 = E27 − E35, A14 = E27 + E46,

where Eij ∈ so(7) (i < j) is the skew-symmetric matrix such that (Eij)ij = 1,(Eij)ji = −1 and (Eij)kl = 0 for other k and l.

Consider the linear map P ∈ Hom(R7, g2) defined as

P (e1) = A6, P (e2) = A4 + A5, P (e3) = A1 + A7, P (e4) = A1,

P (e5) = A4, P (e6) = −A5 + A6, P (e7) = A7.

It can be checked that P ∈ P(g2). Moreover, the elements A1, A4, A5, A6, A7 ∈ g2

generate the Lie algebra g2.The holonomy algebra of the metric

g = 2dx0dx8 +7∑

i=1

(dxi)2 + 27∑

i=1

uidxidx8,

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where

u1 =23(2x2x3 + x1x4 + 2x2x4 + 2x3x5 + x5x7),

u2 =23(−x1x3 − x2x3 − x1x4 + 2x3x6 + x6x7),

u3 =23(−x1x2 + (x2)2 − x3x4 − (x4)2 − x1x5 − x2x6),

u4 =23(−(x1)2 − x1x2 + (x3)2 + x3x4),

u5 =23(−x1x3 − 2x1x7 − x6x7),

u6 =23(−x2x3 − 2x2x7 − x5x7),

u7 =23(x1x5 + x2x6 + 2x5x6),

at the point 0 ∈ R9 is g2,g2 ⊂ so(1, 8).

Using computer it can be checked that dimP(g2) = 64. This means that we canconstruct quite a big number of metrics with the holonomy algebra g2,g2 ⊂ so(1, 8).

Example 13 (Metric with the holonomy algebra g2,spin(7) ⊂ so(1, 9)).Consider the Lie subalgebra spin(7) ⊂ so(8). The vector subspace spin(7) ⊂ so(8)is spanned by the following matrices [3]

A1 = E12 + E34, A2 = E13 − E24, A3 = E14 + E23, A4 = E56 + E78,

A5 = −E57 + E68, A6 = E58 + E67, A7 = −E15 + E26, A8 = E12 + E56,

A9 = E16 + E25, A10 = E37 − E48, A11 = E38 + E47, A12 = E17 + E28,

A13 = E18 − E27, A14 = E35 + E46, A15 = E36 − E45, A16 = E18 + E36,

A17 = E17 + E35, A18 = E26 − E48, A19 = E25 + E38, A20 = E23 + E67,

A21 = E24 + E57.

Consider the linear map P ∈ Hom(R8, spin(7)) defined as

P (e1) = 0, P (e2) = −A14, P (e3) = 0, P (e4) = A21,

P (e5) = A20, P (e6) = A21 − A18, P (e7) = A15 − A16, P (e7) = A14 − A17.

It can be checked that P ∈ P(spin(7)). Moreover, the elements A14, A15 −A16, A17, A18, A20, A21 ∈ spin(7) generate the Lie algebra spin(7).

The holonomy algebra of the metric

g = 2dx0dx9 +8∑

i=1

(dxi)2 + 28∑

i=1

uidxidx9,

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1036 A. S. Galaev

where

u1 = −43x7x8, u2 =

23((x4)2 + x3x5 + x4x6 − (x6)2),

u3 = −43x2x5, u4 =

23(−x2x4 − 2x2x6 − x5x7 + 2x6x8),

u5 =23(x2x3 + 2x4x7 + x6x7), u6 =

23(x2x4 + x2x6 + x5x7 − x4x8),

u7 =23(−x4x5 − 2x5x6 + x1x8), u8 =

23(−x4x6 + x1x7),

at the point 0 ∈ R9 is g2,spin(7) ⊂ so(1, 9).

Note that dimP(spin(7)) = 112.

Proof of Theorem 10. We will prove the theorem for the case of algebras oftype 4. For other types the proof is analogous.

Since the coefficients of the metric g are polynomial functions, the Levi-Civitaconnection given by g is analytic and the Lie algebra hol0 is generated by theoperators

R(X, Y )0,∇R(X, Y ; Z1)0,∇2R(X, Y ; Z1; Z2)0, . . . ∈ so(1, n + 1),

where ∇rR(X, Y ; Z1; . . . ; Zr) = (∇Zr · · · ∇Z1R)(X, Y ) and X , Y , Z1, Z2, . . . arevectors at the point 0.

To prove the theorem we will compute the components of the curvature tensorand of its derivatives.

The non-zero Christoffel symbols for the metric g given by (8) are the following

Γ00n+1 =

12

∂f

∂x0,

Γ0n+1 n+1 =

12

−

n0∑i=1

ui

(2

∂ui

∂xn+1− ∂f

∂xi

)+

∂f

∂xn+1

+

∂f

∂x0

f −

n0∑i=1

(ui)2

,

Γ0ij

=12

(∂ui

∂xj+

∂uj

∂xi

),

Γ0in+1

=12

n0∑j=1

uj

(∂ui

∂xj− ∂uj

∂xi

)+

12

∂f

∂xi,

Γ0ˆin+1

=12

∂f

∂xˆi,

Γijn+1

=12

(∂ui

∂xj− ∂uj

∂xi

),

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Γin+1 n+1 =

12

(−2

∂ui

∂xn+1− ∂f

∂xi+ ui ∂f

∂x0

),

Γîn+1 n+1 = −1

2∂f

∂xî,

Γn+1n+1 n+1 = −1

2∂f

∂x0.

Suppose that dim z(h) ≥ n − m. Let f = 2ψαii

xix˜i(xn+1)α−1 +

∑mi=n0+1(x

i)2. Wemust prove that hol0 = g4,h,m,ψ. We have the following non-zero Christoffel symbols

Γ0n+1 n+1 = −

n0∑i=1

ui((α − 1)aiαjk

xjxk(xn+1)α−2 − ψαii

x˜i(xn+1)α−1)

+ (α − 1)ψαii

xix˜i(xn+1)α−2, (12)

Γ0ij

=12(ai

αjk+ aj

αik)xk(xn+1)α−1, (13)

Γ0i,n+1

=1

(α − 1)!ujP i

αjkxk(xn+1)α−1 + ψ

αiix˜i(xn+1)α−1, (14)

Γijn+1

=1

(α − 1)!P i

αjkxk(xn+1)α−1, (15)

Γ0in+1

= xi, (16)

Γ0˜in+1

= ψαii

xi(xn+1)α−1, (17)

Γin+1 n+1 = −xi, (18)

Γ˜in+1 n+1 = −ψ

αiixi(xn+1)α−1. (19)

In particular, note the following

Γkij = Γn+1

bc = Γb0c = Γi

în+1= 0. (20)

For r ≥ 0 let Rbc,d,f ;f1;···;fr

be the functions such that ∇rR( ∂∂xd , ∂

∂xf ; ∂∂xf1 ; · · · ;

∂∂xfr

) ∂∂xc = Rb

c,d,f ;f1;···;fr

∂∂xb . One can compute the following components of the

curvature tensor

Rkji n+1

=1

(α − 1)!P k

αji(xn+1)α−1, R

ˆkji n+1

= 0, (21)

Rkjbc = 0 if (b, c) /∈ {1, . . . , n0} × {n + 1} ∪ {n + 1} × {1, . . . , n0}, (22)

R0˜iin+1

= ψαii

(xn+1)α−1, (23)

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1038 A. S. Galaev

R0˜ibc

= 0 if (b, c) /∈ {1, . . . , n0} × {n + 1} ∪ {n + 1} × {1, . . . , n0}, (24)

R0kij

=1

(α − 1)!P j

αik(xn+1)α−1, (25)

R0iin+1

= 1, Rin+1 in+1

= −1,

Rbcin+1

= 0 if (b, c) �= (0, i) and (b, c) �= (i, n + 1), (26)

R00bc = 0. (27)

Using (20), we get

ΓhbfRf

cdg = ΓhbiR

icdg, Γf

bhRcfdg = Γi

bhRcidg,

ΓfbhRc

dfg = ΓibhRc

dig, ΓfbhRc

dgf = ΓibhRc

dgi.(28)

From equalities (12)–(19) it follows that

ΓibcR

dief = Γi

bcRdief

, ΓibcR

deif = Γi

bcRdeif

,

ΓibcR

defi = Γi

bcRdefi

if b �= n + 1 or c �= n + 1,(29)

ΓbicR

idef = Γb

icRi

def if b �= 0 or c �= n + 1. (30)

Proof of the inclusion g4,h,m,ψ ⊂ hol0.

Lemma 14. For any 1 ≤ r ≤ N we have

(a) Rkjin+1;n+1;···;n+1︸︷︷︸

r−1 times

=∑N

α=r1

(α−r)!Pkαji

(xn+1)α−r + ykαjir

, where ykαjir

are func-

tions such that ykαjir

(0) =(

∂ykαjir

∂xn+1

)(0) = · · · = 0;

(b) R0˜iin+1;n+1;···;n+1︸︷︷︸

r−1 times

=∑N

α=r(α−1)!(α−r)!ψαii

(xn+1)α−r+zkαjir

, where zkαjir

are functions

such that zkαjir

(0) =(

∂zkαjir

∂xn+1

)(0) = · · · = 0.

Proof. We will prove this lemma by induction over r. For r = 1 the lemma followsfrom (21) and (23). Fix r0 > 1 and assume that the lemma is true for all r < r0.We must prove that the lemma is true for r = r0. We have

Rkjin+1;n+1;···;n+1︸︷︷︸

r times

=∂Rk

jin+1;n+1;···;n+1(r−1 times)

∂xn+1+ Γk

n+1 lRljin+1;n+1;···;n+1︸︷︷︸

r−1 times

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−Γln+1 j

Rklin+1;n+1;···;n+1︸︷︷︸

r−1 times

− Γln+1 i

Rkjln+1;n+1;···;n+1︸︷︷︸

r−1 times

−Γln+1 n+1R

kjil;n+1;···;n+1︸︷︷︸

r−1 times

− Γln+1 n+1R

kjin+1;l;n+1;···;n+1︸︷︷︸

r−2 times

− · · · − Γln+1 n+1R

kjin+1;n+1;···;n+1︸︷︷︸

r−2 times

;l

=N∑

α=r+1

1(α − r − 1)!

P kαji

(xn+1)α−r−1 +∂yk

αjir

∂xn+1+ yk

αjir

=N∑

α=r+1

1(α − r − 1)!

P kαji

(xn+1)α−r−1 + ykαjir+1

.

Claim (a) follows from the fact that all Christoffel symbols and all their derivativeswith respect to xn+1 vanish at the point 0. The proof of claim (b) is analogous.The lemma is proved.

From Lemma 14 it follows that for any 1 ≤ r ≤ N we have

Rkjin+1;n+1;···;n+1︸︷︷︸

r−1 times

(0) = P krji

, (31)

R0˜iin+1;n+1;···;n+1︸︷︷︸

r−1 times

(0) = (r − 1)!ψrii

. (32)

Similarly we can prove that from (21) it follows that

Rˆkiin+1;n+1;···;n+1︸︷︷︸

r−1 times

(0) = 0. (33)

From (11), (31), (32) and (33) it follows that

R(ei, en+1; en+1;···;en+1︸︷︷︸r−1 times

)0 = (0, Pr(ei), Xri + ψ(Pr(ei))), (34)

where Xri ∈ span{e1, . . . , em}. Since the elements Pr(ei) generate the Lie algebrah and prso(n) hol0 is a Lie algebra, from (34) we get

h ⊂ prso(n) hol0. (35)

Using (25) we can prove that for all 1 ≤ r ≤ N we have

R0kij;n+1;···;n+1︸︷︷︸

r−1 times

(0) = P j

rik.

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1040 A. S. Galaev

This means that

(0, 0, P j

rikej) ∈ hol0. (36)

Recall that we have decompositions (1), (2) and (3). Since h is generated by theimages of the elements Pα, for any 2 ≤ i ≤ s there exist α, i, j, k such that n1+ · · ·+ni−1 + 1 ≤ j ≤ n1 + · · · + ni and P j

αik�= 0. Combining this with (36), we get

{(0, 0, X)|X ∈ Rni} ∩ hol0 �= {0}.

Since hi ⊂ so(ni) is an irreducible subalgebra, hi ⊂ prso(n) hol0, and for any A ∈ h,Z ∈ R

n, Y ∈ Rni holds

[(0, A, Z), (0, 0, Y )] = (0, 0, AY ) ∈ {(0, 0, X)|X ∈ Rni} ∩ hol0,

we see that

{(0, 0, X)|X ∈ Rni} ⊂ hol0,

hence,

{(0, 0, X)|X ∈ span{e1, . . . , en0}} ⊂ hol0. (37)

From (26) it follows that R(ei, q) = (0, 0, ei), hence,

{(0, 0, X)|X ∈ span{en0+1, . . . , em}} ⊂ hol0. (38)

From (34), (37), (38) and the fact that h is generated by the elements Pα(ei) itfollows that

g4,h,m,ψ ⊂ hol0.

Proof of the inclusion hol0 ⊂ g4,h,m,ψ.

Lemma 15. For any r ≥ 0 we have

(a) Rkjil;f1 ;···;fr

= 0;

(b) Rˆijin+1;f1 ;···;fr

= 0;

(c) R0˜iil;f1 ;···;fr

= 0;

(d) Rk

jˆin+1;f1;···;fr

= 0;

(e) R0˜iin+1;f1;···;fr

= 0;

(f) R00bc;f1;···;fr

= 0;

(g) Rkjin+1;f1;···;fr

=∑

t∈Tif1...frztA

ktj

, where Tif1...fris a finite set of indices, zt

are functions and At ∈ h.

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(h) R0˜iin+1;f1;···;fr

=∑

t∈Tif1...fr

ztψti, where ψ

tiare numbers such that ψ(At) =∑n

˜i=m+1ψ

tie˜i.

Proof. We will prove the claims of the lemma by induction over r. For r = 0 theclaims follow from (21)–(27). Fix r > 0 and assume that the lemma is true for r.We must prove that the lemma is true for r + 1.

(a) We have

Rkjil;f1 ;···;fr ;l1

=∂Rk

jil;f1 ;···;fr

∂xl1+ Γk

l1l2Rl2jil;f1 ;···;fr

− Γl2l1jR

kl2il;f1;···;fr

−Γl2l1iR

kjl2l;f1;···;fr

− Γl2l1lR

kjil2 ;f1;···;fr

−Γl2l1f1

Rkjil;l2;f2:···;fr

− · · · − Γl2l1fr

Rkjil;f1 :···;fr−1;l2 .

From (20) and the inductive hypothesis it follows that

Rkjil;f1 ;···;fr ;l1 = 0.

Similarly,

Rkjil;f1 ;···;fr ;n+1

=∂Rk

jil;f1 ;···;fr

∂xn+1+ Γk

n+1 l2Rl2jil;f1 ;···;fr

− Γl2n+1 jR

kl2il;f1;···;fr

− Γl2n+1 iR

kjl2l;f1;···;fr

−Γl2n+1 lR

kjil2 ;f1;···;fr

− Γl2n+1 f1

Rkjil;l2 ;f2:···;fr

− · · · − Γl2n+1 fr

Rkjil;f1 :···;fr−1;l2 =0.

Claim (a) is proved. The proofs of claims (b)–(f) are analogous.

(g) We have

Rkjin+1;f1;···;fr ;l

=∂Rk

jin+1;f1;···;fr

∂xl+ Γk

ll1Rl1jin+1;f1;···;fr

− Γl1lj

Rkl1 in+1;f1;···;fr

− Γl1li

Rkjl1n+1;f1;···;fr

−Γl1ln+1R

kjil1;f1;···;fr

− Γl1lf1

Rkjin+1;l1;f2:···;fr

− · · · − Γl1lfr

Rkjin+1;f1:···;fr−1;l1

.

From (20), claim (a) of the lemma and the inductive hypothesis it follows that

Rkjin+1;f1;···;fr ;l

=∑

t∈Tif1...fr

∂zt

∂xlAk

tj−

n∑l1=1

∑t∈Til1f2...fr

ztΓl1lf1

Aktj− · · · −

n∑l1=1

∑t∈Tif1...fr−1l1

ztΓl1lfr

Aktj

.

(39)

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1042 A. S. Galaev

Furthermore,

Rkjin+1;f1;···;fr ;n+1

=∂Rk

jin+1;f1;···;fr

∂xn+1+ Γk

n+1 l1Rl1jin+1;f1;···;fr

− Γl1n+1 j


−Γl1n+1 i

Rkjl1n+1;f1;···;fr

− Γl1n+1 n+1R

kjil1;f1;···;fr

−Γl1n+1 f1

Rkjin+1;l1;f2:···;fr

− · · · − Γl1n+1 fr

Rkjin+1;f1:···;fr−1;l1

.

From (15) and (20) it follows that

Γkn+1 l1R

l1jin+1;f1;···;fr

− Γl1n+1 j


=1

(α − 1)!xl2(xn+1)α−1[Pα(el2

), R(ei, en+1; ef1 ; . . . ; efr)]kj.

Using this, claim (a) and inductive hypothesis, we get

Rkjin+1;f1;···;fr ;n+1

=∑

t∈Tif1...fr

∂zt

∂xn+1Ak

tj

+1

(α − 1)!xl2(xn+1)α−1[Pα(el2

), R(ei, en+1; ef1 ; . . . ; efr)]kj

−n∑

l1=1

∑t∈Tl1f1...fr

ztΓl1n+1i

Aktj−

n∑l1=1

∑t∈Til1f2...fr

ztΓl1n+1f1

Aktj

− · · · −n∑

l1=1


ztΓl1n+1fr

Aktj

. (40)

This proves claim (g).

(h) We have

R0˜iin+1;f1;···;fr ;l

=∂R0

˜iin+1;f1;···;fr

∂xl+ Γ0

ll1Rl1˜iin+1;f1;···;fr

− Γl1

liR0

l1 in+1;f1;···;fr

−Γl1li

R0˜il1n+1;f1;···;fr

− Γl1ln+1R

0˜iil1;f1;···;fr

−Γl1lf1

R0˜iin+1;l1;f2:···;fr

− · · · − Γl1lfr

R0˜iin+1;f1:···;fr−1;l1

.

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Using this, (20), claims (b) and (c), and the inductive hypothesis, we get

R0˜iin+1;f1;···;fr ;l

=∑

t∈Tif1...fr

∂zt

∂xlψ

ti−

n∑l1=1

∑t∈Til1f2···fr

ztΓl1lf1

ψti

− · · · −n∑

l1=1

∑t∈Tif1···fr−1l1

ztΓl1lfr

ψti. (41)

Finally,

R0˜iin+1;f1;···;fr ;n+1

=∂R0

˜iin+1;f1;···;fr

∂xn+1+ Γ0

n+1 l1Rl1˜iin+1;f1;···;fr

−Γl1

n+1˜iR0

l1 in+1;f1;···;fr

−Γl1n+1 i

R0˜il1n+1;f1;···;fr

−Γl1n+1 n+1R

0˜iil1;f1;···;fr

−Γl1n+1 f1

R0˜iin+1;l1;f2:···;fr

− · · · − Γl1n+1 fr

R0˜iin+1;f1:···;fr−1;l1

.

Using this, (20), claims (b) and (c), and the inductive hypothesis, we get

R0˜iin+1;f1;···;fr;n+1

=∑

t∈Tif1...fr

∂zt

∂xn+1ψ

ti−

n∑l1=1

∑t∈Tl1f1...fr

ztΓl1n+1i

ψti

−n∑

l1=1

∑t∈Til1f2...fr

ztΓl1n+1f1

ψti

− · · · −n∑

l1=1


ztΓl1n+1fr

ψti. (42)

Combining (39) with (41) and (40) with (42) and using the fact that ψ|h′ = 0, wesee that claim (h) is true. The lemma is proved.

From Lemma 15 it follows that

hol0 ⊂ g4,h,m,ψ.

Thus,

hol0 = g4,h,m,ψ.

The theorem is proved.

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1044 A. S. Galaev

Acknowledgments

I would like to thank Dmitri Vladimirovich Alekseevsky for introducing me tothe Lorentzian holonomy algebras and for his careful attention to my work duringthe last four years. I am grateful to Charles Boubel, who took my attention to theproblem of construction of metrics. I thank the Erwin Schrodinger Institute, wherethe work on this paper was finished.

References

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[2] W. Ambrose and I. M. Singer, A theorem on holonomy, Trans. Amer. Math. Soc. 79(1953) 428–443.

[3] H. Baum and I. Kath, Parallel spinors and holonomy groups on pseudo-Riemannianspin manifolds, (1997) SFB 288 Preprint No 276.

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manifolds, Diff. Geom. Appl. 22 (2005) 1–18.[15] A. S. Galaev, Isometry groups of Lobachevskian spaces, similarity transfor-

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[21] T. Leistner, Berger algebras, weak-Berger algebras and Lorentzian holonomy, (2002)sfb 288-preprint no. 567.

[22] T. Leistner, Towards a classification of Lorentzian holonomy groups, (2003) arXiv:math.DG/0305139.

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[24] T. Leistner, Holonomy and parallel spinors in Lorentzian geometry, PhD thesis,Humboldt-Universitat zu Berlin, 2003.

[25] K. Sfetsos and D. Zoakos, Supersymmetry and Lorentzian holonomy in various dimen-sions, J. High Energy Phys. 9 (2004) 10.

[26] E. B. Vinberg and A. L. Onishchik, Seminar on Lie Groups and Algebraic Groups,Moscow, URSS (1995).

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Abh. Math. Semin. Univ. Hambg. (2009) 79: 47–78DOI 10.1007/s12188-008-0015-7

Holonomy of supermanifolds

Anton S. Galaev

Received: 8 May 2008 / Published online: 8 January 2009© Mathematisches Seminar der Universität Hamburg and Springer 2008

Abstract Holonomy groups and holonomy algebras for connections on locally free sheavesover supermanifolds are introduced. A one-to-one correspondence between parallel sectionsand holonomy-invariant vectors, and a one-to-one correspondence between parallel locallydirect subsheaves and holonomy-invariant vector supersubspaces are obtained. As the spe-cial case, the holonomy of linear connections on supermanifolds is studied. Examples of par-allel geometric structures on supermanifolds and the corresponding holonomies are given.For Riemannian supermanifolds an analog of the Wu theorem is proved. Berger superalge-bras are defined and their examples are given.

Keywords Supermanifold · Superconnection · Holonomy algebra · Berger superalgebra

Mathematics Subject Classification (2000) 58A50 · 53C29

1 Introduction

The holonomy groups play a big role in the study of connections on vector bundles oversmooth manifolds. They link geometric and algebraic properties. In particular, they allow usto find parallel sections in geometric vector bundles associated to the manifold, such as thetangent bundle, tensor bundles, or the spin bundle, as holonomy-invariant objects, see [2, 3,7, 9, 10, 12, 13].

In the present paper we introduce holonomy groups for connections on supermanifolds.

Communicated by V. Cortés.

Supported from the Basic Research Center no. LC505 (Eduard Cech Center for Algebra and Geometry)of Ministry of Education, Youth and Sport of Czech Republic.

A.S. Galaev (�)Department of Algebra and Geometry, Masaryk University in Brno, Kotlárská 2, 611 37 Brno,Czech Republice-mail: [email protected]

mailto:[email protected]

48 A.S. Galaev

In Sect. 2 some necessary preliminaries on supermanifolds are given. Section 3 is anintroduction to the theory of holonomy for connections on vector bundles over smooth man-ifolds. In Sect. 4 we define the holonomy for connections on locally free sheaves over super-manifolds. First we define the holonomy algebra, for this we generalize the Ambrose-Singertheorem and use the covariant derivatives of the curvature tensor and parallel displacements.Then we define the holonomy group as a Lie supergroup. In Sect. 5 we define the infinitesi-mal holonomy algebra and show that in the analytic settings it coincides with the holonomyalgebra. In Sect. 6 we study parallel sections of sheaves over supermanifolds. It is shownthat any parallel section is uniquely defined by its value at any point, in spite of the factthat generally the sections of sheaves over supermanifolds are not defined by their valuesat all points. After this we obtain a one-to-one correspondence between parallel sectionsand holonomy-invariant vectors, as in the case of vector bundles over smooth manifolds.The definition of the holonomy was motivated by this correspondence. In Sect. 7 a one-to-one correspondence between parallel locally direct subsheaves and holonomy-invariantvector supersubspaces is obtained. Then we turn to study holonomy of linear connectionson supermanifolds. In Sect. 8 a one-to-one correspondence between parallel tensors on asupermanifold and holonomy-invariant tensors at one point is obtained. We consider exam-ples of parallel structures on supermanifolds and give the equivalent conditions in terms ofholonomy. In Sect. 9 Berger superalgebras are introduced. These superalgebras generalizethe usual Berger algebras and they can be considered as candidates to holonomy algebrasof linear torsion-free connections on supermanifolds. In Sect. 10 holonomy of locally sym-metric supermanifolds is considered. In Sect. 11 the case of the Levi-Civita connections onRiemannian supermanifolds is studied. Kählerian, special Kählerian, hyper-Kählerian andquaternionic-Kählerian supermanifolds are characterized by their holonomy. It is shownthat special Kählerian supermanifolds are Ricci-flat and, conversely, Ricci-flat simply con-nected Kählerian supermanifolds are special Kählerian. A generalization of the Wu theoremis proved. In Sect. 12 we give examples of complex Berger superalgebras.

Thus holonomy of supermanifolds that is introduced in the present paper is an appro-priate generalization of the usual holonomy of smooth manifolds, as many properties arepreserved.

2 Supermanifolds

In this section we give some necessary preliminaries on supermanifolds. An introduction tolinear superalgebra and to the theory of supermanifolds can be found in [6, 14–16, 23].

A real smooth (analytic) supermanifold M of dimension n|m is a pair (M, O M), whereM is a Hausdorff topological space and O M is a sheaf of commutative superalgebras withunity over R locally isomorphic to R

n|m = (Rn, ORn|m = ORn ⊗ �η1,...,ηm), where ORn is thesheaf of smooth (analytic) functions on R

n and �η1,...,ηm is the Grassmann superalgebra ofm generators. The sections of the sheaf O M are called superfunctions (or just functions) onM. The ideal (O M)1 ⊕ ((O M)1)

2 consists of the nilpotent elements of O M , and the sheafOM defined as the quotient O M/((O M)1 ⊕ ((O M)1)

2) furnish M with the structure of areal smooth (analytic) manifold. We get the canonical projection ∼: O M → OM , f �→ f .The value of a superfunction f at a point x ∈ M is by definition f (x). If m = 0, thenM = M is a smooth (analytic) manifold. Any isomorphism as above O M(U) � ORn|m(U1),where U ⊂ M and U1 ⊂ R

n, defines on M a local coordinate system (U,x1, . . . , xn+m)

(x1, . . . , xn+m are the functions corresponding to the functions y1, . . . , yn, η1, . . . , ηm, wherey1, . . . , yn are the standard coordinates on U1 ⊂ R

n). It is known [1] that there exists an atlas

Holonomy of supermanifolds 49

on M consisting of local coordinate systems (U,x1, . . . , xn+m) such that (U,x1, . . . , xn)

are local coordinate systems on M . We will always use such coordinates. We will use thefollowing convention about the ranks of the indices i, j, k = 1, . . . , n, α,β, γ = 1, . . . ,m

and a, b, c = 1, . . . , n + m. We will use the Einstein rule for sums. Let (U,x1, . . . , xn+m) beas above and denote xn+α by ξα . For any f ∈ O M(U) we get

f =m∑

r=0

∑

α1<···<αr

fα1...αr ξα1 · · · ξαr , (1)

where fα1...αr ∈ OM(U) and f∅ = f . For any α1 < · · · < αr and permutation σ :{α1, . . . , αr} → {α1, . . . , αr} we assume that fσ(α1)...σ (αr ) = signσ fα1...αr . If two of the num-bers α1, . . . , αr are equal, we assume that fα1...αr = 0.

Denote by T M the tangent sheaf, i.e. the sheaf of superderivatives of the sheaf O M . If(U,xi, ξα) is a system of local coordinates, then the vector fields ∂xi , ∂ξα form a basis of thesupermodule T M(U) over the superalgebra O M(U). The vector fields ∂xi and ∂ξα act on afunction f of the form (1) by the rule

∂xi f =m∑

r=0

∑

α1<···<αr

∂xi fα1...αr ξα1 · · · ξαr , (2)

∂ξαf =m∑

r=1

r∑

s=1

∑

α1<···<αs−1<αs=α<αs+1<···<αr

(−1)s−1fα1...αr ξα1 · · · ξαs−1ξαs+1 · · · ξαr . (3)

We will denote the vector field ∂xa just by ∂a .Let M = (M, O M) be a supermanifold and E a locally free sheaf of O M -supermodules

on M, e.g. T M. Let p|q be the rank of E , then locally there exists a basis(eI , h)I=1,...,p;=1,...,q of sections of E . We denote such basis also by eA, where ep+ = h.We will always assume that A,B,C = 1, . . . , p + q . For a point x ∈ M consider the vectorspace Ex = E (V )/(O M(V )x E (V )), where V ⊂ M is an open subset and O M(V )x is theideal in O M(V ) consisting of functions vanishing at the point x. The vector space Ex doesnot depend on choice of V ; it is a real vector superspace of dimension p|q . For any opensubset V ⊂ M we have the projection map from E (V ) onto Ex . For example, if E = T M ,then (T M)x is the tangent space Tx M to M at the point x.

A connection on E is an even morphism ∇ : T M ⊗R E → E of sheaves of supermodulesover R such that

∇f Y X = f ∇Y X and ∇Y f X = (Yf )X + (−1)|Y ||f |f ∇Y X

for all homogeneous functions f , vector fields Y on M and sections X of E , here | · | ∈ Z2 ={0, 1} denotes the parity. In particular, |∇Y X| = |Y |+ |X|. Locally we get the superfunctions�A

aB such that ∇∂a eB = �AaBeA. Obviously, |�A

aB | = |a| + |A| + |B|, where |a| = |∂xa | and|A| = |eA|.

The curvature tensor of the connection ∇ is given by

R(Y,Z) = [∇Y ,∇Z] − ∇[Y,Z], (4)

50 A.S. Galaev

where Y and Z are vector fields on M. Let ∇ be a connection on T M . Define the covariantderivatives of R with respect to ∇ as follows

∇rYr ,...,Y1

R(Y,Z)X = ∇Yr (∇r−1Yr−1,...,Y1

R(Y,Z)X) − ∇r−1∇Yr Yr−1,...,Y1

R(Y,Z)X

− (−1)|Yr ||Yr−1|∇r−1Yr−1,∇Yr Yr−2,...,Y1

R(Y,Z)X

− · · · − (−1)|Yr |(|Yr−1|+···+|Y2|)∇r−1Yr−1,...,Y2,∇Yr Y1

R(Y,Z)X

− (−1)|Yr |(|Yr−1|+···+|Y1|)∇r−1Yr−1,...,Y1

R(∇Yr Y,Z)X

− (−1)|Yr |(|Yr−1|+···+|Y1|+|Y |)∇r−1Yr−1,...,Y1

R(Y, ∇Yr Z)X

− (−1)|Yr |(|Yr−1|+···+|Y1|+|Y |+|Z|)∇r−1Yr−1,...,Y1

R(Y,Z)∇Yr X, (5)

here r ≥ 1, Yr, . . . , Y1, Y,Z ∈ T M(M) are homogeneous and X ∈ E (M). We assume that∇0R = R. It holds |∇r

Yr ,...,Y1R(Y,Z)| = |Yr |+ · · ·+ |Y1|+ |Y |+ |Z|. If E = T M and ∇ = ∇ ,

then we get the usual covariant derivatives of R.Let r ≥ 0. Define the components ∇r

ar ,...,a1RA

Bab of ∇rR by the condition

∇r∂ar ,...,∂a1

R(∂a, ∂b)eB = ∇rar ,...,a1

RABabeA.

Then, |∇rar ,...,a1

RABab| = |ar |+· · ·+|a1|+|a|+|b|+|A|+|B|. It is easy to show the following

RABab = ∂a�

AbB + (−1)|a|(|b|+|B|+|C|)�C

bB�AaC − (−1)|a||b|(∂b�

AaB + (−1)|b|(|a|+|B|+|C|)�C

aB�AbC)

(6)and

∇rar ,...,a1

RABab = ∂ar (∇r−1

ar−1,...,a1RA

Bab) + (−1)|ar |(|ar−1|+···+|a1|+|a|+|b|+|B|+|C|)∇r−1ar−1,...,a1

RCBab�

AarC

− �car ar−1

∇r−1c,ar−2,...,a1

RABab − (−1)(|c|+|ar−2|)|ar−1|�c

ar ar−2∇r−1

ar−1,c,ar−3,...,a1RA

Bab

− · · · − (−1)(|c|+|a1|)(|ar−1|+···+|a2|)�car a1

∇r−1ar−1,...,a2,cR

ABab

− (−1)(|c|+|a|)(|ar−1|+···+|a1|)�car a

∇r−1ar−1,...,a1

RABcb

− (−1)(|c|+|b|)(|ar−1|+···+|a1|+|a|)�car b

∇r−1ar−1,...,a1

RABac

− (−1)(|C|+|B|)(|ar−1|+···+|a1|+|a|+|b|)�CarB

∇r−1ar−1,...,a1

RACab. (7)

Let � denote the parity change functor. For example, if V = V0 ⊕ V1 is a vector super-space, then �(V ) is the vector superspace with (�(V ))0 = V1 and (�(V ))1 = V0 .

3 Holonomy of smooth manifolds

In this section we recall some standard facts about holonomy of connections on vector bun-dles over smooth manifolds, see e.g. [2, 12, 13].

Let E be a vector bundle over a connected smooth manifold M and ∇ a connectionon E. It is known that for any smooth curve γ : [a, b] ⊂ R → M and any X0 ∈ Eγ(a) thereexists a unique section X of E defined along the curve γ and satisfying the differentialequation ∇γ (s)X = 0 with the initial condition Xγ(a) = X0. Consequently, for any smooth


curve γ : [a, b] ⊂ R → M we obtain the isomorphism τγ : Eγ(a) → Eγ(b) defined by τγ :X0 �→ Xγ(b). The isomorphism τγ is called the parallel displacement along the curve γ .The parallel displacement can be defined in the obvious way also for piecewise smoothcurves. Let x ∈ M . The holonomy group Hol(∇)x of the connection ∇ at the point x isthe subgroup of GL(Ex) that consists of parallel displacements along all piecewise smoothloops at the point x ∈ M . If we consider only null-homotopic loops, we get the restrictedholonomy group Hol(∇)0

x . Obviously, Hol(∇)0x ⊂ Hol(∇)x is a subgroup. If the manifold

M is simply connected, then Hol(∇)0x = Hol(∇)x . It can be proved that the group Hol(∇)x

is a Lie subgroup of the Lie group GL(Ex) and the group Hol0x is the connected identity

component of the Lie group Hol(∇)x . The Lie algebra hol(∇)x of the Lie group Hol(∇)x

(and of Hol(∇)0x ) is called the holonomy algebra of the connection ∇ at the point x. Since the

manifold M is connected, the holonomy groups of ∇ at different points of M are isomorphic.Remark that by the holonomy group (resp. holonomy algebra) we understand not just

the Lie group Hol(∇)x (resp. Lie algebra hol(∇)x ), but the Lie group Hol(∇)x with therepresentation Hol(∇)x ↪→ GL(Ex) (resp. the Lie algebra hol(∇)x with the representationhol ↪→ gl(Ex)). These representations are called the holonomy representations.

The theorem of Ambrose and Singer states that the holonomy algebra hol(∇)x coincideswith the vector subspace of gl(Ex) spanned by the elements of the form

τ−1γ ◦ Ry(Y,Z) ◦ τγ ,

where R is the curvature tensor of the connection ∇ , γ is any curve in M beginning at thepoint x; y ∈ M is the end-point of the curve γ and Y,Z ∈ TyM .

Note that if E = T M is the tangent bundle of M , then

τ−1γ ◦ ∇r

Yr ,...,Y1Ry(Y,Z) ◦ τγ ∈ hol(∇)x,

where Y,Z,Y1, . . . , Yr ∈ TyM .A section X ∈ �(E) is called parallel if ∇X = 0. This is equivalent to the condition that

X is parallel along all curves in M , i.e. for any curve γ : [a, b] → M holds τγ Xγ (a) = Xγ(b).The following theorem is one of the main applications of the holonomy.

Theorem 3.1 Let M be a smooth manifold, x ∈ M , E a vector bundle over M , and ∇a connection on E. Then there is a one-to-one correspondence between parallel sectionsX ∈ �(E) and vectors Xx ∈ Ex preserved by the holonomy group Hol(∇)x .

Proof Having a parallel section X ∈ �(E) it is enough to take the value Xx ∈ Ex . Since X

is invariant under the parallel displacements, the vector Xx is invariant under the paralleldisplacements along the loops at the point x, i.e. under the holonomy representation. Con-versely, for a given vector Xx ∈ Ex preserved by Hol(∇)x define the section X ∈ �(E) suchthat at any point y ∈ M it holds Xy = τγ Xx , where γ is any curve beginning at x and endingat y. It is clear that Xy does not depend on the choice of the curve γ . �

A vector subbundle F ⊂ E is called parallel if for all Y ∈ �(T M) and all X ∈ �(F)

holds ∇Y X ∈ �(F). This is equivalent to the condition that F is parallel along all curves inM , i.e. for any curve γ : [a, b] → M holds τγ Fγ (a) = Fγ(b).

Theorem 3.2 Let M be a smooth manifold, E a vector bundle over M and ∇ a connectionon E. Then there exists a one-to-one correspondence between parallel vector subbundlesF ⊂ E of some rank p and subspaces Fx ⊂ Ex of dimension p preserved by the holonomygroup Hol(∇)x .

52 A.S. Galaev

4 Definition of the holonomy for a connection on a sheaf over a supermanifold

Let (M, O M) be a supermanifold, E a locally free sheaf of O M -supermodules of rankp|q on M and ∇ a connection on E . Consider the vector bundle E over M defined asE = ⋃

x∈M Ex . The rank of E is p + q . Define the subbundles E0 = ⋃x∈M(Ex)0 and E1 =⋃

x∈M(Ex)1 of E. Obviously, the restriction ∇ = (∇|�(T M)⊗�(E))∼ : �(T M) ⊗ �(E) →

�(E) is a connection on E. Since ∇ is even, the subbundles E0,E1 ⊂ E are parallel.Let γ : [a, b] ⊂ R → M be a curve and τγ : Eγ(a) → Eγ(b) the parallel displacementalong γ . Since the subbundles E0,E1 ⊂ E are parallel, we have τγ (E0)γ (a) = (E0)γ (b) andτγ (E1)γ (a) = (E1)γ (b). We get the even isomorphism

τγ : Eγ (a) → Eγ (b)

of vector superspaces. We call this isomorphism the parallel displacement in E along γ .

Remark 4.1 In [8] a parallel displacement in T M along supercurves γ : R1|1 → M is in-

troduced. The proof of the existence of the parallel displacement (p. 569) shows that theparallel displacements in T M along a supercurve γ : R

1|1 → M and along the underlyingcurve γ : R → M coincide and they coincide with our definition for the case E = T M .

Denote by gl(Ex) the Lie superalgebra of all endomorphisms of the vector superspacegl(Ex). Fixing a basis in Ex , we get an isomorphism

gl(Ex) � gl(n|m,R) ={(

A B

C D

)},

where

gl(n|m,R)0 ={(

A 00 D

)}, gl(n|m,R)1 =

{(0 B

C 0

)}.

Definition 4.1 Let (M, O M) be a supermanifold, E a locally free sheaf of O M -super-modules on M and ∇ a connection on E . The holonomy algebra hol(∇)x of the connection∇ at a point x ∈ M is the supersubalgebra of the Lie superalgebra gl(Ex) generated by theoperators of the form

τ−1γ ◦ ∇r

Yr ,...,Y1Ry(Y,Z) ◦ τγ : Ex → Ex,

where γ is any curve in M beginning at the point x; y ∈ M is the end-point of the curve γ ,r ≥ 0, Y,Z,Y1, . . . , Yr ∈ Ty M and ∇ is a connection on T M|U for an open neighbourhoodU ⊂ M of y.

Proposition 4.1 The definition of the holonomy algebra hol(∇)x does not depend on thechoice of the connection ∇ .

Proof Let γ be a curve in M beginning at the point x and ending at a point y ∈ M . LetU ⊂ M be an open neighbourhood of the point y. For any connection ∇ on T M|U and anyinteger t ≥ 0 define the vector space

L(∇)t = span{∇rYr ,...,Y1

Ry(Y,Z)|0 ≤ r ≤ t, Y,Z,Y1, . . . , Yr ∈ Ty M}.


Clearly, L(∇)t does not depend on the choice of U . Let (U,xa) be a system of local coor-dinates such that y ∈ U . Let ∇ be a connection on T M|U . Denote by �∇ the connection onT M|U such that �∇∂a = 0. To prove the proposition it is enough to show that for any t ≥ 0we have L(∇)t = L( �∇)t . This will follow from the following lemma.

Lemma 4.1 For any t ≥ 0 it holds

∇ t∂at ,...,∂a1

R(∂a0 , ∂a−1) = �∇ t∂at ,...,∂a1

R(∂a0 , ∂a−1)

+t−1∑

s=0

∑

(bs ,...,b−1)

Bat ...a−1,bs ...b−1�∇s

∂bs ,...,∂b1R(∂b0 , ∂b−1),

where Bat ...a−1,bs ...b−1 ∈ O M(U).

Proof We will prove the lemma by the induction over t . For t = 0 there is nothing to prove.Fix t > 0. Suppose that the lemma is true for all r < t and prove it for r = t .

Using (7) and the induction hypothesis, we get

∇r∂ar ,...,∂a1

R(∂a0 , ∂a−1)

= [∇∂r , ∇r−1∂ar−1 ,...,∂a1

R(∂a0 , ∂a−1)]

−r−1∑

l=−1

(−1)(|c|+|al |)(|ar−1|+···+|al+1|)�car al

∇r−1∂ar−1 ,...,∂al+1 ,∂c,∂al−1 ,...,∂a1

R(∂a0 , ∂a−1)

=[∇∂r ,

�∇r−1∂ar−1 ,...,∂a1

R(∂a0 , ∂a−1) +r−2∑

s=0

∑

(bs ,...,b−1)

Bar−1...a−1,bs ...b−1�∇s

∂bs ,...,∂b1R(∂b0 , ∂b−1)

]

−r−1∑

l=−1

(−1)(|c|+|al |)(|ar−1|+···+|al+1|)�car al

∇r−1∂ar−1 ,...,∂al+1 ,∂c,∂al−1 ,...,∂a1

R(∂a0 , ∂a−1)

= [∇∂r ,�∇r−1

∂ar−1 ,...,∂a1R(∂a0 , ∂a−1)]

+r−2∑

s=0

∑

(bs ,...,b−1)

∂ar (Bar−1...a−1,bs ...b−1)�∇s

∂bs ,...,∂b1R(∂b0 , ∂b−1)

+r−2∑

s=0

∑

(bs ,...,b−1)

(−1)|ar ||Bar−1 ...a−1,bs ...b−1 |

Bar−1...a−1,bs ...b−1 [∇∂ar, �∇s

∂bs ,...,∂b1R(∂b0 , ∂b−1)]

−r−1∑

l=−1

(−1)(|c|+|al |)(|ar−1|+···+|al+1|)�car al

∇r−1∂ar−1 ,...,∂al+1 ,∂c,∂al−1 ,...,∂a1

R(∂a0 , ∂a−1)

= �∇r∂ar ,...,∂a1

R(∂a0 , ∂a−1) +r−2∑

s=0

∑

(bs ,...,b−1)

∂ar (Bar−1...a−1,bs ...b−1)�∇s

∂bs ,...,∂b1R(∂b0 , ∂b−1)

54 A.S. Galaev

+r−2∑

s=0

∑

(bs ,...,b−1)

(−1)|ar ||Bar−1 ...a−1,bs ...b−1 |

Bar−1...a−1,bs ...b−1�∇s+1

∂ar ∂bs ,...,∂b1R(∂b0 , ∂b−1)

−r−1∑

l=−1

(−1)(|c|+|al |)(|ar−1|+···+|al+1|)�car al

∇r−1∂ar−1 ,...,∂al+1 ,∂c,∂al−1 ,...,∂a1

R(∂a0 , ∂a−1).

The proof of the lemma follows from the induction hypothesis applied to the last term. �

The proposition is proved. �

The next proposition simplifies the expression for the holonomy algebra. In particular itshows that it is not necessary to take the covariant derivatives of the curvature tensor in thedirections of the vectors tangent to M .

Proposition 4.2 The holonomy algebra hol(∇)x coincides with the supersubalgebra of theLie superalgebra gl(Ex) generated by the operators of the form

τ−1γ ◦ Ry(∂i, ∂j ) ◦ τγ and τ−1

γ ◦ ∇r∂γr ,...,∂γ1

Ry(∂γ , ∂a) ◦ τγ ,

where γ is any curve starting at the point x; y is the end-point of the curve γ , r ≥ 0,γr > · · · > γ1, xa are local coordinates on M over an open neighbourhood U of the point y

and ∇ is a connection on T M|U .

Proof Let g be the supersubalgebra of gl(Ex) generated by the operators as in the for-mulation of the proposition without the assumption γr > · · · > γ1. We will prove thathol(∇)x = g. Fix a curve γ starting at the point x. Let y be the end-point of γ , U ⊂ M

an open neighbourhood of y, xa local coordinates on M over U and ∇ a connection onT M|U . It is enough to show that any element η = τ−1

γ ◦ ∇r∂ar ,...,∂a1

Ry(∂a, ∂b) ◦ τγ ∈ hol(∇)x

belongs to g. We will prove this statement by the induction over r . By Lemma 4.1, we mayassume that the connection ∇ is flat such that ∇∂a = 0. For r = 0 there is nothing to prove.Fix t > 0 suppose that the statement is true for all r < t and prove it for r = t .

Lemma 4.2 Let r ≥ 2 and Y1, . . . , Yr , Y,Z ∈ T M(U) be matually commuting homogeneousparallel vector fields. Then

∇rYr ,...,Y1

R(Y,Z)

= (−1)|Ys ||Ys−1|∇rYr ,...,Ys+1,Ys−1,Ys ,Ys−2,...,Y1

R(Y,Z)

+∑

I⊂{s+2,...,r}(−1)S(I)

[∇[I ]

Yi1 ,...,Yi[I ]R(Ys+1, Ys), ∇r−2−[I ]

Yi′1

,...,Yi′[I ′ ]

,Ys−1,...,Y1R(Y,Z)

], (8)

where I = {i1, . . . , i[I ]} ⊂ {s + 2, . . . , r} is a subset with i1 > · · · > i[I ], [I ] is the number ofelements in I , I ′ = {i ′

1, . . . , i′[I ′]} = {s + 2, . . . , r}\I , i ′

1 > · · · > i ′[I ′] and S(I) is the sign of

the permutation (i1, . . . , i[I ], s + 1, s, i ′1, . . . , i

′[I ′]) of the numbers r, . . . ,1.

Proof Using (5), the assumptions of the lemma and the Jacobi super identity, we get

∇rYr ,...,Y1

R(Y,Z) = [∇Yr , [∇Yr−1 , . . . , [∇Ys+1 , [∇Ys , ∇s−1Ys−1,...,Y1

R(Y,Z)]] . . .]]


= [∇Yr , [∇Yr−1 , . . . , [[∇Ys+1 ,∇Ys ], ∇s−1Ys−1,...,Y1

R(Y,Z)] . . .]]+ (−1)|Ys ||Ys−1|[∇Yr , [∇Yr−1 , . . . , [∇Ys , [∇Ys+1 , ∇s−1

Ys−1,...,Y1R(Y,Z)]] . . .]]

= [∇Yr , [∇Yr−1 , . . . , [R(Ys+1, Ys), ∇s−1Ys−1,...,Y1

R(Y,Z)] . . .]]+ (−1)|Ys ||Ys−1|∇r

Yr ,...,Ys+1,Ys−1,Ys ,Ys−2,...,Y1R(Y,Z).

Applying the Jacobi super identity r − s times to the first term of the last equality, we getthe proof of the lemma. �

Note that for a curve μ(t) in M such that μ(0) = y it holds

∇rμ(0),∂ar−1 ,...,∂a1

Ry(∂a, ∂b) = d

dt

∣∣∣t=0

((τμ)t0)

−1 ◦ ∇r−1∂ar−1 ,...,∂a1

Rμ(t)(∂a, ∂b) ◦ (τμ)t0, (9)

where (τμ)t0 is the parallel displacement along the curve μ|[0,t]. Fix an element η =

τ−1γ ◦ ∇r

∂ar ,...,∂a1Ry(∂a, ∂b) ◦ τγ ∈ hol(∇)x . If ar ≤ n, then there exists a curve μ(t) with

μ(0) = y and μ(t) = ∂ar (μ(t)). From (9) and the induction hypothesis it follows that η ∈ g.Suppose that there exists s such that r − 1 ≥ s ≥ 1 and as ≤ n. Let s be maximal withthis property. Applying to η several times (8) and using the induction hypothesis, we getη − (−1)r−sτ−1

γ ◦ ∇r∂as ,∂ar ,...,∂as+1 ,∂as−1 ,...,∂a1

Ry(∂a, ∂b) ◦ τγ ∈ g. By the above argumentation,η ∈ g. Now we may assume that ar, . . . , a1 > n. If a ≤ n or b ≤ n, then using the sec-ond Bianchi super identity we get η = −(−1)|a1||a|τ−1

γ ◦ ∇r∂ar ,...,∂a2 ,∂a

Ry(∂b, ∂a1) ◦ τγ − τ−1γ ◦

∇r∂ar ,...,∂a2 ,∂b

Ry(∂a1 , ∂a) ◦ τγ ∈ g. Thus, hol(∇)x = g. Equality (8) shows that g coincideswith the supersubalgebra of gl(Ex) generated by the operators as in the formulation of theproposition. �

Let E be the vector bundle over M and ∇ the connection on E as above. Then theholonomy algebra hol(∇)x is contained in (hol(∇)x)0, but these Lie algebras must not co-incide, this shows the following example.

Example 4.1 Consider the supermanifold R0|1 = ({0},�ξ ). Define the connection ∇ on TR0|1

by ∇∂ξ∂ξ = ξ∂ξ . Then, hol(∇)0 = {0} and (hol(∇)0)0 = hol(∇)0 = gl(0|1).

Now we define the holonomy group. Recall that a Lie supergroup G = (G, O G ) is a groupobject in the category of supermanifolds. The underlying smooth manifold G is a Lie group.The Lie superalgebra g of G can be identified with the tangent space to G at the identitye ∈ G. The Lie algebra of the Lie group G is the even part g0 of the Lie superalgebra g.

Any Lie supergroup G is uniquely given by a pair (G,g) (Harish-Chandra pair), whereG is a Lie group, g = g0 ⊕ g1 is a Lie superalgebra such that g0 is the Lie algebra of the Liegroup G and there exists a representation Ad of G on g that extends the adjoint representa-tion of G on g0 and the differential of Ad coincides with the Lie superbracket of g restrictedto g0 × g1, see [6, 8].

Denote by Hol(∇)0x the connected Lie subgroup of the Lie group GL((Ex)0)×GL((Ex)1)

corresponding to the Lie subalgebra (hol(∇)x)0 ⊂ gl((Ex)0) ⊕ gl((Ex)1) ⊂ gl(Ex). LetHol(∇)x be the Lie subgroup of the Lie group GL((Ex)0) × GL((Ex)1) generated by theLie groups Hol(∇)0

x and Hol(∇)x . Clearly, the Lie algebra of the Lie group Hol(∇)x is(hol(∇)x)0 . Let Ad′ be the representation of the connected Lie group Hol(∇)0

x on hol(∇)x

such that the differential of Ad′ coincides with the Lie superbracket of hol(∇)x restricted

56 A.S. Galaev

to (hol(∇)x)0 × (hol(∇)x)1. Define the representation Ad′′ of the Lie group Hol(∇)x onhol(∇)x by the rule

Ad′′τμ

(τ−1γ ◦ ∇r

Yr ,...,Y1Ry(Y,Z) ◦ τγ ) = τμ ◦ τ−1

γ ◦ ∇rYr ,...,Y1

Ry(Y,Z) ◦ τγ ◦ τ−1μ .

Note that Ad′ |Hol(∇)0x∩Hol(∇)x

= Ad′′ |Hol(∇)0x∩Hol(∇)x

. Consequently, we get a representation

Ad of the group Hol(∇)x on hol(∇)x . It is obvious that Hol(∇)0x ∩ Hol(∇)x = Hol(∇)0

x andif M is simply connected, then Hol(∇)x ⊂ Hol(∇)0

x and Hol(∇)x = Hol(∇)0x .

Definition 4.2 The Lie supergroup Hol(∇)x given by the Harish-Chandra pair (Hol(∇)x ,hol(∇)x) is called the holonomy group of the connection ∇ at the point x. The Lie super-group Hol(∇)0

x given by the Harish-Chandra pair (Hol(∇)0x,hol(∇)x) is called the restricted

holonomy group of the connection ∇ at the point x.

5 Infinitesimal holonomy algebras

In this section we define the infinitesimal holonomy algebra and show that in the analyticcase it coincides with the holonomy algebra.

Definition 5.1 Let (M, O M) be a supermanifold, E a locally free sheaf of O M -super-modules on M and ∇ a connection on E . The infinitesimal holonomy algebra hol(∇)

infx

of the connection ∇ at a point x ∈ M is the supersubalgebra of the Lie superalgebra gl(Ex)

generated by the operators of the form

∇rYr ,...,Y1

Rx(Y,Z),

where r ≥ 0, Y,Z,Y1, . . . , Yr ∈ Tx M and ∇ is a connection on T M|U for an open neigh-bourhood U ⊂ M of x.

Lemma 4.1 shows that the definition of hol(∇)infx does not depend on the choice of the

connection ∇ .

Theorem 5.1 Let M = (M, O M) be an analytic supermanifold, E a locally free sheaf ofO M -supermodules on M and ∇ an analytic connection on E . Then

hol(∇)x = hol(∇)infx .

Proof First we prove the following lemma.

Lemma 5.1 If γ : [0,1] ⊂ R → M is a piecewise analytic curve starting at x ∈ M andending at y ∈ M , then

τ−1γ ◦ hol(∇)inf

y ◦ τγ = hol(∇)infx .

Proof Consider some partition 0 = t0 < t1 < · · · < tk = 1. Obviously, if the statement of thelemma holds for each curve γ |[ti ,ti+1], then the lemma is true. Consequently we may assumethat the image of γ is contained in some coordinate neighbourhood U ⊂ M . Fix an analyticconnection ∇ on T M|U .


Fix an element ∇r

Yr ,...,Y1Ry(Y , Z) ∈ hol(∇)

infy . It is enough to prove that τ−1

γ ◦ ∇r

Yr ,...,Y1×

Ry(Y , Z) ◦ τγ ∈ hol(∇)infx . Let Y,Z,Y1, . . . , Yr ∈ Tx M be the vectors such that Y =

τγ Y, Z = τγ Z, Y1 = τγ Y1, . . . , Yr = τγ Yr . For any s, t ∈ [0,1] with s ≤ t denote by (τγ )ts :

Eγ (s) → Eγ (t) the parallel displacement along the curve γ |[s,t]. Consider the endomorphism

F(t) = ((τγ )t0)

−1 ◦ ∇r

(τγ )t0Yr ,...,(τγ )t0Y1Rγ(t)((τγ )t

0Y, (τγ )t0Z) ◦ (τγ )t

0 : Ex → Ex .

We must prove that F(1) ∈ hol(∇)infx . Fix a basis of Ex . Without loss of generality we my

assume that the elements FAB (t) of the matrix of F(t) are analytic functions of t , i.e. FA

B (t) =∑∞k=0 FA

Bktk for some real numbers FA

Bk . For each k ≥ 0 denote by Fk the endomorphismof Ex with the matrix FA

Bk . Since FAB0 = FA

B (0), we have F0 = F(0) = ∇rYr ,...,Y1

Rx(Y,Z) ∈hol(∇)

infx . Further,

d

dtF (t) = lim

s→0

F(t + s) − F(t)

s

= ((τγ )t0)

−1 ◦ ∇r+1(τγ )t0Yr+1,(τγ )t0Yr ,...,(τγ )t0Y1

Rγ(t)((τγ )t0Y, (τγ )t

0Z) ◦ (τγ )t0,

where Yr+1 = ((τγ )t0)

−1γ (t). Consequently, F1 = ddt

F (0) ∈ hol(∇)infx . Similarly for each

k > 1 Fk = dk

(dt)kF (0) ∈ hol(∇)

infx . Thus, F(1) = ∑∞

k=0 Fk ∈ hol(∇)infx . The lemma is

proved. �

Let now τ−1γ ◦ ∇r

Yr ,...,Y1Ry(Y,Z) ◦ τγ ∈ hol(∇)x , where γ is any piecewise smooth curve

beginning at x ∈ M and ending at y ∈ M . Let μ be a piecewise analytic curve beginning atx ∈ M and ending at y ∈ M such that the loop γ ∗ μ−1 is null-homotopic. We have

τ−1γ ◦ ∇r

Yr ,...,Y1Ry(Y,Z) ◦ τγ = τ−1

μ ◦((τμ ◦ τ−1

γ ) ◦ ∇rYr ,...,Y1

Ry(Y,Z) ◦ (τγ ◦ τ−1μ )

)◦ τμ

= τ−1μ ◦

(Ad′′

τμ∗γ−1

∇rYr ,...,Y1

Ry(Y,Z))

◦ τμ.

Here τμ∗γ −1 ∈ Hol(∇)0y acts on the element η = ∇r

Yr ,...,Y1Ry(Y,Z) ∈ hol(∇)

infy ⊂ hol(∇)y

as it was defined above. Note that the connection ∇ is analytic. By the classical result, thegroup Hol(∇)0

y coincides with the infinitesimal holonomy group Hol(∇)infy . Consequently,

τμ∗γ −1 = exp ξ for some ξ ∈ hol(∇)infy ⊂ hol(∇)

infy . Finally,

Ad′′exp ξ η = (exp(d Ad′′)ξ )η = (exp adξ )η =

∞∑

k=0

adkξ

k! η ∈ hol(∇)infy .

By the above lemma, τ−1γ ◦ ∇r

Yr ,...,Y1Ry(Y,Z) ◦ τγ ∈ hol(∇)

infx . Thus, hol(∇)x ⊂ hol(∇)

infx .

The inverse inclusion is trivial. The theorem is proved. �

6 Parallel sections

Let (M, O M) be a supermanifold, E a locally free sheaf of O M -supermodules on M and∇ a connection on E . A section X ∈ E (M) is called parallel if ∇X = 0. Let E be thevector bundle on M and ∇ the connection on E as above. For the section X ∈ �(M) we get

58 A.S. Galaev

∇X = 0. Hence for any curve γ : [a, b] ⊂ R → M we have τγ Xγ (a) = Xγ (b). Consequently,τγ Xγ (a) = Xγ(b), where τγ : Eγ (a) → Eγ (b), i.e. X is parallel along curves in M .

Consider a system of local coordinates (U,xa) and a basis eA of E (U). Let X ∈ E (U),then

X = XAeA, ∇∂aX = ∂aXAeA + (−1)|a||XA|XA�B

aAeB.1

Thus the condition ∇X = 0 is equivalent to the following condition in local coordinates

∂aXA + (−1)|a||XB |XB�A

aB = 0 (10)

or to the conditions

∂iXA + XB�A

iB = 0, (11)

∂γ XA + (−1)|XB |XB�AγB = 0. (12)

Equations (11) and (12) are equivalent to

(∂γr . . . ∂γ1(∂iXA + XB�A

iB))∼ = 0, (13)

(∂γr . . . ∂γ1(∂γ XA + (−1)|XB |XB�AγB))∼ = 0, (14)

where 0 ≤ r ≤ m. These equations can be written as

∂iXAγ1...γr

+ signγ1,...,γr

r∑

l=0

∑

{α1,...,αr }={γ1,...,γr }α1<···<αl ,αl+1<···<αr

signα1,...,αrXB

α1...αl�A

iBαl+1...αr= 0, (15)

XAγγ1...γr

+ signγ1,...,γr

r∑

l=0

∑

{α1,...,αr }={γ1,...,γr }α1<···<αl ,αl+1<···<αr

signα1,...,αr(−1)lXB

α1...αl�A

γBαl+1...αr= 0. (16)

Using this, we can prove the following proposition.

Proposition 6.1 Let M = (M, O M) be a supermanifold, E a locally free sheaf of O M -supermodules on M and ∇ a connection on E . Then a parallel section X ∈ E (M) is uniquelydefined by its value at any point x ∈ M .

Proof Let ∇X = 0, x ∈ M and Xx be the value of X at the point x. Since X is parallel alongcurves in M , using Xx , we can find the values of X at all points of M . Consider the localcoordinates as above. As we know the values of X at all points, we know the functions XA.Using (16) for r = 0, we can find the functions XA

γ . Namely, XAγ = −XB�A

γB . Using (16)

for r = 1, we get XAγγ1

= −XB�AγBγ1

+ XBγ1

�AγB . In the same way we can find all functions

XAγγ1...γr

, i.e. we know the functions XA and we reconstruct the section X in any coordinatesystem. The proposition is proved. �

1We assume that (−1)|a||XA|XA�BaA

= (−1)|a|XA0

�BaA

− (−1)|a|XA1

�BγA

, where XA = XA0

+ XA1

is the

decomposition of XA in the sum of the even and odd parts.


Theorem 6.1 Let M = (M, O M) be a supermanifold, x ∈ M , E a locally free sheaf ofO M -supermodules on M and ∇ a connection on E . Then there exists a one-to-one cor-respondence between parallel sections X ∈ E (M) and vectors Xx ∈ Ex annihilated by theholonomy algebra hol(∇)x and preserved by the group Hol(∇)x .

Proof of Theorem 6.1 Suppose that a section X ∈ E (M) is parallel, i.e. ∇X = 0. Let U ⊂ M

be an open subset and ∇ a connection on T M|U . From (5) it follows that

∇rYr ,...,Y1

R(Y,Z)X = 0 (17)

for any vector fields Y,Z,Y1, . . . , Yr ∈ T M(U). Since X is parallel, for any curve γ in M

beginning at the point x ∈ M and ending at a point y ∈ M , we have Xy = τγ Xx . Hence tosee that Xx is annihilated by hol(∇)x , it is enough to consider (17) at the point y.

Conversely, suppose there exists a non-zero vector Xx ∈ Ex annihilated by the holonomyalgebra hol(∇)x and preserved by the group Hol(∇)x . From Theorem 3.1 it follows thatthere exists a section X0 ∈ �(E) such that ∇X0 = 0 and (X0)x = Xx . Fix a coordinateneighborhood (U,xa) on M and a local basis eA of E (U). Then eA is a local basis of�(U,E) and we get the functions XA

0 ∈ OM(U) such that X0 = XA0 eA on U . Using (16)

and XA0 , as in the proof of Proposition 6.1, define functions XA

γγ1...γr∈ OM(U) for all γ <

γ1 < · · · < γr , 0 ≤ r ≤ m − 1. This gives us functions XA ∈ O M(U) such that XA = XA0 .

Consider the section X = XAeA ∈ E (U). We claim that ∇X = 0. To prove this it is enoughto show that the functions XA satisfy (13) and (14) for all γ1 < · · · < γr , 0 ≤ r ≤ m and anyγ , then XA will satisfy (13) and (14) for all γ1, . . . , γr and γ . Note that, by construction, thefunctions XA satisfy (14) for γ < γ1 < · · · < γr .

For the proof we use the induction over r . Parallel to this we will prove that

(∂ar . . . ∂as+1((−1)(|A|+|B|)|XB |∇s−1as ,...,a2

RABa1aX

B))∼ = 0 for all r ≥ 1 and 1 ≤ s ≤ r. (18)

For r = 0 (13) follows from the fact that ∇X0 = 0; (14) follows from the definition offunctions XA

γ . Fix r0 > 0. Suppose that (13) and (14) hold for all r < r0 and check this forr = r0.

Lemma 6.1 It holds

(∂γr . . . ∂γ1(∂iXA + XB�A

iB))∼ = (∂γr . . . ∂γ2((−1)(|A|+|B|)|XB |RABγ1iX

B))∼, (19)

(∂γr . . . ∂γ1(∂γ XA + (−1)|XB |XB�AγB))∼

=

⎧⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

0, if γ < γ1;(−1)s−1 1

2 (∂γr . . . ∂γs+1∂γs−1 . . . ∂γ1((−1)(|A|+|B|)|XB |RABγsγs

XB))∼,

if γ = γs for some s,1 ≤ s ≤ r;(∂γr . . . ∂γ2((−1)(|A|+|B|)|XB |RA

Bγ1γ XB))∼,

if γs < γ < γs+1 for some s,1 ≤ s ≤ r.

(20)

Proof We have

(∂γr . . . ∂γ1(∂iXA + XB�A

iB))∼

= (∂γr . . . ∂γ2(∂i∂γ1XA + (∂γ1X

B)�AiB + (−1)|XB |XB∂γ1�

AiB))∼

60 A.S. Galaev

= (∂γr . . . ∂γ2(∂i(−(−1)|XB |XB�Aγ1B) − (−1)|XC |XC�B

γ1C�AiB + (−1)|XB |XB∂γ1�

AiB))∼

= (∂γr . . . ∂γ2(−(−1)|XB |(∂iXB)�A

γ1B − (−1)|XB |XB∂i�Aγ1B

− (−1)|XC |XC�Bγ1C�A

iB + (−1)|XC |XC∂γ1�AiC))∼

= (∂γr . . . ∂γ2((−1)|XB |XC�BiC�A

γ1B − (−1)|XC |XC∂i�Aγ1C


iB + (−1)|XC |XC∂γ1�AiC))∼

= (∂γr . . . ∂γ2((−1)|XC |+|B|+|C|XC�BiC�A

γ1B − (−1)|XC |XC∂i�Aγ1C


iB + (−1)|XC |XC∂γ1�AiC))∼

= (∂γr . . . ∂γ2((−1)|XC |XCRACγ1i ))

∼ = (∂γr . . . ∂γ2((−1)|XC |(|A|+|C|)RACγ1iX

C))∼

= (∂γr . . . ∂γ2((−1)(|A|+|B|)|XB |RABγ1iX

B))∼.

Here we used the induction hypotheses, (6) and the fact that the induction hypotheses imply

(∂γr . . . ∂γ2(−(−1)|XB |∂iXB))∼ = (∂γr . . . ∂γ2((−1)|XB |XC�B

iC))∼

= (∂γr . . . ∂γ2((−1)|XC |+|B|+|C|XC�BiC))∼.

This proves (19).Let us prove (20). First, if γ < γ1, then (∂γr . . . ∂γ1(∂γ XA + (−1)|XB |XB�A

γB))∼ = 0 bythe definition of XA. As above, using the induction hypotheses, for any t , 1 ≤ t ≤ r we get

(∂γr . . . ∂γ1((−1)|XB |XB�AγB))∼

= (−1)t−1(∂γr . . . ∂γt+1∂γt−1 . . . ∂γ1∂γt ((−1)|XB |XB�AγB))∼

= (−1)t−1(∂γr . . . ∂γt+1∂γt−1 . . . ∂γ1

× ((−1)(|A|+|B|)|XB |(−(−1)|B|+|C|�CγtB

�AγC + ∂γt �

AγB)XB))∼. (21)

Further, if γ = γs for some s, 1 ≤ s ≤ r , then ∂γr . . . ∂γ1∂γ XA = 0 and

(∂γr . . . ∂γ1(∂γ XA + (−1)|XB |XB�AγB))∼

= (−1)s−1(∂γr . . . ∂γs+1∂γs−1 . . . ∂γ1

× ((−1)(|A|+|B|)|XB |(−(−1)|B|+|C|�CγsB

�AγsC

+ ∂γs �AγsB

)XB))∼

= (−1)s−1 1

2(∂γr . . . ∂γs+1∂γs−1 . . . ∂γ1((−1)(|A|+|B|)|XB |RA

BγsγsXB))∼,

where we used (21) for t = s and (6).In the remaining case γs < γ < γs+1 for some s, 1 ≤ s ≤ r . Then,

(∂γr . . . ∂γ1∂γ XA)∼

= (−1)s(∂γr . . . ∂γs+1∂γ ∂γs . . . ∂γ1XA)∼


= (−1)s(∂γr . . . ∂γs+1∂γ ∂γs . . . ∂γ2(−(−1)|XB |XB�Aγ1B))∼

= (∂γr . . . ∂γ2∂γ ((−1)|XB |XB�Aγ1B))∼

= (∂γr . . . ∂γ2((−1)|XB |∂γ XB�Aγ1B + XB∂γ �A

γ1B))∼

= (∂γr . . . ∂γ2(−(−1)|XB |(−1)|XC |XC�BγC�A

γ1B + XB∂γ �Aγ1B))∼

= (∂γr . . . ∂γ2(−(−1)|C|+|B|+(|A|+|B|)|XB |�CγB�A

γ1CXB + (−1)(|A|+|B|)|XB |∂γ �Aγ1BXB))∼,

where we used the definition of XA and the induction hypotheses. Combining this with (21)for t = 1 and (6), we get

(∂γr . . . ∂γ1(∂γ XA + (−1)|XB |XB�AγB))∼ = (∂γr . . . ∂γ2((−1)(|A|+|B|)|XB |RA

Bγ1γ XB))∼.

The lemma is proved. �

Since ∇X0 = 0, for any curve γ in U beginning at the point x ∈ M and ending at a pointy ∈ U , we have Xy = τγ Xx . From this and (ii) it follows that

(∇sYs ,...,Y1

R(Y,Z))∼X = 0 (22)

for all s ≥ 0 and Y,Z,Y1, . . . , Yr ∈ T M(U). Consequently,

(∇ t−1at ,...,a2

RABa1aX

B)∼ = 0 for all t ≥ 1. (23)

Lemma 6.1 and (23) prove (13), (14) and (18) for r = 1. Suppose now that (18) holds forall r < r0 and check it together with (13) and (14) for r = r0.

Lemma 6.2 It holds

(∂ar . . . ∂as+1((−1)(|A|+|B|)|XB |∇s−1as ,...,a2

RABa1aX

B))∼

= (∂ar . . . ∂as+2((−1)(|A|+|B|)|XB |∇sas+1,...,a2

RABa1aX

B))∼

for all s, 1 ≤ s ≤ r .

Proof We have

(∂ar . . . ∂as+1((−1)(|A|+|B|)|XB |∇s−1as ,...,a2

RABa1aX

B))∼

= (∂ar . . . ∂as+2((−1)(|A|+|B|)|XB |(∂as+1(∇s−1as ,...,a2

RABa1a)X

B

+ (−1)|as+1|(|as |+···+|a1|+|a|+|A|+|B|)∇s−1as ,...,a2

RABa1a∂as+1X

B)))∼.

Using (7) and the induction hypotheses, we get

(∂ar . . . ∂as+2((−1)(|A|+|B|)|XB |∂as+1(∇s−1as ,...,a2

RABa1a)X

B))∼

= (∂ar . . . ∂as+2((−1)(|A|+|B|)|XB |∇sas+1,...,a2

RABa1aX

B

+ (−1)(|A|+|B|)|XB |(−1)(|C|+|B|)(|ar |+···+|a1|+|a|)�Cas+1B∇s−1

as ,...,a2RA

Ca1aXB))∼.

62 A.S. Galaev

Furthermore,

(∂ar . . . ∂as+2((−1)(|A|+|B|)|XB |(−1)|as+1|(|as |+···+|a1|+|a|+|A|+|B|)∇s−1as ,...,a2

RABa1a∂as+1X

B))∼

= −(∂ar . . . ∂as+2((−1)(|A|+|B|)|XB |(−1)|as+1|(|as |+···+|a1|+|a|+|A|+|B|)

× ∇s−1as ,...,a2

RABa1a(−1)|as+1||XC |XC�B

as+1C)∼

= −(∂ar . . . ∂as+2((−1)(|A|+|B|)(|XC |+|B|+|C|)+|as+1|(|as |+···+|a1|+|a|+|A|+|B|)+|as+1||XC |

× ∇s−1as ,...,a2

RABa1aX

C�Bas+1C))∼

= −(∂ar . . . ∂as+2((−1)(|A|+|C|)(|XB |+|C|+|B|)+|as+1|(|as |+···+|a1|+|a|+|A|+|C|)+|as+1||XB |

× ∇s−1as ,...,a2

RACa1aX

B�Cas+1B))∼

= −(∂ar . . . ∂as+2((−1)σ ∇s−1as ,...,a2

RACa1aX

B�Cas+1B))∼

= −(∂ar . . . ∂as+2((−1)σ (−1)(|C|+|as+1|+|B|)(|as |+···+|a1|+|a|+|C|+|XB |)

× �Cas+1B∇s−1

as ,...,a2RA

Ca1aXB))∼,

= −(−1)(|A|+|B|)|XB |(−1)(|C|+|B|)(|ar |+···+|a1|+|a|)�Cas+1B∇s−1

as ,...,a2RA

Ca1aXB))∼.

Here we used the fact that by the induction hypotheses

(∂ar . . . ∂as+2(∂as+1XB))∼ = −(∂ar . . . ∂as+2((−1)|as+1||XC |XC�B

as+1C))∼

and that (−1)|as+1||XC |XC�Bas+1C = (−1)|as+1|(|XB |+|B|+|C|)XC�B

as+1C .Thus,

(∂ar . . . ∂as+1((−1)(|A|+|B|)|XB |∇s−1as ,...,a2

RABa1aX

B))∼

= (∂ar . . . ∂as+2((−1)(|A|+|B|)|XB |∇sas+1,...,a2

RABa1aX

B))∼.

The lemma is proved. �

Using Lemma 6.2, (23) and the fact that if for a function f ∈ O M(U) holds |f | = 1, thenf = 0, we get

(∂ar . . . ∂as+1((−1)(|A|+|B|)|XB |∇s−1as ,...,a2

RABa1aX

B))∼

= ((−1)(|A|+|B|)|XB |∇r−1ar ,...,a2

RABa1aX

B)∼ = (∇r−1ar ,...,a2

RABa1aX

B)∼ = 0.

Further,

(∂γr . . . ∂γ1(∂iXA + XB�A

iB))∼

= (∂γr . . . ∂γ2((−1)(|A|+|B|)|XB |RABγ1iX

B))∼

= ((−1)(|A|+|B|)|XB |∇r−1γr ,...,γ2

RABγ1iX

B)∼ = (∇r−1γr ,...,γ2

RABγ1iX

B)∼ = 0.

Similarly, (∂γr . . . ∂γ1(∂γ XA + (−1)|XB |XB�AγB))∼ = 0.


Thus the functions XA satisfy (13) and (14). Consequently, ∇X = 0. From Proposi-tion 6.1 it follows that X does not depend on the choice of coordinates over U . Hence foreach coordinate neighbourhood U ⊂ M we have a unique parallel section X ∈ E (U). Thuswe get a parallel section X ∈ E (M). The theorem is proved. �

Recall that the connection ∇ is called flat if E admits local bases of parallel sections.

Corollary 6.1 Let M = (M, O M) be a supermanifold, E a locally free sheaf of O M -supermodules on M and ∇ a connection on E . Then the following conditions are equivalent:(i) ∇ is flat; (ii) R = 0; (iii) hol(∇)x = 0.

7 Parallel subsheaves

Let M = (M, O M) be a supermanifold, E a locally free sheaf of rank p|q of O M -supermodules on M. For fixed integers 0 ≤ p1 ≤ p and 0 ≤ q1 ≤ q we assume that

A, B, C = 1, . . . , p1,p + 1, . . . , p + q1 and A, B, C = p1 + 1, . . . , p,p + q1 + 1, . . . , q.

Recall that a subsheaf F ⊂ E of O M -supermodules (of rank p1|q1) is called a locally directsubsheaf if locally there exists a basis eA of E (U) such that eA is a basis of F (U). Forexample, a locally direct subsheaf of T M is called a distribution on M.

Let ∇ be a connection on E . A locally direct subsheaf F ⊂ E is called parallel if for anyopen subset U ⊂ M and any Y ∈ T M(U) and X ∈ F (U) it holds ∇Y X ∈ F (U).

The following theorem is a generalization of Theorem 3.2.

Theorem 7.1 Let M = (M, O M) be a supermanifold, x ∈ M , E a locally free sheaf ofO M -supermodules on M and ∇ a connection on E . Then there exists a one-to-one corre-spondence between parallel locally direct subsheaves F ⊂ E of some rank p1|q1 and vectorsupersubspaces Fx ⊂ Ex of dimension p1|q1 preserved by hol(∇)x and Hol(∇)x .

Proof of Theorem 7.1 Let F ⊂ E be a parallel locally direct subsheaf of rank p1|q1. Considerthe vector subbundle F = ⋃

y∈M Fy ⊂ E. Since F ⊂ E is parallel, the subbundle F ⊂ E isparallel. In particular, F is invariant under the parallel displacements in M and for anycurve γ in M beginning at the point x ∈ M and ending at a point y ∈ M , we have Fy =τγ Fx . Let U ⊂ M be an open subset. Since F ⊂ E is parallel, for any X ∈ F (U) and anyY,Z,Y1, . . . , Yr ∈ T M(U) we have

∇rYr ,...,Y1

R(Y,Z)X ∈ F (U).

Consequently, (∇rYr ,...,Y1

R(Y,Z))yX ∈ Fy for all X ∈ Fy and Y,Z,Y1, . . . , Yr ∈ Ty M. Thisand the fact that F is invariant under the parallel displacements show that Fx is preservedby hol(∇)x .

Conversely, suppose that there exists a vector supersubspace Fx ⊂ Ex preserved byhol(∇)x and Hol(∇)x . Then Hol(∇)x preserves the vector subspaces (Fx)0, (Fx)1 ⊂ Ex ,where (Fx)0 and (Fx)1 are the even and odd parts of Fx , respectively. Consequently, we getparallel vector subbundles F0,F1 ⊂ E on M . Recall that E0,E1 ⊂ E are also parallel. LetU,xa be a system of local coordinates on M and let eA be a basis of �(U,E) such thate1, . . . , ep1 ; ep+1, . . . , ep+q1 ; e1, . . . , ep and ep+1, . . . , ep+q are bases of �(U,F0), �(U,F1),

64 A.S. Galaev

�(U,E0) and �(U,E1), respectively. In particular, eA is a basis of E (U). Since the vectorsubbundle F ⊂ E is parallel, we get

�B

iA= 0. (24)

Let f B

A∈ O M(U) be functions. Consider the sections

fA = eA + f B

AeB ∈ E (U).

We will prove the existence and uniqueness of functions f B

Aunder the conditions f B

A= 0,

|f B

A| = |B| + |A| and the condition that there exist functions XB

aA∈ O M(U) such that

∇∂a fA = XB

aAfB . (25)

Then the supersubmodule

F (U) = O M(U) ⊗ spanR{fA} ⊂ E (U)

will be parallel, i.e. for all Y ∈ T M(U) and X ∈ F (U) it holds ∇Y X ∈ F (U). Equation (25)is equivalent to the following two equations

�B

aA+ (−1)|a|(|A|+|C|)f C

A�B

aC= XB

aA, (26)

�B

aA+ ∂af

B

A+ (−1)|a|(|A|+|C|)f C

A�B

aC= XD

aAf B

D. (27)

Combining (26) and (27), we get

�B

aA+ ∂af

B

A+ (−1)|a|(|A|+|C|)f C

A�B

aC− (�D

aA+ (−1)|a|(|A|+|C|)f C

A�D

aC)f B

D= 0. (28)

We will show that (28) has a unique solution f B

Asatisfying the conditions f B

A= 0 and

|f B

A| = |B|+ |A|, then (25) will hold for the functions XB

aAgiven by (26). Equation (28) can

be written as

(∂γr . . . ∂γ1(�B

iA+ ∂if

B

A+ f C

A�B

iC− (�D

iA+ f C

A�D

iC)f B

D))∼ = 0, (29)

(∂γr . . . ∂γ1(�B

γ A+ ∂γ f B

A+ (−1)|A|+|C|f C

A�B

γ C− (�D

γ A+ (−1)|A|+|C|f C

A�D

γ C)f B

D))∼

= 0, (30)

where 0 ≤ r ≤ m. The further proof is similar to the proof of Theorem 6.1. As in Sect. 6, wecan use (30) to define functions f B

Aγ γ1...γr(γ < γ1 < · · · < γr , 0 ≤ r ≤ m − 1) and f B

Ausing

the condition f B

A= 0. Then these functions satisfy |f B

A| = |B| + |A|. We must prove that

f B

Asatisfy (29) and (30) for γ1 < · · · < γr , 0 ≤ r ≤ m and all γ . We will prove this by the

induction over r . Parallel to this we will prove that

(∂ar . . . ∂as+1(∇s−1as ,...,a2

RB

Aa1a+ (−1)|A|+|C|f C

A∇s−1

as ,...,a2RB

Ca1a

− ∇s−1as ,...,a2

RD

Aa1af B

D− (−1)|A|+|C|f C

A∇s−1

as ,...,a2RD

Ca1af B

D))∼ = 0 (31)

for all r ≥ 1 and 1 ≤ s ≤ r . For r = 0 equation (29) follows from (24) and the conditionf B

A= 0; equation (30) follows from the definition of the functions f B

Aγ.


Lemma 7.1 For r ≥ 1 it holds

(∂γr . . . ∂γ1(�B

iA+ ∂if

B

A+ f C

A�B

iC− (�D

iA+ f C

A�D

iC)f B

D))∼

= (∂γr . . . ∂γ2(RB

Aγ1i+ (−1)|A|+|C|f C

ARB

Cγ1i− RD

Aγ1if B

D− (−1)|A|+|C|f C

ARD

Cγ1if B

D))∼.

A similar holds for (∂γr . . . ∂γ1(�B

γ A+ ∂γ f B

A+ (−1)|A|+|C|f C

A�B

γ C− (�D

γ A+ (−1)|A|+|C| ×

f C

A�D

γ C)f B

D))∼ (see Lemma 6.1).

Proof The proof is similar to the proof of Lemma 6.1. �

Lemma 7.2 For r ≥ 1 it holds

(∇r−1ar ,...,a2

RB

Aa1a)∼ = 0.

Proof The proof follows from the facts that the holonomy algebra hol(∇)x preserves thevector subspace Fx ⊂ Ex and that the distribution F ⊂ E is parallel along all curves in M . �

Lemma 7.2 proves (29), (30) and (31) for r = 1. Fix r0 ≥ 2. Suppose that (29), (30) and(31) hold for all r < r0 and check this for r = r0.

Lemma 7.3 It holds

(∂ar . . . ∂as+1(∇s−1as ,...,a2

RB

Aa1a+ (−1)|A|+|C|f C

A∇s−1

as ,...,a2RB

Ca1a

− ∇s−1as ,...,a2

RD

Aa1af B

D− (−1)|A|+|C|f C

A∇s−1

as ,...,a2RD

Ca1af B

D))∼

= (∂ar . . . ∂as+2(∇sas+1,...,a2

RB

Aa1a+ (−1)|A|+|C|f C

A∇s

as+1,...,a2RB

Ca1a

− ∇sas+1,...,a2

RD

Aa1af B

D− (−1)|A|+|C|f C

A∇s

as+1,...,a2RD

Ca1af B

D))∼

for all s, 1 ≤ s ≤ r .

Proof The proof is similar to the proof of Lemma 6.2. �

Now (29), (30) and (31) follow from the above lemmas and the induction hypotheses.We have proved that the supersubmodule F (U) = O M(U) ⊗ span

R{fA} ⊂ E (U) is par-

allel. We claim that F (U) does not depend on the choice of the basis eA. Suppose that wehave another basis e′

A of �(U,E) with the same property as above. Then there exist func-tions HB

A,HB

A∈ OM(U) such that e′

A= HB

AeB and e′

A= HB

AeB . Furthermore,

f ′A

= e′A

+ f ′BAe′

B= HB

AeB + f ′B

AHB

BeB = (H B

A+ f ′C

AH B

C)eB + f ′C

AH B

CeB .

Since (H B

A+ f ′C

AH B

C)∼ = HB

Ais an invertible matrix, the matrix HB

A+ f ′C

AH B

Cis also in-

vertible. Denote by Y B

Aits inverse matrix. Then, Y A

Cf ′

A= eC + Y A

Cf ′C

AH B

CeB . Moreover,

∇∂a (YA

Cf ′

A) = (∂aY

B

C+ (−1)

|a||Y AC

|Y A

CX′B

aA)f ′B

= (∂aYB

C+ (−1)

|a||Y AC

|Y A

CX′B

aA)(H D

B+ f ′C

BH D

C)(Y A

Df ′

A).

66 A.S. Galaev

From the uniqueness of the functions f B

Cit follows that f B

C= Y A

Cf ′C

AH B

C. Consequently,

fC = Y A

Cf ′

A. Similarly, there exist functions Y ′A

Csuch that f ′

C= Y ′A

CfA. This proves that

F (U) does not depend on the choice of the basis eA. Note that the sections fA, eA forma basis of E (U). Thus we get a parallel locally direct subsheaf F ⊂ E of rank p1|q1. Thetheorem is proved. �

8 Holonomy of linear connections over supermanifolds

Let M = (M, O M) be a supermanifold of dimension n|m. In this section we consider aconnection ∇ on the tangent sheaf T M of M. Then in Definition 4.1 of the holonomyalgebra hol(∇)x we may choose ∇ = ∇ . If we put E = T M , then in the above nota-tion, E = ⋃

y∈M Ty M = T M. In particular, E0 = T M is the tangent bundle over M . We

get the connections ∇ and ∇|T M on the vector bundles T M and T M , respectively. Weidentify the holonomy algebra hol(∇)x and the group Hol(∇)x with a supersubalgebrahol(∇) ⊂ gl(n|m,R) and a Lie subgroup Hol(∇) ⊂ GL(n,R) × GL(m,R), respectively.

On the cotangent sheaf T ∗M = HomO M (T M, O M) of M we get the connection ∇∗ de-

fined as follows

(∇∗Xϕ)Y = Xϕ(Y ) − (−1)|X||ϕ|ϕ(∇XY ),

where U ⊂ M is an open subset, X,Y ∈ T M(U) and ϕ ∈ T ∗M(U) are homogeneous. The

curvature tensor R∗ of the connection ∇∗ is given by

(R∗(Y,Z)ϕ)X = −(−1)|ϕ|(|Y |+|Z|)ϕ(R(Y,Z)X),

where U ⊂ M is an open subset, X,Y ∈ T M(U) and ϕ ∈ T ∗M(U) are homogeneous. Let

γ : [a, b] ⊂ R → M be a curve, τγ : Tγ (a)M → Tγ (b)M and τ ∗γ : T ∗

γ (a)M → T ∗γ (b)M be the

parallel displacements of the connections ∇ and ∇∗, respectively. Then,

(τ ∗γ ϕ)X = ϕ(τ−1

γ X),

where ϕ ∈ T ∗γ (a)M and X ∈ Tγ (b)M.

Consider the sheaf of tensor fields of type (r, s) over M,

T r,sM = ⊗r

O MT M

⊗

O M

⊗sO M

T ∗M.

Define the connection ∇r,s on this sheaf by

∇r,sX (X1 ⊗ · · · ⊗ Xr ⊗ ϕ1 ⊗ · · · ⊗ ϕs)

=r∑

i=1

(−1)|X|(|X1|+···+|Xi−1|)X1 ⊗ · · · ⊗ Xi−1 ⊗ ∇XXi ⊗ Xi+1 ⊗ · · · ⊗ Xr ⊗ ϕ1 ⊗ · · · ⊗ ϕs

+s∑

j=1

(−1)|X|(|X1|+···+|Xr |+|ϕ1|+···+|ϕj−1|)X1 ⊗ · · · ⊗ Xr ⊗ ϕ1 ⊗ · · ·

⊗ ϕj−1 ⊗ ∇∗Xϕj ⊗ ϕj+1 ⊗ · · · ⊗ ϕs,


where U ⊂ M is an open subset, X,X1, . . . ,Xr ∈ T M(U) and ϕ1, . . . , ϕr ∈ T ∗M(U) are

homogeneous. For the curvature tensor Rr,s of this connection we get

Rr,s(Y,Z)(X1 ⊗ · · · ⊗ Xr ⊗ ϕ1 ⊗ · · · ⊗ ϕs)

=r∑

i=1

(−1)(|Y |+|Z|)(|X1|+···+|Xi−1|)X1 ⊗ · · · ⊗ Xi−1 ⊗ R(Y,Z)Xi ⊗ Xi+1 ⊗ · · ·

⊗ Xr ⊗ ϕ1 ⊗ · · · ⊗ ϕs

+s∑

j=1

(−1)(|Y |+|Z|)(|X1|+···+|Xr |+|ϕ1|+···+|ϕj−1|)X1 ⊗ · · · ⊗ Xr ⊗ ϕ1 ⊗ · · ·

⊗ ϕj−1 ⊗ R∗(Y,Z)ϕj ⊗ ϕj+1 ⊗ · · · ⊗ ϕs,

where U ⊂ M is an open subset, Y,Z,X1, . . . ,Xr ∈ T M(U) and ϕ1, . . . , ϕs ∈ T ∗M(U) are

homogeneous. For the parallel displacement τ r,sγ of the connections ∇r,s along a curve γ :

[a, b] ⊂ R → M it holds τ r,sγ (X1 ⊗· · ·⊗Xr ⊗ϕ1 ⊗· · ·⊗ϕs) = (τγ X1 ⊗· · ·⊗τγ Xr ⊗τ ∗

γ ϕ1 ⊗· · · ⊗ τ ∗

γ ϕs), where X1, . . . ,Xr ∈ Tγ (a)M and ϕ1, . . . , ϕs ∈ T ∗γ (a)M.

Thus we see that the holonomy algebra hol(∇r,s)x of the connection ∇r,s at a point x ∈ M

and the group Hol(∇r,s)x coincide with the tensor extension of the representation of theholonomy algebra hol(∇)x and with the tensor extension of the representation of the groupHol(∇)x , respectively. From Theorem 6.1 we immediately read the following.

Theorem 8.1 Let M = (M, O M) be a supermanifold, x ∈ M , and ∇ a connection onT M . Then there exists a one-to one correspondence between parallel tensors P ∈ T r,s

M (M)

and tensors Px ∈ T r,sx M annihilated by the tensor extension of the representation of the

holonomy algebra hol(∇)x and preserved by tensor extension of the representation of thegroup Hol(∇)x .

To consider the examples, let us recall the definitions of some classical Lie superalgebras.First of all, sl(n|m,R) = {ξ ∈ gl(n|m,R)| str ξ = 0}, where str

(A B

C D

) = trA − trD. Let g

be an even non-degenerate supersymmetric form on the vector superspace Rn ⊕ �(R2k),

i.e. g(Rn,�(R2k)) = g(�(R2k),Rn) = 0, the restriction of g to R

n is non-degenerate andsymmetric (with some signature (p, q), p + q = n), and the restriction of g to �(R2k) isnon-degenerate and skew-symmetric. The orthosymplectic Lie superalgebra is defined asthe supersubalgebra of gl(n|2k,R) preserving g,

osp(p, q|2k)i = {ξ ∈ gl(n|2k,R)i |g(ξx, y) + (−1)|x|ig(x, ξy) = 0

for all x, y ∈ Rn ∪ �(R2k)}, i ∈ Z2.

In particular, if the restriction of g to Rn is positive definite, choose a basis with respect to

which the matrix of g has the form

⎛

⎝1n 0 00 0 1k

0 −1k 0

⎞

⎠ .

68 A.S. Galaev

Then,

osp(n|2k,R) =

⎧⎪⎨

⎪⎩

⎛

⎜⎝A B1 B2

Bt2 C1 C2

−Bt1 C3 −Ct

1

⎞

⎟⎠

∣∣∣∣∣∣∣At = −A,Ct

2 = C2,Ct3 = C3

⎫⎪⎬

⎪⎭.

Similarly, the Lie superalgebra ospsk(2k|p,q), p+q = m, is defined as the supersubalgebraof gl(2k|m,R) preserving an even non-degenerate super skew-symmetric form on the vectorsuperspace R

2k ⊕�(Rm), in this case the restriction of g to R2k is non-degenerate and skew-

symmetric, and the restriction of g to �(Rm) is non-degenerate and symmetric (with thesignature (p, q), p + q = m). For example,

ospsk(2k|m,R) =

⎧⎪⎨

⎪⎩

⎛

⎜⎝C1 C2 B1

C3 −Ct1 B2

Bt2 −Bt

1 A

⎞

⎟⎠

∣∣∣∣∣∣∣At = −A,Ct

2 = C2,Ct3 = C3

⎫⎪⎬

⎪⎭.

Consider an odd non-degenerate supersymmetric form g on Rn ⊕ �(Rn), i.e. g(Rn,R

n) =g(�(Rn),�(Rn)) = 0, and g(x0, x1) = g(x1, x0) for all x0 ∈ R

n, x1 ∈ �(Rn). There ex-ists a basis of R

n ⊕ �(Rn) such that matrix of g has the form( 0 1n

1n 0

). The periplectic Lie

superalgebra pe(n,R) is the subalgebra of gl(n|n,R) preserving g, then,

pe(n,R) ={(

A B

C −At

)∣∣∣∣B = −Bt,C = Ct

}.

Similarly, consider an odd non-degenerate super skew-symmetric form g on Rn ⊕ �(Rn).

There exists a basis of Rn ⊕ �(Rn) with respect to which the matrix of g has the form( 0 1n

−1n 0

). The Lie superalgebra pesk(n,R) is the subalgebra of gl(n|n,R) preserving g, in

particular,

pesk(n,R) =

{(A B

C −At

)∣∣∣∣B = Bt,C = −Ct

}.

Finally, let J be an odd complex structure on Rn ⊕ �(Rn), i.e. J is an odd isomorphism

of Rn ⊕ �(Rn) with J 2 = − id. The queer Lie superalgebra q(n,R) is the subalgebra of

gl(n|n,R) commuting with J . There exists a basis of Rn ⊕ �(Rn) such that the matrix of J

has the form( 0 1n

−1n 0

). Then, q(n,R) = {(

A B

B A

)}.

Example 8.1 In Table 1 we give equivalent conditions for existance of some parallel tensorson supermanifolds and inclusions of the holonomy.

The torsion of the connection ∇ is given by the formula

T (X,Y ) = ∇XY − (−1)|X||Y |∇Y X − [X,Y ], (32)

where U ⊂ M is open and X,Y ∈ T M(U) are homogeneous. If the connection ∇ is torsion-free, i.e. T = 0, then the curvature tensor R satisfies the Bianchi identity

R(X,Y )Z + (−1)|X|(|Y |+|Z|)R(Y,Z)X + (−1)|Z|(|X|+|Y |)R(Z,X)Y = 0 (33)

for U ⊂ M open and all homogeneous X,Y,Z ∈ T M(U).


Table 1 Examples of parallel structures and the corresponding holonomy

Parallel structure on M hol(∇) is Hol(∇) is Restriction


Riemannian supermetric, osp(p, q|2k) O(p, q) × Sp(2k,R) n = p + q,m = 2k

i.e. even non-degenerate


Even non-degenerate ospsk(2k|p,q) Sp(2k,R) × O(p, q) n = 2k,m = p + q

super skew-symmetric metric

Odd non-degenerate pe(n,R){(A 0

0 (At )−1)∣∣A ∈ GL(n,R)

}m = n


Odd non-degenerate super pesk(n,R){(A 0

0 (At )−1)∣∣A ∈ GL(n,R)

}m = n

skew-symmetric metric

Complex structure gl(k|l,C) GL(k,C) × GL(l,C) n = 2k,m = 2l

Odd complex structure, q(n,R){(A 0

0 A

)∣∣A ∈ GL(n,R)}

m = n

i.e. odd automorphism

J of T M with J 2 = − id

Recall that a distribution on M is a locally direct subsheaf F ⊂ T M . The distributionF is called involutive, if [F (U), F (U)] ⊂ F (U) for all open subsets U ⊂ M . Theorem 7.1gives us a one-to-one correspondence between parallel distributions on M and vector su-persubspaces F ⊂ Tx M preserved by hol(∇)x and Hol(∇)x . From (32) it follows that if ∇is torsion-free, then any parallel distribution on M is involutive.

9 Berger superalgebras

Let V be a real or complex vector superspace and g ⊂ gl(V ) a supersubalgebra. The spaceof algebraic curvature tensors of type g is the vector superspace R(g) = R(g)0 ⊕ R(g)1,

where

R(g) =

⎧⎪⎨

⎪⎩R ∈ V ∗ ∧ V ∗ ⊗ g

∣∣∣∣∣∣∣

R(X,Y )Z + (−1)|X|(|Y |+|Z|)R(Y,Z)X

+ (−1)|Z|(|X|+|Y |)R(Z,X)Y = 0

for all homogeneous X,Y,Z ∈ V

⎫⎪⎬

⎪⎭.

Obviously, R(g) is a g-module with respect to the action

A · R = RA,(34)

RA(X,Y ) = [A,R(X,Y )] − (−1)|A||R|R(AX,Y ) − (−1)|A|(|R|+|X|)R(X,AY),

where A ∈ g, R ∈ R(g) and X,Y ∈ V are homogeneous. If M is a supermanifold and ∇ is alinear torsion-free connection on T M, then applying covariant derivatives to (33), we get that(∇r

Yr ,...,Y1R)x ∈ R(hol(∇)x) for all r ≥ 0 and Y1, . . . , Yr ∈ TxM . Moreover, |(∇r

Yr ,...,Y1R)x | =

|Y1| + · · · + |Yr |, whenever Y1, . . . , Yr are homogeneous.

70 A.S. Galaev


L(R(g)) = span{R(X,Y )|R ∈ R(g), X,Y ∈ V } ⊂ g.

From (34) it follows that L(R(g)) is an ideal in g. We call a supersubalgebra g ⊂ gl(V ) aBerger superalgebra if L(R(g)) = g.

Proposition 9.1 Let M be a supermanifold of dimension n|m with a linear torsion-freeconnection ∇ . Then its holonomy algebra hol(∇) ⊂ gl(n|m,R) is a Berger superalgebra.

Proof The proof follows from Definition 4.1 and (33). �


R∇(g) =

⎧⎪⎨

⎪⎩S ∈ V ∗ ⊗ R(g)

∣∣∣∣∣∣∣

SX(Y,Z) + (−1)|X|(|Y |+|Z|)SY (Z,X)

+ (−1)|Z|(|X|+|Y |)SZ(X,Y ) = 0

for all homogeneous X,Y,Z ∈ V

⎫⎪⎬

⎪⎭.

If M is a supermanifold and ∇ is a linear torsion-free connection on T M , then applying co-variant derivatives to the second Bianchi identity, we get that (∇r

Yr ,...,Y2,·R)x ∈ R∇(hol(∇)x)

for all r ≥ 1 and Y2, . . . , Yr ∈ TxM . Moreover, |(∇rYr ,...,Y2,·R)x | = |Y2|+ · · ·+ |Yr |, whenever

Y2, . . . , Yr are homogeneous.

10 Holonomy of locally symmetric superspaces

Let M be a supermanifold of dimension n|m with a linear connection ∇ . The supermani-fold (M,∇) is called locally symmetric if ∇ is torsion-free and ∇R = 0. Note that in thissituation the underlying manifold (M, ∇|T M) is locally symmetric as well. Let x ∈ M . Theo-rem 8.1 implies that hol(∇)x annihilates the value Rx ∈ R(hol(∇)x) and Hol(∇)x preservesRx . We get that hol(∇)x = span{Rx(X,Y )|X,Y ∈ Tx M}.

More generally, let V be a vector superspace and suppose that g ⊂ gl(V ) is a subalgebrathat annihilates an R ∈ R(g)0 . Consider the Lie superalgebra h = g+V with the Lie brackets

[x, y] = −R(x, y), [A,x] = Ax, [A,B] = A ◦ B − (−1)|A||B|B ◦ A,

where x, y ∈ V and A,B ∈ g. We get that h = g + V is a symmetric decomposition ofthe Lie superalgebra h [5]. All such decompositions for h simple are described in [22]. Inparticular, this allows to find all possible irreducible holonomy algebras of locally symmetricsupermanifolds.

A Berger superalgebra g is called symmetric if R∇(g) = 0.

Proposition 10.1 Let M be a supermanifold with a torsian free connection ∇ . If hol(∇) isa symmetric Berger superalgebra, then (M,∇) is locally symmetric.

Proof We need to prove that ∇R = 0. Since R∇(hol(∇)y) = 0 for all y ∈ M , we get(∇r

Yr ,...,Y1R)∼ = 0 for all r ≥ 0 and all vector fields Y1, . . . , Yr on M. Using this, (7) and dou-

ble induction over r and s it is easy to get that (∂ar . . . ∂as+1∇s+1as ,...,a1,f Rd

cab)∼ = 0 for all r and

s such that r ≥ s ≥ 0. In particular, (∂γr . . . ∂γ1∇f Rdcab)

∼ = 0 for all r ≥ 0. Thus, ∇R = 0. �



Proposition 10.2 Let g ⊂ gl(V ) be an irreducible Berger superalgebra. If g annihilates themodule R(g), then g is a symmetric Berger superalgebra.

11 Holonomy of Riemannian supermanifolds

A Riemannian supermanifold (M, g) is a supermanifold M of dimension n|m, m = 2k en-dowed with an even non-degenerate supersymmetric metric g [4]. In particular, the value gx

of g at a point x ∈ M satisfies: gx |(Tx M)0,(Tx M)1= 0, gx |(Tx M)0×(Tx M)0

is non-degenerate,symmetric and gx |(Tx M)1×(Tx M)1

is non-degenerate, skew-symmetric. The metric g definesa pseudo-Riemannian metric g on the manifold M . Note that g is not assumed to be pos-itively defined. The supermanifold (M, g) has a unique linear connection ∇ such that ∇is torsion-free and ∇g = 0. This connection is called the Levi-Civita connection. We de-note the holonomy algebra of the connection ∇ by hol(M, g). As we have already noted,hol(M, g) ⊂ osp(p0, q0|2k) and Hol(∇) ⊂ O(p0, q0)×Sp(2k,R), where (p0, q0) is the sig-nature of the pseudo-Riemannian metric g.

The Kählerian, hyper-Kählerian and quaternionic-Kählerian supermanifolds are definedin the natural way, see e.g. [4]. We define special Kählerian or Calabi-Yau supermanifoldsby the condition from Table 2.

Before we consider the examples, let us recall the definitions of the following Lie super-algebras. Suppose that n and m are even and suppose that we have an even non-degeneratesupersymmetric form on the vector superspace R

n ⊕ �(Rm) such that the restriction of g toR

n has signature (2p0,2q0), 2p0 + 2q0 = n. Suppose that we have also a complex structureJ on R

n ⊕ �(Rm) such that g(Jx,Jy) = g(x, y) for all x, y ∈ Rn ⊕ �(Rm). Note that the

restriction of g(J ·, ·) to �(Rm) is symmetric and non-degenerate and let this form have thesignature (2p1,2q1), 2p1 + 2q1 = m. By definition,

u(p0, q0|p1, q1) = {ξ ∈ osp(2p0,2q0|m)|[ξ, J ] = 0},su(p0, q0|p1, q1) = {ξ ∈ u(p0, q0|p1, q1)| str(J ◦ ξ) = 0}.

Similarly, suppose that n and m are integers divided by 4 and suppose that we have an evennon-degenerate supersymmetric form on the vector superspace R

n ⊕�(Rm) such that the re-striction of g to R

n has the signature (4p,4q), 4p+4q = n. Suppose that we have a quater-nionic structure I, J,K on R

n ⊕�(Rm) (i.e. I, J,K are complex structures and they gener-ate the Lie algebra isomorphic to sp(1)) such that g(Ix, Iy) = g(Jx,Jy) = g(Kx,Ky) =

Table 2 Special geometries of Riemannian supermanifolds and the corresponding holonomies

Type of (M, g) hol(M, g) is Hol(∇) is Restriction


Kählerian u(p0, q0|p1, q1) U(p0, q0) × U(p1, q1) n = 2p0 + 2q0,

m = 2p1 + 2q1

Special Kählerian su(p0, q0|p1, q1) U(1)(SU(p0, q0) × SU(p1, q1)) n = 2p0 + 2q0,

m = 2p1 + 2q1

Hyper-Kählerian hosp(p, q|m) Sp(p, q) × SO∗(k) n = 4p + 4q, m = 4k

Quaternionic- sp(1) ⊕ hosp(p, q|m) Sp(1)(Sp(p, q) × SO∗(k)) n = 4p + 4q ≥ 8,

Kählerian m = 4k

72 A.S. Galaev

g(x, y) for all x, y ∈ Rn ⊕ �(Rm). By definition,

hosp(p, q|m) = {ξ ∈ osp(4p,4q|m)|[ξ, I ] = [ξ, J ] = [ξ,K] = 0}.Note that the normalizer of sp(1) in osp(4p,4q|m) coincides with sp(1) ⊕ hosp(p, q|m).Remark also that hosp(p, q|m)0 ∩ sp(m,R) = so∗(k), where 4k = m, and so∗(k) is thesubalgebra of gl(k,H) acting on R

m = Hk and preserving the skew-quaternionic-Hermitian

form

w(·, ·) + iw(I ·, ·) + jw(J ·, ·) + kw(K·, ·),here w is the restriction of g to �(Rm) considered as a skew-symmetric bilinear form on R

m.

Example 11.1 In Table 2 we give equivalent conditions for special geometries of (M, g)

and inclusions of the holonomy.

Define the Ricci tensor Ric of the supermanifold (M, g) by the formula

Ric(Y,Z) = str(X �→ (−1)|X||Z|R(Y,X)Z), (35)

where U ⊂ M is open and X,Y,Z ∈ T M(U) are homogeneous.

Proposition 11.1 Let (M, g) be a Kählerian supermanifold, then Ric = 0 if and only ifhol(M, g) ⊂ su(p0, q0|p1, q1). In particular, if (M, g) is special Kählerian, then Ric = 0; ifM is simply connected, (M, g) is Kählerian and Ric = 0, then (M, g) is special Kählerian.

Proof If R is an algebraic curvature tensor of type u(p0, q0|p1, q1), then Ric(R) is definedby (35). The following formula can be proved as in the usual case up to additional signs

Ric(R)(Y,Z) = 1

2str(J ◦ R(JY,Z)), (36)

where J is the complex structure. Recall that

su(p0, q0|p1, q1) = {ξ ∈ u(p0, q0|p1, q1)| str(J ◦ ξ) = 0}. (37)

Combining (5) and (7) and using the equality

(−1)|ar |(|ar−1|+···+|a1|+|a|+|b|+|d|+|c|)+|b|(|d|+1)∇r−1ar−1,...,a1

Rcdab�

bar c

− (−1)(|c|+|b|)(|ar−1|+···+|a1|+|a|)+|b|(|d|+1)�car b

∇r−1ar−1,...,a1

Rbdac = 0,

we get

Ric(∇s∂as ,...,∂a1

R)(∂a, ∂d) = ∂as (Ric(∇s−1∂as−1 ,...,∂a1

R)(∂a, ∂d)) − Ric(∇s−1∇∂as

∂as−1 ,...,∂a1R)(∂a, ∂d)

− (−1)|as ||as−1| Ric(∇s−1∂as−1 ,∇∂as

∂as−2 ,...,∂a1R)(∂a, ∂d)

− · · · − (−1)|as |(|as−1|+···+|a2|) Ric(∇s−1∂as−1 ,...,∂a2 ,∇∂as

∂a1R)(∂a, ∂d)

− (−1)|as |(|as−1|+···+|a1|) Ric(∇s−1∂as−1 ,...,∂a1

R)(∇∂as∂a, ∂d)

− (−1)|as |(|as−1|+···+|a1|+|a|) Ric(∇s−1∂as−1 ,...,∂a1

R)(∂a,∇∂as∂d). (38)


Suppose that Ric = 0. Using (38) and induction it is easy to prove that Ric(∇s∂as ,...,∂a1

R)(∂a ,∂d) = 0 for all s ≥ 0. Consequently, (Ric(∇s

∂as ,...,∂a1R)(∂a , ∂d))

∼ = 0 for all s ≥ 0. This, (36)and (37) yield hol(M, g) ⊂ su(p0, q0|p1, q1).

Suppose that hol(M, g) ⊂ su(p0, q0|p1, q1). Then (Ric(∇s∂as ,...,∂a1

R)(∂a, ∂d))∼ = 0 for

all s ≥ 0. Using this, (38) and double induction over r and s, it is easy to show that

(∂ar . . . ∂as+1(Ric(∇s∂as ,...,∂a1

R)(∂a, ∂d)))∼ = 0

for all r and s such that r ≥ s ≥ 0. In particular, (∂γr · · · ∂γ1(Ric(∂a, ∂d)))∼ = 0 for all r ≥ 0,

i.e. Ric = 0. The proposition is proved. �

Corollary 11.1 Let (M, g) be a quaternionic-Kählerian supermanifold, then Ric = 0 if andonly if hol(M, g) ⊂ hosp(p0, q0|p1, q1). In particular, if (M, g) is hyper-Kählerian, thenRic = 0; if M is simply connected, (M, g) is quaternionic-Kählerian and Ric = 0, then(M, g) is hyper-Kählerian.

We call a supersubalgebra g ⊂ osp(n|2k) weakly-irreducible if it does not preserve anynon-degenerate vector supersubspace of R

n ⊕ �(R2k).Let M and N be supermanifolds. Recall the definition of the product M × N =

(M × N, O M×N ). Let U ⊂ M and V ⊂ N be open subsets, and (U,x1, . . . , xn, ξ 1, . . . , ξm)

and (V , y1, . . . , yp, η1, . . . , ηq) coordinate systems on M and N , respectively. Then by de-finition, O M×N (U × V ) = OM×N(U × V ) ⊗ �ξ1,...,ξm,η1,...,ηq and this condition defines thesheaf O M×N uniquely [15, 16]. Let (M, g) and (N , h) be Riemannian supermanifolds andlet ∇g and ∇h be the corresponding Levi-Civita connections. Then g + h is a Riemanniansupermetric on M × N and, obviously, hol(M × N , g +h)(x,y) = hol(M, g)x ⊕hol(N , h)y

and Hol(∇g+h)(x,y) = Hol(∇g)x × Hol(∇h)y , where x ∈ M and y ∈ N . The following theo-rem generalizes the Wu theorem [24].

Theorem 11.1 Let (M, g) be a Riemannian supermanifold such that the pseudo-Riemann-ian manifold (M, g) is simply connected and geodesically complete. Then there exist Rie-mannian supermanifolds (M0, g0), (M1, g1), . . . , (Mr , gr ) such that

(M, g) = (M0 × M1 × · · · × Mr , g0 + g1 + · · · + gr), (39)

the supermanifold (M0, g0) is flat and the holonomy algebras of the supermanifolds(M1, g1), . . . , (Mr , gr ) are weakly-irreducible. In particular,

hol(M, g) = hol(M1, g1) ⊕ · · · ⊕ hol(Mr , gr ). (40)


Proof The proof of the local version of this theorem is similar to the proof of the lo-cal version of the Wu theorem. Let x ∈ M . Suppose that hol(M, g)x is not weakly-irreducible, then hol(M, g)x preserves a non-degenerate vector supersubspace F1 ⊂ Tx M.Let F2 ⊂ Tx M be its orthogonal complement. Then hol(M, g)x preserves the decom-position F1 ⊕ F2 = Tx M. By Theorem 7.1, there exist parallel distributions F1 andF2 over on M defined over an open neighbourhood U of the point x (at the mo-ment we do not assume that M is simply connected and that Hol(∇)x preserves F1

and F2). As we have noted, the distributions F1 and F2 are involutive. Hence, there

74 A.S. Galaev

exist maximal integral submanifolds M1 and M2 of M passing through the point x

and corresponding to the distributions F1 and F2, respectively [14]. Moreover, there ex-ist local coordinates x1, . . . , xn, ξ 1, . . . , ξm (resp., y1, . . . , yn, η1, . . . , ηm) on M such thatx1, . . . , xn1 , ξ 1, . . . , ξm1 (resp., y1, . . . , yn−n1 , η1, . . . , ηm−m1 ) are coordinates on M1 (resp.,on M2). Consequently, x1, . . . , xn1 , y1, . . . , yn−n1 , ξ 1, . . . , ξm1 , η1, . . . , ηm−m1 are coordi-nates on M and we see that M is locally isomorphic to a domain in the product M1 × M2.Since F1 and F2 are non-degenerate, the restrictions g1 and g2 of g to F1 and F2, respec-tively, are non-degenerate. It is easy to check that g1 and g2 do not depend on the coordinatesy1, . . . , yn−n1 , η1, . . . , ηm−m1 and x1, . . . , xn1 , ξ 1, . . . , ξm1 , respectively. Thus, (M1, g1) and(M2, g2) are Riemannian supermanifolds and g = g1 +g2. The local version of the theoremis proved.

Suppose that the pseudo-Riemannian manifold (M, g) is simply connected and geodesi-cally complete. Let F1, F2, F1, F2 M1 and M2 be as above. Obviously, the vector subspaces(F0)0, (F1)0 ⊂ TxM are non-degenerate and preserved by the holonomy group Hol(M, g)x

of the pseudo-Riemannian manifold (M, g). By the Wu theorem, M is diffeomorphic to theproduct M1 × M2, where M1 and M2 are integral submanifolds passing through the point x

and corresponding to the parallel distributions defined by the vector subspaces (F0)0 ⊂ TxM

and (F1)0 ⊂ TxM , respectively. It is obvious that the underlying manifolds of the supermani-folds M1 and M2 are M1 and M2, respectively. From the local part of the theorem it followsthat M = M1 × M2 and g = g1 + g2. The theorem is proved. �

12 Examples of Berger superalgebras

In this section we give examples of complex Berger superalgebras. We use results and de-notations of [17–20].

Let g−1 denote a complex vector superspace and let g0 ⊂ gl(g−1) be a supersubalgebra.The k-th prolongation (k ≥ 1) gk of g0 is defined as for representations of usual Lie algebrasup to additional signs. Consider the Cartan prolong g∗ = g∗(g−1,g0) = ⊕k≥−1gk . Note thatg∗ has a structure of Lie superalgebra. By analogy with [21] we get the following exactsequence

0 −→ g2 −→ g∗−1 ⊗ g1 −→ R(g0) −→ H 2,2

g0−→ 0, (41)

where H 2,2g0

is the (2,2)-th Spencer cohomology group (note that this group is denoted in[21] by H 1,2

g0). The second map in the sequence is given by

Rφ⊗α(x, y) = φ(x)α(y) − (−1)|x||y|φ(y)α(x). (42)

In [17, 19, 20] examples of irreducible subalgebras g0 ⊂ gl(g−1) with g1 �= 0 are givenand for the most of them the groups H 2,2

g0are computed.

Let us set some notation. The complex Lie superalgebras gl(n|m), sl(n|m), osp(n|2m),ospsk(2n|m), pe(n), pesk(n), q(n) are defined in the same way as their real analogs. Let τ =( 1n 0

0 −1n

), z = 12n. For a Lie superalgebra g let cg = g ⊕ Cz, pg = g/Cτ (if g containes τ ),

sg = g/Cz (if g containes z and is not contained in q(n)), and sq(n) = {( A B

B A

) ∈ q(n)∣∣ trB =

0}. Let vect(n|m) and svect(n|m) denote, respectively, the Lie superalgebra of all vectorfields and the Lie superalgebra of the divergence-free vector fields on C

n|m. Let h(2n|m)

denote the Lie superalgebra of the Hamiltonian vector fields on C2n|m. The other notations

are explained in [17–20].


Table 3 Examples of irreducible subalgebras g0 ⊂ gl(g−1) with g1 �= 0 and g2 = 0

g0 g0 : g−1 g0 : g1 g∗ Restriction

c(sl(n − p|q) ⊕ sl(p|m − q)) id∗ ⊗ id g∗−1 sl(n|m) n �= m,n − p + q ≥ 2,

m − q + p ≥ 2

sl(n − p|q) ⊕ sl(p|n − q) id∗ ⊗ id g∗−1 psl(n|n) n ≥ 3, n − p + q ≥ 2,

m − q + p ≥ 2

cosp(n|2k) id id osp(n + 2|2k)

gl(l|k) �2 id g∗−1 osp(2l|2k) (l, k) �= (3,0)

ps(q(p) ⊕ q(n − p)) id∗ ⊗ id g∗−1 psq(n) n ≥ 3, n − 1 ≥ p ≥ 1

sl(p|n − p) �(S2 id) �(�2 id∗) spe(n) n ≥ 3, n ≥ p ≥ 0

gl(1|2) V(1+α)ε1 V−αε1 D(α)

gl(1|2) V 1+αα ε1

V− 1α ε1

D(α)

gl(1|2) V α1+α

ε1V 1

1+αε1

D(α)

cosp(2|4) V−ε1+δ1+δ2 V3ε1 ab(3)

Proposition 12.1 The following Lie superalgebras are Berger superalgebras:

(1) c(sl(n − p|q) ⊕ sl(p|m − q)) and sl(n − p|q) ⊕ sl(p|m − q) if n �= m,n − p + q ≥2,m − q + p ≥ 2, sl(n − p|q) ⊕ sl(p|n − q) if n ≥ 3, n − p + q ≥ 2, n − q + p ≥2, cosp(n|2k), osp(n|2k), ps(q(p) ⊕ q(n − p)) and p(sq(p) ⊕ sq(n − p)) with theirstandard representations;

(2) gl(l|k) and sl(l|k) acting on �2(Rl ⊕ �(Rk));(3) sl(p|n − p) acting on both �(S2(Rp ⊕ �(Rn−p))) and �(�2(Rp ⊕ �(Rn−p)));(4) gl(1|2) and sl(1|2) acting on each of V(1+α)ε1 , V−αε1 , V 1+α

α ε1, V− 1

α ε1V α

1+αε1 and V 1

1+αε1

;

(5) cosp(2|4) and osp(2|4) acting on both V−ε1+δ1+δ2 and V3ε1 .

Proof For the proof we use Table 3. In all cases g∗ is simple. Hence, [g−1,g1] = g0. Thisand the exact sequence (41) yield that g0 is a Berger superalgebra: take x, y ∈ g−1, α ∈ g1

and φ ∈ g∗−1 such that α(x) �= 0, φ(y) = 1 and φ(x) = 0, then Rφ⊗α(x, y) = α(x); on the

other hand, these elements span g0. Suppose that g0 is not semisimple and g0 �= ps(q(p) ⊕q(n − p)), then g0 = g0 ⊕ C for an ideal g0 ⊂ g0. Let α ∈ (g1)0. Then there exists a non-zero x ∈ (g−1)0 such that (g−1)0 = Cx ⊕ ker pr

Cα|(g−1)0

. Take a non-zero φ ∈ (g−1)0 suchthat φ(x) = 0. Than 0 �= Rφ⊗α ∈ R(g0). Consequently, if g0 is simple then it is a Bergersuperalgebra. If g0 is not simple, then it is clear that all elements of R(g0) can not take imagein one of the simple summands of g0. The case g0 = ps(q(p) ⊕ q(n − p)) is similar. �

Proposition 12.2 Let n ≥ 3. The Lie subalgebras spe(n),pe(n), cspe(n), cpe(n), spe(n) �

〈τ + nz〉, spe(n) � 〈aτ + bz〉 (a, b ∈ C, ba

�= n) of gl(n|n) are Berger superalgebras.

Proof If g0 = spe(n),pe(n), cspe(n) or spe(n) � 〈aτ + bz〉, then g1 = 0. Moreover,R(spe(n)) � H

2,2spe(n) and there are the following non-split exact sequences

0 −→ Vε1+ε2 −→ H2,2spe(n) −→ �(V2ε1+2ε2) −→ 0,

76 A.S. Galaev

where n ≥ 4, and

0 −→ V −→ H2,2spe(3) −→ �(V3ε1) −→ 0,

where V is determined from the following non-split exact sequence

0 −→ Vε1+ε2 −→ V −→ �(V2ε1+2ε2) −→ 0.

Since spe(n) is simple, it is a Berger superalgebra. For each of the last three values of g0

there is an injective non-surjective map from H2,2spe(n) to H 2,2

g0. Since each of these three Lie

superalgebras has dimension dim spe(n) + 1, they are Berger superalgebras.Further, if n ≥ 4, then H

2,2spe(n)�〈τ+nz〉 � �(V2ε1+2ε2) and there is the following non-split

exact sequence

0 −→ �(V2ε1+2ε2) −→ H2,2spe(3)�〈τ+3z〉 −→ �(V3ε1) −→ 0.

From this, (41) and the exact sequence

0 −→ �(C) −→ S2g

∗−1 −→ Vε1+ε2 −→ 0

it follows that R(spe(n) � 〈τ + nz〉) �= R(spe(n)). Hence spe(n) � 〈τ + nz〉 is a Berger su-peralgebra. Finally, there is an injective non-surjective map from H

2,2spe(n)�〈τ+nz〉 to H

2,2cpe(n).

This, Table 4 and the sequences (41) written for the both Lie superalgebras show thatR(spe(n) � 〈τ + nz〉) �= R(cpe(n)). Since dim cpe(n) = dim spe(n) � 〈τ + nz〉 + 1, we getthat cpe(n) is a Berger superalgebra. �

Proposition 12.3 The Lie subalgebras gl(n|m), sl(n|m), ospsk(n|2m) and spesk(k) (k ≥ 3)with their standard representations are Berger superalgebras.

Proof The proof follows from (41) and Table 5. �

Table 4 Examples of irreducible subalgebras g0 ⊂ gl(g−1) with g1 �= 0 and g2 �= 0

g0 g0 : g−1 g0 : g1 g0 : g2 g∗ Restriction

cpe(n) id id∗ �(C) pe(n + 1) n ≥ 3

spe(n) � 〈τ + nz〉 id id∗ �(C) spe(n + 1) n ≥ 3

Table 5 Examples of irreducible subalgebras g0 ⊂ gl(g−1) whose Cartan prolongs are vectorial Lie super-algebras

g0 g0 : g−1 g∗ H2,2g0 Restriction

gl(n|m) id vect(n|m) 0

sl(n|m) id svect(n|m) 0 (n,m) �= (0,2)

sl(0|2) id svect(0|2) �(C)

ospsk(2n|m) id h(2n|m) 0

spesk(n) id sle(n) �(C) n ≥ 3


Proposition 12.4 Let g0 be a simple complex Lie superalgebra, g−1 = �(g0) and g0 acton g−1 vie the adjoint representation. Then g1 = Cϕ1 and g2 = 0, where ϕ1 : �(g0) =�((g0)1) ⊕ �((g0)0) → g0 = (g0)0 ⊕ (g0)1 , ϕ1(x) = (−1)|x|�(x) for all homogeneousx ∈ �(g0). In particular, for any simple complex Lie superalgebra g, the subalgebrag ⊂ gl(�(g)) is a Berger superalgebra.

Proof We have

g1 = {ϕ ∈ �(g0)∗ ⊗ g0| [ϕ(x), y] = (−1)|x||y|[ϕ(y), x], x, y ∈ �(g0)}

= �{ϕ ∈ g∗0 ⊗ g0| [ϕ(x), y] = (−1)(|x|+1)(|y|+1)[ϕ(y), x], x, y ∈ g0}. (43)

Let us first find all ϕ ∈ g1 annihilated by g0. Suppose that 0 �= ϕ ∈ (g1)0 is annihi-lated by g0, i.e. [x,ϕ(y)] = ϕ([x, y]) for all x, y ∈ g0. Consequently, the kernel ofϕ is an ideal in g0 and it must be trivial, i.e. ϕ is injective. On the other hand,for x, y ∈ g0 we have ϕ([ϕ(x), y]) = [ϕ(x),ϕ(y)] = −(−1)(|x|+1)(|y|+1)[ϕ(y),ϕ(x)] =−(−1)(|x|+1)(|y|+1)ϕ([ϕ(y), x]), since ϕ ∈ (�(g0)

∗ ⊗ g0)0 = (g∗0 ⊗ g0)1. Hence,

ϕ([ϕ(x), y]) = 0. Since ϕ is injective, [ϕ(x), y] = 0. But this yields ϕ = 0 and we get acontradiction. Let ϕ1 be as in the statement of the proposition. Obviously, ϕ1 ∈ (g1)1 and ϕ1

is annihilated by g0. From the above and the Schur Lemma it follows that the subset of g1

annihilated by g0 coincides with Cϕ1.If g1 is not equal to Cϕ1, then there exists a non-trivial g0-irreducible submodule W ⊂ g1.

Consider the Z-graded Lie superalgebra h = g−1 ⊕ g0 ⊕ W ⊕ W 2 ⊕ W 3 ⊕ · · · . By the con-struction of h and a proposition from [11], h is simple. On the other hand, all simple Z-graded Lie superalgebras of depth 1 are listed in [17] and the case g−1 = �(g0) does notoccur there. Thus, g1 = Cϕ1. �

Remark Note that Proposition 12.4 shows in particular that any simple vectorial Lie super-algebra g acting on �(g) is a Berger superalgebra. These examples have no analogs in thecase of the usual Berger algebras [21].

References

1. Batchelor, M.: The structure of supermanifolds. Trans. Am. Math. Soc. 253, 329–338 (1979)2. Besse, A.L.: Einstein Manifolds. Springer, Berlin-Heidelberg-New York (1987)3. Bryant, R.: Recent advances in the theory of holonomy. Séminaire Bourbaki 51 éme année. 1998–99.

no 8614. Cortés, V.: A new construction of homogeneous quaternionic manifolds and related geometric structures.

Mem. Am. Math. Soc. 147(700), viii+63 (2000)5. Cortés, V.: Odd Riemannian symmetric spaces associated to four-forms. Math. Scand. 98(2), 201–216

(2006)6. Deligne, P., Morgan, J.W.: Notes on supersymmetry (following Joseph Bernstein). In: Quantum Fields

and Strings: A Course for Mathematicians, Vols. 1, 2, Princeton, NJ, 1996/1997, pp. 41–97. Am. Math.Soc., Providence (1999)

7. Galaev, A., Leistner, T.: Recent developments in pseudo-Riemannian holonomy theory. In: Handbookof Pseudo-Riemannian Geometry. IRMA Lectures in Mathematics and Theoretical Physics (2009, toappear)

8. Goertsches, O.: Riemannian supergeometry. Math. Z. 260(3), 557–593 (2008)9. Joyce, D.: Compact Manifolds with Special Holonomy. Oxford University Press, London (2000)

10. Joyce, D.: Riemannian Holonomy Groups and Calibrated Geometry. Oxford University Press, London(2007)

11. Kac, V.G.: Lie superalgebras. Adv. Math. 26, 8–96 (1977)

78 A.S. Galaev

12. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. 1. Interscience Wiley, New York(1963)

13. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. 2. Interscience Wiley, New York(1967)

14. Manin, Y.I.: Gauge Field Theory and Complex Geometry. Grundlehren, vol. 289. Springer, Berlin(1988). First appeared as Kalibrovochnye polya i kompleksnaya geometriya, Nauka, Moscow (1984)

15. Leites, D.A.: Introduction to the theory of supermanifolds. Usp. Mat. Nauk 35(1), 3–57 (1980). Trans-lated in Russ. Math. Surv. 35(1), 1–64 (1980)

16. Leites, D.A.: Theory of Supermanifolds. Petrozavodsk (1983) (in Russian)17. Leites, D.A., Poletaeva, E., Serganova, V.: On Einstein equations on manifolds and supermanifolds.

J. Nonlinear Math. Phys. 9(4), 394–425 (2002)18. Leites, D.A. (ed.): Supersymmetries. Algebra and Calculus. Springer (2009, to appear)19. Poletaeva, E.: Analogues of Riemann tensors for the odd metric on supermanifolds. Acta Appl. Math.

31(2), 137–169 (1993)20. Poletaeva, E.: The analogs of Riemann and Penrose tensors on supermanifolds. arXiv:math/051016521. Schwachhöfer, L.J.: Connections with irreducible holonomy representations. Adv. Math. 160(1), 1–80

(2001)22. Serganova, V.V.: Classification of simple real Lie superalgebras and symmetric superspaces. Funct. Anal.

Appl. 17(3), 200–207 (1983) (in Russian)23. Varadarajan, V.S.: Supersymmetry for Mathematicians: An Introduction. Courant Lecture Notes, vol. 11.

Am. Math. Soc., Providence (2004)24. Wu, H.: Holonomy groups of indefinite metrics. Pac. J. Math. 20, 351–382 (1967)

http://arxiv.org/abs/arXiv:math/0510165

Differential Geometry and its Applications 27 (2009) 743–754

Contents lists available at ScienceDirect

Differential Geometry and its Applications

www.elsevier.com/locate/difgeo

Irreducible complex skew-Berger algebras

Anton S. Galaev 1

Department of Mathematics and Statistics, Faculty of Science, Masaryk University in Brno, Kotlárská 2, 611 37 Brno, Czech Republic

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 November 2008Available online 9 October 2009Communicated by S. Merkulov

MSC:58A5053C29

Keywords:Holonomy algebra of a supermanifoldBerger superalgebraSkew-Berger algebraSkew-symmetric prolongation

Irreducible skew-Berger algebras g ⊂ gl(n,C), i.e. algebras spanned by the images of thelinear maps R :�2

Cn → g satisfying the Bianchi identity, are classified. These Lie algebras

can be interpreted as irreducible complex Berger superalgebras contained in gl(0|n,C).© 2009 Elsevier B.V. All rights reserved.

1. Introduction

The classification of irreducible holonomy algebras of linear torsion-free connections is well known [5,6,13,19]. The firststep to this classification was to find candidates to these algebras, these candidates are called Berger algebras. These algebrasg ⊂ gl(n,R) are spanned by the images of the linear maps R : Λ2

Rn → g satisfying the Bianchi identity.

Recently in [8] holonomy algebras of connections on supermanifolds were introduced. The natural problem is to classifyirreducible holonomy algebras of linear torsion-free connections on supermanifolds. In [8] were defined Berger superal-gebras g ⊂ gl(m|n,R), which are the generalization of the usual Berger algebras, since gl(m|0,R) = gl(m,R) and Bergersuperalgebras are the same as Berger algebras in this case. In the present paper we study the mirror case to the classicalone: we classify irreducible complex Berger superalgebras contained in gl(0|n,C). These Lie superalgebras are the sameas irreducible skew-Berger algebras g ⊂ gl(n,C), i.e. algebras spanned by the images of the linear maps R : �2

Cn → g

(the skew-curvature tensors) satisfying the Bianchi identity. The reduction to the real skew-Berger algebras is a standardprocedure, see Proposition 2.4 below.

The paper has the following structure. In Section 2 we give the necessary preliminaries. In Section 3 we formulate themain theorem, where we classify irreducible skew-Berger algebras g ⊂ gl(n,C). In Section 4 we classify irreducible sub-algebras g ⊂ gl(n,C) admitting skew-curvature tensors R such that R(Cn,C

n) = g and g annihilates R . This classificationimmediately follows from the classification of simple complex Lie superalgebras. Using this list, we obtain examples ofskew-Berger algebras. In Section 5 we classify irreducible subalgebras g ⊂ gl(n,C) with non-trivial first skew-symmetricprolongations g[1] = {ϕ ∈ Hom(Cn,g) | ϕ(x)y = −ϕ(y)x for all x, y ∈ C

n} and get examples of skew-Berger algebras. In Sec-

E-mail address: [email protected] Supported from the Basic Research Center no. LC505 (Eduard Cech Center for Algebra and Geometry) of Ministry of Education, Youth and Sport of Czech

Republic.

0926-2245/$ – see front matter © 2009 Elsevier B.V. All rights reserved.doi:10.1016/j.difgeo.2009.09.001

http://www.ScienceDirect.com/

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mailto:[email protected]

http://dx.doi.org/10.1016/j.difgeo.2009.09.001

744 A.S. Galaev / Differential Geometry and its Applications 27 (2009) 743–754

tion 6 we finish the proof of the main theorem. In Section 7 we explain how this classification can be used to study someclasses of Berger superalgebras g ⊂ osp(n|2m,C).

The methods of the paper are mostly taken from [19]. In fact, a big number of results of [19] can be applied to our casewithout change. In the same time some particular cases required new ideas.

2. Preliminaries

First we give several definitions and facts from [8].Let V = V 0 ⊕ V 1 be a real or complex vector superspace and g ⊂ gl(V ) a supersubalgebra. The space of algebraic curvature

tensors of type g is the vector superspace R(g) = R(g)0 ⊕ R(g)1 , where

R(g) ={

R ∈ Λ2 V ∗ ⊗ g

∣∣∣ R(X, Y )Z + (−1)|X |(|Y |+|Z |)R(Y , Z)X + (−1)|Z |(|X |+|Y |) R(Z , X)Y = 0for all homogeneous X, Y , Z ∈ V

}.

Here |·| ∈ Z2 denotes the parity. The identity that satisfy the elements R ∈ R(g) is called the Bianchi super identity. Obviously,R(g) is a g-module with respect to the action

A · R = R A, R A(X, Y ) = [A, R(X, Y )

] − (−1)|A||R|R(A X, Y ) − (−1)|A|(|R|+|X |)R(X, AY ), (1)

where A ∈ g, R ∈ R(g) and X, Y ∈ V are homogeneous.If M is a supermanifold and ∇ is a linear torsion-free connection on the tangent sheaf T M with the holonomy algebra

hol(∇)x at some point x, then for the covariant derivatives of the curvature tensor we have (∇rYr ,...,Y1

R)x ∈ R(hol(∇)x)

for all r � 0 and tangent vectors Y1, . . . , Yr ∈ Tx M. Moreover, |(∇rYr ,...,Y1

R)x| = |Y1| + · · · + |Yr |, whenever Y1, . . . , Yr arehomogeneous.


L(

R(g)) = span

{R(X, Y )

∣∣ R ∈ R(g), X, Y ∈ V} ⊂ g.

From (1) it follows that L(R(g)) is an ideal in g. We call a supersubalgebra g ⊂ gl(V ) a Berger superalgebra if L(R(g)) = g.If V is a vector space, which can be considered as a vector superspace with the trivial odd part, then g ⊂ gl(V ) is a

usual Lie algebra, which can be considered as a Lie superalgebra with the trivial odd part. Berger superalgebras in this caseare the same as the usual Berger algebras.

Proposition 2.1. (See [8].) Let M be a supermanifold of dimension n|m with a linear torsion-free connection ∇ . Then its holonomyalgebra hol(∇) ⊂ gl(n|m,R) is a Berger superalgebra.

Thus real Berger superalgebras are candidates to the holonomy algebras of linear torsion-free connections on superman-ifolds. The classification of irreducible complex and real Berger algebras is well known [5,6,13,19].


R∇(g) ={

S ∈ V ∗ ⊗ R(g)

∣∣∣ S X (Y , Z) + (−1)|X |(|Y |+|Z |) SY (Z , X) + (−1)|Z |(|X |+|Y |) S Z (X, Y ) = 0for all homogeneous X, Y , Z ∈ V

}.

If M is a supermanifold and ∇ is a linear torsion-free connection on T M , then (∇rYr ,...,Y2,·R)x ∈ R∇(hol(∇)x) for all r � 1

and Y2, . . . , Yr ∈ Tx M. Moreover, |(∇rYr ,...,Y2,·R)x| = |Y2| + · · · + |Yr |, whenever Y2, . . . , Yr are homogeneous.

A Berger superalgebra g is called symmetric if R∇(g) = 0. This is a generalization of the usual symmetric Berger algebras,see e.g. [19], and the following is a generalization of the well-known fact about smooth manifolds.

Proposition 2.2. (See [8].) Let M be a supermanifold with a torsion-free connection ∇ . If hol(∇) is a symmetric Berger superalgebra,then (M,∇) is locally symmetric (i.e. ∇R = 0). If (M,∇) is a locally symmetric superspace, then its curvature tensor at any point isannihilated by the holonomy algebra at this point and its image coincides with the holonomy algebra.


Proposition 2.3. Let g ⊂ gl(V ) be an irreducible Berger superalgebra. If g annihilates the module R(g), then g is a symmetric Bergersuperalgebra.

In this paper we consider the case when the vector space V is complex and purely odd, i.e. its even part is trivial. Inthis case a supersubalgebra g ⊂ gl(V ) is just usual Lie algebra. We may consider the representation g ⊂ gl(Π(V )), where Π

it the parity changing functor and Π(V ) becomes a usual vector space.For a vector space V and a subalgebra g ⊂ gl(V ) define the space of skew-curvature tensors (or just curvature tensors

for short) of type g:

A.S. Galaev / Differential Geometry and its Applications 27 (2009) 743–754 745

R(g) ={

R ∈ �2 V ∗ ⊗ g

∣∣∣ R(X, Y )Z + R(Y , Z)X + R(Z , X)Y = 0for all X, Y , Z ∈ V

}.

Obviously, R(g) = R(g acting on Π(V )). A subalgebra g ⊂ gl(V ) is called skew-Berger if g = L(R(g)), where

L(

R(g)) = span

{R(X, Y )

∣∣ R ∈ R(g), X, Y ∈ V} ⊂ g.

We see that g ⊂ gl(V ) is a skew-Berger algebra if and only if g ⊂ gl(Π(V )) is a Berger superalgebra. Let R∇(g) =Π(R∇(g acting on Π(V ))). A skew-Berger algebra g ⊂ gl(V ) is called symmetric if R∇(g) = 0, i.e. g ⊂ gl(Π(V )) is sym-metric.

We will use the following fact

R(g) = ker(∂ : �2 V ∗ ⊗ g → �3 V ∗ ⊗ V

), (2)

where ∂ is the symmetrisation map. Note that the map ∂ is g-equivariant.Let us explain now how to obtain a classification of irreducible real skew-Berger algebras using the results of this paper.

Let V be a real vector space and g ⊂ gl(V ) an irreducible subalgebra. Consider the complexifications VC = V ⊗R C andgC = g ⊗R C ⊂ gl(VC). It is easy to see that

R(gC) = R(g) ⊗R C and R∇(gC) = R∇(g) ⊗R C.

Recall that the subalgebra g ⊂ gl(V ) is called absolutely irreducible if gC ⊂ gl(VC) is irreducible and it is called not absolutelyirreducible otherwise. The last situation appears if and only if there exists a complex structure J on V commuting withthe elements of g. Then V can be considered as a complex vector space and g ⊂ gl(V ) can be considered as a complexirreducible subalgebra. Consider also the natural representation i : gC → gl(V ) in the complex vector space V . The followingproposition is the analog of Proposition 3.1 from [19].

Proposition 2.4. Let V be a real vector space and g ⊂ gl(V ) an irreducible subalgebra.

1. If the subalgebra g ⊂ gl(V ) is absolutely irreducible, then g ⊂ gl(V ) is a skew-Berger algebra if and only if gC ⊂ gl(VC) is askew-Berger algebra.

2. If the subalgebra g ⊂ gl(V ) is not absolutely irreducible and if (i(gC))[1] = 0, then g ⊂ gl(V ) is a skew-Berger algebra if and onlyif Jg = g and g ⊂ gl(V ) is a complex irreducible skew-Berger algebra.

From this and Proposition 3.1 from [19] it follows that if the subalgebra g ⊂ gl(V ) is absolutely irreducible and gC ⊂gl(VC) is both a skew-Berger and a Berger algebra, then the subalgebra g ⊂ gl(V ) is a skew-Berger algebra if and only if itis a Berger algebra. Similarly, if the subalgebra g ⊂ gl(V ) is not absolutely irreducible, (i(gC))[1] = (i(gC))(1) = 0 ((i(gC))(1)

denotes the first prolongation), and g ⊂ gl(V ) is both a complex irreducible skew-Berger and a Berger algebra, then g ⊂gl(V ) is a real skew-Berger algebra if and only if it is a real Berger algebra. Thus it is left to consider not absolutelyirreducible real subalgebras g ⊂ gl(V ) such that the corresponding representation i : gC → gl(V ) in the complex vectorspace V is one of the entries 1–10 of Table 1 below. This will be done in another paper.

3. The main theorem

Theorem 3.1. Let V be a complex vector space. The irreducible complex skew-Berger subalgebras g ⊂ gl(V ) are exhausted by therepresentations of Table 1. The representations 1–8 and 15 have non-trivial first skew-symmetric prolongations; the representations7 with z = 0, 11–14 and 19–22 are symplectic; the representations 8 and 15–18 are orthogonal. If g admits an element R ∈ R(g)

annihilated by g and such that its image coincides with g, then g is either 7 with z = 0, or 11, or 19, or 20, or 21, or 22, or 8 withg = sl(n,C). The representations 19–24 are symmetric skew-Berger algebras. Any irreducible non-symmetric skew-Berger subalgebrag ⊂ gl(V ) coincides with one of the subalgebras 1–18. The absolutely irreducible real forms of the last two representations cannotappear as the holonomy algebras of linear torsion-free connections on purely odd supermanifolds.

Remark 3.1. From the proof of the theorem it follows that if g ⊂ gl(V ) satisfies R(g) = 0, then g ⊂ gl(V ) appears in Table 1,or g = C ⊕ h, where h ⊂ gl(V ) appears in Table 1.

Remark 3.2. We do not find the space R∇(g) for the representations 1–18. If for some of these representation this spaceis trivial, then absolutely irreducible real forms of this representation cannot appear as the holonomy algebras of lineartorsion-free connections on purely odd supermanifolds.

Remark 3.3. The list of representations from Proposition 6.4 below mostly coincides with the list of representations g ⊂gl(V ) of simple Lie algebras g such that dim g > dim V [3]. We see that for the representations 1–18 of Table 1 it holdsdim g � dim V . This proves the following statement.


Table 1Irreducible skew-Berger subalgebras g ⊂ gl(V ) (z denotes 0 or C).

g V Restriction

1 z ⊕ sl(n,C) Cn n � 3

2 z ⊕ sl(n,C) ⊕ sl(m,C) Cn ⊗ C

m , n,m � 2, n = m3 sl(n,C) ⊕ sl(n,C) C

n ⊗ Cn , n � 3

4 sl(n,C) Λ2C

n n � 65 z ⊕ sl(5,C) Λ2

C5

6 sl(n,C) �2C

n n � 37 z ⊕ sp(2n,C) C

2n n � 28 g g g is a simple Lie algebra

9 z ⊕ spin(10,C) �+10 = C

16

10 f6 C27

11 sl(2,C) ⊕ so(n,C) C2 ⊗ C

n n � 312 spin(12,C) �+

12 = C32

13 sl(6,C) Λ3C

6 = C20

14 sp(6,C) Vπ3 = C14

15 so(n,C) Cn n � 3

16 g2 C7

17 spin(7,C) C8

18 sl(2,C) ⊕ sp(2n,C) C2 ⊗ C

2n n � 2

19 sl(2,C) C2

20 so(n,C) ⊕ sp(2m,C) Cn ⊗ C

2m n � 3, m � 221 g2 ⊕ sl(2,C) C

7 ⊗ C2

22 spin(7,C) ⊕ sl(2,C) C8 ⊗ C

2

23 so(n,C) ⊕ sl(m,C) Cn ⊗ C

m n,m � 324 sp(2n,C) ⊕ sl(m,C) C

2n ⊗ Cm n � 2, m � 3

Table 2Simple Lie superalgebras f with f0 acting irreducibly on f1 .

f f0 f1

osp(n|2m,C), n � 1, n = 2, m � 1 so(n,C) ⊕ sp(2m,C) Cn ⊗ C

2m

q(n), n � 3 sl(n) sl(n)

F (4) spin(7,C) ⊕ sl(2,C) C8 ⊗ C

2

G(3) g2 ⊕ sl(2,C) C7 ⊗ C

2

D(α) sl(2,C) ⊕ sl(2,C) ⊕ sl(2,C) C2 ⊗ C

2 ⊗ C2

Let g ⊂ gl(V ) be an irreducible skew-Berger algebra. If dimg < dim V , then g ⊂ gl(V ) is symmetric.In fact, nearly the same holds for Berger algebras:Let g ⊂ gl(V ) be an irreducible Berger algebra. If dim g � dim V , then g ⊂ gl(V ) is symmetric.This is the analog of the statement of the Berger holonomy theorem:Let G ⊂ SO(n,R) be the holonomy group of a Riemannian manifold (M, g). If G does not act transitively on the (n − 1)-

dimensional sphere, then (M, g) is locally symmetric.This formulation follows from the list of possible connected holonomy groups of Riemannian manifolds obtained by

M. Berger in [4]. In [18] J. Simens gave a direct proof of this statement. And recently C. Olmos obtained a more simple andgeometric proof of this fact [15].

4. Complex odd symmetric superspaces and the associated skew-Berger algebras

In this section we classify irreducible subalgebras g ⊂ gl(n,C) admitting elements R ∈ R(g) such that R(Cn,Cn) = g and

g annihilates R . This classification immediately follows from the classification of simple complex Lie superalgebras. Then weobtain examples of skew-Berger algebras.

Having such g ⊂ gl(n,C) and R ∈ R(g). Define the Lie superalgebra f = g ⊕ Π(Cn) with the superbrackets [ξ,η] = [ξ,η],[ξ,Π(x)] = Π(ξx) and [Π(x),Π(y)] = R(x, y), where ξ,η ∈ g and x, y ∈ C

n . We get an irreducible infinitesimal symmetricsuperspace (f,g,Π(Cn)) [7,20]. The Proposition 1.2.7 from [10] implies that f is a simple Lie superalgebra. In Table 2 we listsimple Lie superalgebras f with f0 acting irreducibly on f1 .

The proof of the following theorem is similar to the proof of Theorem 3.6 from [19].

Theorem 4.1. Let g ⊂ gl(n,C) be an irreducible subalgebra. Suppose that there exists an irreducible odd infinitesimal symmetricsuperspace((

g ⊕ sl(2,C)) ⊕ Π

(C

n ⊗ C2),g ⊕ sl(2,C),Π

(C

n ⊗ C2)).


Then g ⊂ so(n,C) with respect to some scalar product g on Cn, there exists a g-equivariant map ∧ : Λ2

Cn → g satisfying

(x ∧ y)z + (x ∧ z)y = −2g(y, z)x + g(x, y)z + g(x, z)y

for all x, y, z ∈ Cn. Moreover, for any A ∈ g, the map R A : �2

Cn → g defined by

R A(x, y) = 2g(x, y)A + Ax ∧ y + Ay ∧ x

belongs to R(g) and the map g → R(g), A �→ R A is injective. In particular, g is a skew-Berger algebra.

Corollary 4.1. The following subalgebras are skew-Berger algebras: so(n,C) ⊂ gl(n,C) (n � 3), sp(2m,C)⊕sl(2,C) ⊂ gl(C2m ⊗C2)

(m � 1), spin(7,C) ⊂ gl(8,C) and g2 ⊂ gl(7,C).

Proof. We get the first algebra using osp(n|2,C) and the fact that sp(2,C) sl(2,C). We get the second algebra usingosp(4|2m,C) and the fact that so(4,C) sl(2,C) ⊕ sl(2,C). �Remark 4.1. Note that spin(7,C) and g2 are very important Berger algebras, which turn out to be also skew-Berger algebras.

5. Representations with non-trivial first skew-symmetric prolongations

Let g ⊂ gl(n,C) be a subalgebra. Put g[0] = g. Define the k-th (k � 1) skew-symmetric prolongation of g by the rule

g[k] = {ϕ ∈ Hom

(C

n,g[k−1]) ∣∣ ϕ(x)y = −ϕ(y)x for all x, y ∈ Cn}.

Let g−1 denote a complex vector superspace and let g0 ⊂ gl(g−1) be a supersubalgebra. The k-th prolongation (k � 1) gkof g0 is defined as for the representations of the usual Lie algebras up to additional signs:

gk = {ϕ ∈ Hom(g−1,gk−1)

∣∣ ϕ(x)y = (−1)|x||y|ϕ(y)x for all homogeneous x, y ∈ g−1}.

Consider the Cartan prolong g∗ = g∗(g−1,g0) = ⊕k�−1 gk . Note that g∗ has a structure of Lie superalgebra. In [12,16,17]

examples of irreducible subalgebras g0 ⊂ gl(g−1) with g1 = 0 are given and for the most of them the (2,2)-th Spencercohomology groups H2,2

g0 are computed.It is obvious that for a subalgebra g ⊂ gl(0|n,C) its prolongations coincide with the corresponding skew-symmetric

prolongations of the subalgebra g ⊂ gl(n,C).Let g ⊂ gl(n,C) be a subalgebra. By analogy with [19] we get the following exact sequence

0 −→ g[2] −→ (C

n)∗ ⊗ g[1] −→ R(g) −→ H2,2g −→ 0, (3)

where H2,2g is the (2,2)-th Spencer cohomology group for the representation g ⊂ gl(Π(Cn)). The second map in the se-

quence is given by

Rφ⊗α(x, y) = φ(x)α(y) + φ(y)α(x). (4)

Theorem 5.1. Let V be a complex vector space. All irreducible subalgebras g ⊂ gl(V ) with non-trivial first prolongations and g[1] , g[2] ,H2,2

g for these subalgebras are listed in Table 3.

The spaces (Cn ⊗ Λ2(Cn)∗)0 and (Cn ⊗ Λ3(Cn)∗)0 consist of tensors such that the contraction of the upper index withany down index gives zero. The spaces H2,2

sp(2n,C) and H2,2sp(2n,C)⊕C

are given in [17].

Proof of Theorem 5.1. Suppose that for an irreducible subalgebra g ⊂ gl(V ) we have g[1] = 0. Put g−1 = Π(V ) and g0 =g ⊂ gl(g−1). We get that g1 = 0. It is obvious that the Cartan prolong g∗ is an irreducible transitive Lie superalgebra with theconsistent Z-grading and g1 = 0. This means that g0 acts irreducibly on g−1, the equality [a,g−1] = 0, where a ∈ gk , k � 1,implies a = 0, and (g∗)0 = ⊕∞

k=0 g2k , (g∗)1 = ⊕∞k=0 g2k−1. From [10, Theorem 4] it follows that g∗ must coincide with one

of the following Lie superalgebras:

I sl(n|m,C) (n = m, m,n � 2), psl(n|n,C) (n � 2), osp(2|2n,C) (n � 1), spe(n,C) (n � 3) with the canonical Z-gradings;II vect(0|n,C) (n � 2), svect(0|n,C) (n � 3), h(0|n,C) (n � 4), h(0|n,C) (n � 4) with the canonical Z-gradings;

III g = g−1 ⊕ g0 ⊕ g1, where g0 = g is a simple Lie algebra, g−1 = Π(g), and g1 = C; the non-zero Lie superbrackets arethe following: [x, y] = [x, y], [x,Π(y)] = Π([x, y]), [ξ,Π(x)] = ξx, where x, y ∈ g and ξ ∈ C;

IV f = ∑∞k=−1 fk with f0 = f0 ⊕ C, fk = fk for k = 0 and elements of C acting by the multiplication on fk for k = 0, where f

is of type I, II or III and the center of f0 is trivial.


Table 3Irreducible subalgebras g ⊂ gl(n,C) with g[1] = 0.

g V g[1] g[2] H2,2g

1 sl(n,C) Cn , n � 3 (Cn ⊗ Λ2(Cn)∗)0 (Cn ⊗ Λ3(Cn)∗)0 0

2 gl(n,C) Cn , n � 2 C

n ⊗ Λ2(Cn)∗ Cn ⊗ Λ3(Cn)∗ 0

3 sl(n,C) �2C

n , n � 3 Λ2(Cn)∗ 0 04 gl(n,C) �2

Cn , n � 3 Λ2(Cn)∗ 0 0

5 sl(n,C) Λ2C

n , n � 5 �2(Cn)∗ 0 06 gl(n,C) Λ2

Cn , n � 5 �2(Cn)∗ 0 0 if n � 6

C5 if n = 5

7 sl(n,C) ⊕ sl(m,C) ⊕ C Cn ⊗ C

m , n,m � 2 V ∗ 0 0n = m

8 sl(n,C) ⊕ sl(n,C) Cn ⊗ C

n , n � 3 V ∗ 0 09 sl(n,C) ⊕ sl(n,C) ⊕ C C

n ⊗ Cn , n � 3 V ∗ 0 0

10 so(n,C) Cn , n � 4 Λ3 V ∗ Λ4 V ∗ 0

11 so(n,C) ⊕ C Cn , n � 4 Λ3 V ∗ Λ4 V ∗ 0

12 sp(2n,C) ⊕ C C2n , n � 2 V ∗ 0

13 g is simple g C id 0 H =?14 g ⊕ C, g is simple g C id 0 H

The standard Z-gradings of the Lie superalgebras of type I and II are described in [10]. The Lie superalgebras of type I giveus the entries 3, 5, 7, 8, 12 of the table; the Lie superalgebras of type II give us 1, 2, 10; the Lie superalgebras of type IIIgive us 13, and the Lie superalgebras of type IV give us all the other entries. We write n � 5 for the entries 5 and 6, asΛ2

C3 C

2 and we should consider entries 1 and 2 for n = 3; similarly, sl(4,C) so(6,C) and Λ2C

4 C6. By an analog

reason we assume n � 3 for the entries 8 and 9, and n � 2 for the entry 12. The Spencer cohomology groups H2,2 for theentries 1, 2, 3, 5, 7, 8, 10, 12 are computed in [12,16,17]. The other cohomology groups (except for the entry 13) will becomputed in the following proposition.

Proposition 5.1. All the representations of Table 3 except for the entries 4, the entries 5 for n � 6, the entries 9, 11, 12 and 14are skew-Berger algebras; the representation of sl(n,C) ⊕ sl(m,C) on C

n ⊗ Cm (n,m � 2) and sp(2n,C) ⊂ sl(2n,C) (n � 2) are

skew-Berger algebras.

Proof. The proof of the fact that the entries 1, 2, 3, 5, 7, 8, 10 of the table and also the representation of sl(n,C) ⊕ sl(m,C)

on Cn ⊗ C

m (n,m � 2) and sp(2n,C) ⊂ sl(2n,C) (n � 2) are skew-Berger algebras is similar to the proof of Proposition 12.1from [8], it follows mostly from the exact sequence (3).

Lemma 5.1. Consider the representation of the Lie algebra sl(n,C) ⊕ sl(n,C) ⊕ C on V = Cn ⊗ C

n (n � 3). Then R(sl(n,C) ⊕sl(n,C) ⊕ C) = R(sl(n,C) ⊕ sl(n,C)) and the representation of the Lie algebra sl(n,C) ⊕ sl(n,C) ⊕ C on C

n ⊗ Cn (n � 3) is not a

skew-Berger algebra.

Proof. Let π1, . . . ,πn−1, π1, . . . , πn−1 denote the fundamental weights of the Lie algebra sl(n,C) ⊕ sl(n,C). Recall that thespace R(sl(n,C)⊕sl(n,C)⊕C) can be defined by (2). It can be checked that the sl(n,C)⊕sl(n,C)-module �2 V ∗ �2 V ∗ ⊗C ⊂ �2 V ∗ ⊗ (sl(n,C) ⊕ sl(n,C) ⊕ C) can be decomposed as the direct sum Vπn−2+πn−2 ⊕ V 2πn−1+2πn−1 and each of thedecompositions into irreducible components of the sl(n,C) ⊕ sl(n,C)-modules �2 V ∗ ⊗ (sl(n,C) ⊕ sl(n,C)) and �3 V ∗ ⊗ Vcontain two copies of the modules Vπn−2+πn−2 and V 2πn−1+2πn−1 . Suppose that R(sl(n,C) ⊕ sl(n,C) ⊕ C) = R(sl(n,C) ⊕sl(n,C)). Using the exact sequence (3), we get R(sl(n,C)⊕ sl(n,C)) V ∗ ⊗ V ∗ and this sl(n,C)⊕ sl(n,C)-module containsone of each irreducible components Vπn−2+πn−2 and V 2πn−1+2πn−1 . From (2) it is clear that R(sl(n,C)⊕sl(n,C)⊕C) containsat least one additional component (Vπn−2+πn−2 or V 2πn−1+2πn−1 ). Suppose that 2Vπn−2+πn−2 ⊂ R(sl(n,C) ⊕ sl(n,C) ⊕ C). Letg1 and g2 denote the first and the second summands in sl(n,C) ⊕ sl(n,C). From the symmetry it follows that each of thesl(n,C)⊕sl(n,C)-modules �2 V ∗⊗g1 and �2 V ∗⊗g2 contains one irreducible component Vπn−2+πn−2 . Denote these modulesby U1 and U2. It is clear that any non-zero R ∈ R(sl(n,C) ⊕ sl(n,C)) cannot take values in one of the Lie algebras g1and g2. Hence ∂|U1 and ∂|U2 are injective. Moreover, ∂|U1⊕U2 has the kernel isomorphic to Vπn−2+πn−2 . Denote by W1 ⊕ W2,where W1 and W2 are isomorphic to Vπn−2+πn−2 , the submodule 2Vπn−2+πn−2 ⊂ �3 V ∗ ⊗ V . We may assume that ∂|U1 and∂|U2 take U1 and U2 isomorphically onto W1. Denote by U the submodule Vπn−2+πn−2 ⊂ �2 V ∗ ⊗ C ⊂ �2 V ∗ ⊗ (sl(n,C) ⊕sl(n,C) ⊕ C). It is obviously that ∂|U is injective. Let now S1 + S2 + S3 ∈ R(sl(n,C) ⊕ sl(n,C) ⊕ C) and S1 + S2 + S3 /∈R(sl(n,C) ⊕ sl(n,C)), where S1 + S2 + S3 ∈ U1 ⊕ U2 ⊕ U . We have ∂(S1 + S2 + S3) = 0. Since ∂(S1 + S2) ∈ W1 and∂|U1 : U1 → W1 is an isomorphism, there exists an S ′

1 ∈ U1 such that ∂(S1 + S2) = ∂(S ′1). This implies ∂(S ′

1 + S) = 0, i.e.S ′

1 + S is a non-zero curvature tensor of type sl(n,C) ⊕ sl(n,C) ⊕ C taking values in g1 ⊕ C, which is impossible and weget a contradiction. The module V 2πn−1+2πn−1 can be considered in the same way. �


Lemma 5.2. Consider the representation of the Lie algebra gl(n,C) on V = Λ2C

n. If n � 6, then R(gl(n,C)) = R(sl(n,C)) andgl(n,C) acting on Λ2

Cn is not a Berger algebra.

Proof. We have �2 V ∗ = V 2πn−2 ⊕ Vπn−4 . Suppose that R ∈ R(gl(n,C)) and R /∈ R(sl(n,C)). Suppose that R has weightπn−4. Then R = S +φ, where S ∈ �2 V ∗ ⊗ sl(n,C) and φ ∈ �2 V ∗ ⊗ C have weight πn−4. Let e1, . . . , en be the standard basisof C

n and e∗1, . . . , e∗

n its dual basis. Assume that φ = (e∗n−3 ∧ e∗

n−2) � (e∗n−1 ∧ e∗

n). Consider the Bianchi identity

R(en−3 ∧ en−2, en−1 ∧ en)ei ∧ e j + R(en−1 ∧ en, ei ∧ e j)en−3 ∧ en−2 + R(ei ∧ e j, en−3 ∧ en−2)en−1 ∧ en = 0.

Note that A = S(en−3 ∧ en−2, en−1 ∧ en) has weight 0, i.e. it is an element of the Cartan subalgebra t ⊂ sl(n,C). If 1 � i =j � n − 4, then R(en−1 ∧ en, ei ∧ e j) ∈ sl(n,C) must have weight −εn−3 − εn−2 + εi + ε j but the Lie algebra sl(n,C) has nosuch root, i.e. R(en−1 ∧ en, ei ∧ e j) = 0. Similarly, R(en−3 ∧ en−2, ei ∧ e j) = 0. We get that (εi + ε j)A = −1, i.e. Aii = − 1

2 for1 � i � n − 4 (Aij denote the elements of the matrix of A). By analogy, taking i = n − 3, j = n − 2 and i = n − 1, j = n, weget An−3n−3 + An−2n−2 = −1 and An−1n−1 + Ann = −1. Thus tr A = 0 and we get a contradiction.

The case when R has weight 2πn−2 is similar (we take φ = (e∗n−1 ∧ e∗

n) � (e∗n−1 ∧ e∗

n)). �Consider the representation of the Lie algebra gl(5,C) on V = Λ2

C5. Using the package Mathematica, we find that

dim R(gl(n,C)) = dim R(sl(n,C)) + 5. Hence gl(5,C) acting on Λ2C

5 is a Berger algebra. Moreover, �2 V ∗ = Vπ1 ⊕ V 2π3 ,hence R(gl(n,C)) = R(sl(n,C)) ⊕ U , where U is isomorphic to Vπ1 = C

5.

Lemma 5.3. Consider the representation of the Lie algebra gl(n,C) on V = �2C

n, n � 3. Then R(gl(n,C)) = R(sl(n,C)) andgl(n,C) acting on Λ2

Cn is not a Berger algebra.

Proof. The proof of the statement of the lemma for n � 6 is similar to the proof of the previous lemma (note that we have�2 V ∗ = V 2πn−2 ⊕ V 4πn−1 ). To prove the statement for n = 3,4 and 5 we use the package Mathematica. �

The fact that the subalgebra so(n,C) ⊕ C ⊂ gl(n,C) for n � 2 is not a skew-Berger algebra will be proved in Proposi-tion 6.2 below. Let g be a simple Lie algebra and g ⊂ gl(g) its adjoint representation. From (3) it follows that R(g) containsa component isomorphic to g and hence g ⊂ gl(g) is a skew-Berger algebra. We do not compute the Spencer cohomol-ogy for the adjoint representations. Due to the Cartan–Killing form we have g ⊂ so(g) and hence R(g ⊕ C) = R(g) (seeProposition 6.2 below) and g ⊕ C ⊂ gl(g) is not a skew-Berger algebra. The proposition is proved. �

Consider the representation of the Lie algebra sl(n,C) ⊕ sl(m,C) ⊕ C on the space V = Cn ⊗ C

m , n,m � 2, n = m. Weget that R(sl(n,C) ⊕ sl(m,C) ⊕ C) V ∗ ⊗ V ∗ . To describe this isomorphism we use the structure of the Lie superbracketson sl(n|m,C). For τ ∈ V ∗ ⊗ V ∗ , the corresponding curvature tensor is defined by Rτ (x1 ⊗ x2, u1 ⊗ u2) = A(x1 ⊗ x2, u1 ⊗u2) + B(x1 ⊗ x2, u1 ⊗ u2), where A(x1 ⊗ x2, u1 ⊗ u2) ∈ sl(n,C) ⊕ C, B(x1 ⊗ x2, u1 ⊗ u2) ∈ sl(m,C) ⊕ C, and for v1 ∈ C

n andv2 ∈ C

m we have

A(x1 ⊗ x2, u1 ⊗ u2)v1 = −τ (x1, x2, v1, u2)u1 − τ (u1, u2, v1, x2)x1,

B(x1 ⊗ x2, u1 ⊗ u2)v2 = τ (x1, x2, u1, v2)u2 + τ (u1, u2, x1, v2)x2.

In particular, tr(Rτ (x1 ⊗ x2, u1 ⊗ u2)) = (n − m)(τ (x1, x2, u1, u2) + τ (u1, u2, x1, x2)). Thus, R(sl(n,C) ⊕ sl(m,C)) Λ2 V ∗ .Similarly, R(sl(n,C) ⊕ sl(n,C)) V ∗ ⊗ V ∗ . As we have seen, R(sl(n,C) ⊕ sl(n,C)) = R(sl(n,C) ⊕ sl(n,C) ⊕ C). We willuse this later.

Remark 5.1. In [14] skew-symmetric prolongations of the subalgebras of so(n,R) were considered. In particular it is provedthat the only proper irreducible subalgebras of so(n,R) with non-trivial skew-symmetric prolongations are exhausted bythe adjoint representations of compact simple Lie algebras.

6. Proof of Theorem 3.1

The following two propositions are analogs of Propositions 3.2 and Lemma 3.5 from [19], respectively.

Proposition 6.1. Let g ⊂ sp(2n,C) be a proper irreducible subalgebra. Then R(g ⊕ C) = R(g). In particular, g ⊕ C is not a skew-Berger algebra.

Proposition 6.2. Let g ⊂ so(n,C) be an irreducible subalgebra. Then R(g ⊕ C) = R(g). In particular, g ⊕ C is not a skew-Bergeralgebra.


In the following we use the results from [19]. In fact, most of these result do not depend on what do we consider: Bergeralgebras or skew-Berger algebras (more precisely, it does not matter if we consider a curvature tensor as a map from Λ2 Vor from �2 V ). Some cases require additional considerations.

Let V be a complex vector space and g ⊂ gl(V ) an irreducible subalgebra. Let gs and z denote, respectively, the semi-simple part and the center of g. Then g = gs ⊕ z and either z = C or z = 0. If t ⊂ g is a Cartan subalgebra, let t0 = t ⊕ z.Denote the set of roots of gs by � and the set of weights of the representation g ⊂ gl(V ) by Φ .

Let �0 = � ∪ {0}. For each α ∈ � fix a non-zero Aα in the weight space gα , and let

Φα = {weights of Aα V } = (α + Φ) ∩ Φ.

A triple (λ0, λ1,α), where λ0, λ1 ∈ Φ and α ∈ �, is called a spanning triple if

Φα ⊂ {λ0 + β,λ1 + β | β ∈ �0}.A spanning triple is called extremal if λ0 and λ1 are extremal weights.

Proposition 6.3. (See [19, Proposition 3.10].) Let g ⊂ gl(V ) be an irreducible skew-Berger algebra. Then for every root α ∈ � there isa spanning triple (λ0, λ1,α), a weight element R ∈ R(g) and vectors x0, x1 ∈ V of weights λ0, λ1 such that R(x0, x1) = Aα .

In fact, if R ∈ R(g) is a weight element and if there are weight vectors x0, x1 ∈ V of weights λ0, λ1 such that R(x0, x1) = Aα , then(λ0, λ1,α) is a spanning triple.

Theorem 6.1. Let g ⊂ gl(V ) be an irreducible skew-Berger algebra. Then there is an extremal spanning triple (λ0, λ1,α).

Proof. The proof is the same as the proof of Theorem 3.12 from [19]. As the last step we need to show that so(n,C)

acting on (�2C

n)0 = �2C

n/Cg , where g is the scalar product on Cn , is not a skew-Berger algebra. Consider the inclu-

sion so(n,C) ⊂ sl(n,C) and the representation of the Lie algebra sl(n,C) on the vector space �2C

n . Then, R(so(n,C)) ⊂R(sl(n,C)). From Section 5 we know that R(sl(n,C)) �2(Cn)∗ ⊗ Λ2(Cn)∗ . To describe this isomorphism we use thestructure of the Lie superbrackets of spe(n,C). For τ ∈ �2(Cn)∗ ⊗ Λ2(Cn)∗ , the corresponding curvature tensor is definedby

Rτ (x1 � x2, y1 � y2)z = −2(τ (x1, x2, y1, z)y2 + τ (x1, x2, y2, z)y1 + τ (y1, y2, x1, z)x2 + τ (y1, y2, x2, z)x1

),

where x1, x2, y1, y2, z ∈ Cn . It is not hard to verify that from the condition Rτ (x1 � x2, y1 � y2) ∈ so(n,C) for all

x1, x2, y1, y2 ∈ Cn it follows that Rτ = 0. Thus, R(so(n,C)) = 0. �

Now we will consider the case when gs is simple.

Proposition 6.4. (See [19, Proposition 3.18].) Let g ⊂ gl(V ) be an irreducible subalgebra such that gs is simple. Suppose that thereexists an extremal spanning triple (λ0, λ1,α). Then either the dominant weight is a root, i.e. Φ ⊂ �0 , or the representation of gs on Vis conjugated to one of the following:

(i) •k •0 • · · ·0 •0 •0 with k = 1,2 (ii) •0 •1 • · · ·0 •0 •0

(iii) •1 •0 • · · ·0 •0 >•0 (iv) •1 •0 • · · ·0 •0 <•0

(v) •1 •0 • · · ·0 •0 •0��•0

��•0(vi) •k for k � 3

(vii) •1 •1 •0 (viii) •0 •0 • · · ·1 •0 •0 for n = 5,6

(ix) •0 •0 •0 •1 •0 •0 •0 (x) •1 >•1

(xi) •0 •0 <•1 (xii) •0 •0 •1 <•0

(xiii) •0 •0 •0 <•1

(xiv) •0 •0 • · · ·0 •0 >•1 for n � 7 (xv) •0 •0 • · · ·0 •0 •0��•1

��•0for 5 � n � 8

(xvi) •1 •0 •0 •0 •0

• 0

(xvii) •0 •0 •0 •0 •0

• 0

•1


Note that if Φ ⊂ �0, then gs ⊂ gl(V ) is one of the following:

(xviii) sp(2n,C) acting on (Λ2C

2n)0 = Λ2C

2n/CΩ , where Ω is the symplectic form on C2n;

(xix) f4 acting on V f4π1 = C

26;(xx) g2 ⊂ gl(7,C);

(xxi) the adjoint representation of a simple Lie algebra.

We have already discussed the representations (i)–(v), (xiv) for n = 3, (xx) and (xxi).Now we consider the remaining representations.(vi) (gl(2,C) ⊂ gl(�k

C2), k � 3). The computations using the package Mathematica show that R(gs) = 1,3,0 for k =

1,2,3, respectively, and R(gs) = 6,3,0 for k = 1,2,3. Thus the only skew-Berger algebras are gs and g for k = 1, and gs fork = 2 (in the last case we get the standard representation of so(3,C)).

The representations (vii), (ix), (x), (xii), (xiii), (xv), n = 8, can be dealt with exactly in the same way as in [19] (forsome of these representations it is proved dim R(g) � 1; in the same way we get dim R(g) � 1, but these representationdoes not appear in Table 3, hence dim R(g) = 0).

Recall that the space R(g) can be find from (2) and the map ∂ is g-equivariant. Decompose the g-modules �2 V ∗ ⊗ g

and �3 V ∗ ⊗ V into the direct sums of irreducible components. If a component VΛ appears in �2 V ∗ ⊗ g more often than in�3 V ∗ ⊗ V , then the space R(g) contains a g-submodule isomorphic to VΛ . In particular, R(g) = 0, and if g is simple, thenit is a skew-Berger algebra. We apply this idea to the following 4 cases.

(xv) For n = 5 (the spin representation of the Lie algebra g = so(10,C) on V = �+10). Using the package LiE, we check

that �2 V ∗ ⊗ g contains two copies of Vπ3 , while �3 V ∗ ⊗ V contains only one copy of Vπ3 . Note that dim Vπ3 = 120.Using the program Mathematica we find that dim R(spin(10,C)) = 120. Thus, R(spin(10,C)) Vπ3 . Since the Lie algebraso(10,C) is simple, we get that the representation spin(10,C) ⊂ gl(�+

10) is a skew-Berger algebra. Furthermore, we findthat dim R(spin(10,C) ⊕ C) = 176. Consequently, spin(10,C) ⊕ C ⊂ gl(�+

10) is a skew-Berger algebra.(xv) For n = 6 (the spin representation of the Lie algebra g = so(12,C) on V = �+

12). Using the package LiE, we checkthat �2 V ∗ ⊗ g contains a submodule isomorphic to V 2π1 , while �3 V ∗ ⊗ V does not contain a submodule isomorphic toV 2π1 . Thus spin(12,C) ⊂ gl(�+

12) is a skew-Berger algebra. Since spin(12,C) is contained in the symplectic Lie algebra,spin(12,C) ⊕ C ⊂ gl(�+

12) is not a skew-Berger algebra.(xvi) (e6 ⊂ gl(27,C)). Using the package LiE, we check that �2 V ∗ ⊗ g contains two copies of Vπ5 , while �3 V ∗ ⊗ V

contains only one copy of Vπ5 . Thus e6 acting on C27 is a skew-Berger algebra. We claim that R(e6 ⊕ C) = R(e6), i.e.

e6 ⊕ C acting on C27 is not a skew-Berger algebra. The idea of the proof is similar to the proof of Lemma 5.2. Note that

the representation of g on V has 27 weights. This means that each weight subspace of V is one-dimensional. For eachweight λ choose a non-zero weight vector eλ ∈ V . Then the vectors eλ form a basis of V . Let e∗

λ be the dual basis. Wehave �2 V ∗ = V 2π6 ⊕ Vπ1 . Suppose that S + φ ∈ R(e6 ⊕ C), S and φ have weight π1 and φ = 0. We may assume thatφ = e∗−π6

� e∗−π1+π6. Note that A = S(e−π6 , e−π1+π6 ) has weight 0, i.e. it is an element of the Cartan subalgebra of e6.

Writing down the Bianchi identity for the vectors e−π6 , e−π6 , e−π1+π6 , and using the fact that π1 − 2π6 is not a rootof e6, we get R(e−π6 , e−π1+π6 )e−π6 = 0. Consequently, π6(A) = 1. Similarly, writing the Bianchi identity for the vectorse−π6 , e−π1+π6 , e−π1+π6 and for the vectors e−π6 , eπ1 , e−π1+π6 , we get that (−π1 + π6)A = −1 and π1(A) = −1, whichleads to a contradiction. The elements S + φ ∈ R(e6 ⊕ C) of weight 2π6 can be considered in the same way. We chooseφ = e∗−π6

� e∗−π6and use the Bianchi identity for the vectors e−π6 , e−π6 , e−π6 , for the vectors e−π6 , e−π6 , e−π3+π4 , and for

the vectors e−π6 , e−π6 , eπ3−π4−π6 .(xi) (sp(6,C) ⊂ gl(Λ3

C6)). In this case �2 V ∗ ⊗ g contains a submodule isomorphic to Vπ3 , while �3 V ∗ ⊗ V does not

contain a submodule isomorphic to Vπ3 . Hence sp(6,C) ⊂ gl(14,C) is a skew-Berger algebra. Since this representation issymplectic, g ⊕ C is not a skew-Berger algebra.

(xv) For n = 7 (the spin representation of the Lie algebra gs = so(14,C) on V = �+14). The only spanning triples, up

to the action of the Weyl group, are (π7,π3 − π7,π2) and (π7,π1 − π7,π2) [19]. Consequently, elements of R(g) mayhave, up to the action of the Weyl group, weights π2 − π3 and π2 − π1. Suppose that R ∈ R(g) has weight π2 − π3. Letx, y, z ∈ V be vectors of weights π7, π3 − π7 and −π2 + π3 − π7. Considering the Bianchi identity for these vectors andusing the facts that π3 and π3 − 2π7 are not roots, we get R(y, z) = R(x, z) = 0 and R(x, y)z = 0. If we suppose thatR(x, y) = 0, then since R(x, y) is a root vector of weight π2, z has weight −π2 + π3 − π7, and π3 − π7 belongs to theweights of the representation, we get R(x, y)z = 0. Thus, R(x, y) = 0. Similarly, if x and y have weights π7 and π1 − π7,respectively, then in the same way we show that R(x, y) = 0 (choose z of weight −π2 + π5 − π7). Consider a spanningtriple (λ0, λ1,π2). Let x and y have weights λ0 and λ1, respectively. Then there is an element w in the Weyl group taking(λ0, λ1,π2) either to (π7,π3 − π7,π2) or to (π7,π1 − π7,π2), let us assume the first. Applying w to the system of positiveroots, we get another system of positive roots. In this new system (λ0, λ1,π2) has the same expression as (π7,π3 − π7,π2)

in the old one. Consequently, R(x, y) ∈ gπ2 implies R(x, y) = 0. And there are no weight elements R and x, y such that0 = R(x, y) ∈ gπ2 . Since gs is simple, we conclude that R(g) = 0. Thus spin(14,C) ⊂ gl(�+

14) and spin(14,C) ⊕ C ⊂ gl(�+14)

are not skew-Berger algebras.(xiv) (The spin representation of the Lie algebra gs = so(2n + 1,C) on V = �2n+1, n � 7). We have spin(2n + 1,C) ⊂

spin(2n + 2,C). From the above we get R(spin(2n + 1,C)) = 0 for n = 6,7. We use the package Mathematica to show


that R(spin(2n + 1,C)) = 0 for n = 4,5. The case n = 3 is already considered in Section 4; alternatively, using the packageMathematica we find that dim R(spin(7,C)) = 126. If n = 2, then we get the standard representation of sp(4,C); for n = 1,we get the standard representation of sl(2,C).

(viii) If n = 6, then using the package Mathematica we show that dim R(gs) = 35; in this case the representation issymplectic and R(g) = R(gs). Suppose that n = 7. The only spanning triples, up to the action of the Weyl group, are(ε1 + ε2 + ε3, ε4 + ε5 + ε6, ε1 − ε7) and (ε1 + ε2 + ε3, ε1 + ε4 + ε5, ε1 − ε7) [19]. We will denote a vector ei ∧ e j ∧ ek

by ei jk . Suppose that R ∈ R(g) is a weight vector and R(e123, e456) ∈ gε1−ε7 , i.e. R has weight ε1. There is some a ∈ C

such that R(e123, e456) = aE17, where E17 is the matrix with 1 on the position (1,7) and 0 on the other positions. LetA = R(e237, e456) ∈ g. Then A is an element of the Cartan subalgebra of g. Applying the Bianchi identity to the vectorse123, e456 and e237, and using that R(e123, e237) = 0 (as this element has weight which is not a root of g), we get A11 +A22 + A33 + a = 0. Applying the Bianchi identity to e456, e237 and one of the vectors e12i (4 � i � 6), e237, e147, we getA11 + A22 + Aii = 0, A22 + A33 + A77 = 0, A11 + A44 + A77 = 0. Using these conditions and the traceless of A, we get thata = 0. The spanning triple (ε1 + ε2 + ε3, ε1 + ε4 + ε5, ε1 − ε7) can be considered in the same way. As in the case (xv) forn = 7 we conclude that if R ∈ R(g), x and y are weight elements such that R(x, y) ∈ gε1−ε7 , then R(x, y) = 0, and R(g) = 0.

(xvii): e7 ⊂ gl(56,C) is not a skew-Berger algebra. In [1] there is the following description of this representation. TheLie algebra e7 admits the following structure of Z2-graded Lie algebra: e7 = sl(8,C) ⊕ Λ4

C8. The representation space

C56 is decomposed into the direct sum C

56 = Λ2C

8 ⊕ Λ2(C8)∗ . The elements of sl(8,C) preserve this decompositionand act naturally on each component, and the elements of Λ4

C8 interchange these components. Since e7 ⊂ sp(56,C),

any element R ∈ R(e7) is a symmetric map R : �2C

56 → e7 ⊂ �2C

56 and it is zero on the orthogonal complement toe7 ⊂ �2

C56. Thus R ∈ �2e7. As the e7-module �2e7 can be decomposed as �2e7 = V e7

2π1⊕ V e7

π6 ⊕ C. We have dim V e72π1

=7371 and dim V e7

π6 = 1539. Not that each such component consists of curvature tensors if and only if it contains at leastone non-zero curvature tensor (such idea was used by D.V. Alekseevsky in [2] to find the spaces of curvature tensorsfor irreducible holonomy algebras of Riemannian manifolds). As the sl(8,C)-module �2e7 = �2(sl(8,C) ⊕ Λ4

C8) can be

decomposed as �2e7 = �2sl(8,C) ⊕ �2Λ4C

8 ⊕ W , where W ⊂ �2e7 consists of symmetric maps interchanging sl(8,C)

and Λ4C

8. An important fact is that in the decomposition of �2e7 into the direct sum of irreducible sl(8,C)-modulesthere is only one summand of dimension greater than one that appears twice: �2(sl(8,C)) and �2Λ4 contain a submoduleisomorphic to V sl(8,C)

π2+π6. As a consequence, any R ∈ R(e7) is a sum of elements of R(e7) contained in (�2sl(8,C))/V sl(8,C)

π2+π6,

(�2Λ4C

8)/V sl(8,C)π2+π6

, W and in 2V sl(8,C)π2+π6

⊂ �2sl(8,C) ⊕ �2Λ4C

8. We have �2C

56 = �2(Λ2C

8) ⊕ (Λ2C

8 ⊗ Λ2(C8)∗) ⊕�2(Λ2

C8) = (Λ2

C8 ⊕ Λ4

C8) ⊕ (V sl(8,C)

π2+π6⊕ sl(8,C) ⊕ C) ⊕ (Λ2(C8)∗ ⊕ Λ4

C8), as the sl(8,C)-module. This decomposition

determines the inclusion e7 ⊂ �2C

56 and the behavior of the elements of W . Suppose that R ∈ R(e7) ∩ W . Let x, y ∈�2(Λ2

C8). By the definition of W , we have R(x, y) ∈ sl(8,C). Applying the Bianchi identity to x, y, z ∈ �2(Λ2

C8), we get

R|�2(Λ2C8)⊗Λ2C8 ∈ R(sl(8,C) acting on Λ2C

8). Moreover, since R is symmetric, this element is zero if and only if R = 0.

From the above we know that R(sl(8,C) acting on Λ2C

8) Λ2(C8)∗ ⊗ �2(C8)∗ V sl(8,C)π5+π7

⊕ V sl(8,C)π6+2π7

. Thus R(e7) ∩ W

is isomorphic to a submodule of the sl(8,C)-module V sl(8,C)π5+π7

⊕ V sl(8,C)π6+2π7

. Note that dim V sl(8,C)π5+π7

= 378 and dim V sl(8,C)π6+2π7

=630. Next, �2(sl(8,C)) V sl(8,C)

2π1+2π7⊕ V sl(8,C)

π2+π6⊕ sl(8,C) ⊕ C, �2Λ4

C8 V sl(8,C)

π4 ⊕ V sl(8,C)π2+π6

⊕ C, and dim V sl(8,C)2π1+2π7

= 1232,

V sl(8,C)π2+π6

= 720, dim V sl(8,C)2π4

= 1764. Analyzing the dimensions, we conclude that R(e7) = 0 (in other words, W \R(e7)

contains non-trivial elements from the submodules V e72π1

and V e7π6 ⊂ �2e7, and as a consequence these submodules do not

contain non-zero curvature tensors). Since e7 ⊂ sp(56,C), we get R(e7 ⊕ C) = 0.(xix): f4 ⊂ gl(26,C) is not a skew-Berger algebra. In [1] there is the following description of this representation. The

Lie algebra f4 admits the structure of Z2-graded Lie algebra: f4 = so(9,C) ⊕ �, where � is the representation space forthe spin representation of so(9,C). The representation space C

26 is decomposed into the direct sum C26 = C ⊕ C

9 ⊕ �.The elements of the subalgebra so(9,C) ⊂ f4 preserve these components, annihilate C and act naturally on C

9 and �.Elements of � ⊂ f4 take C and C

9 to � (multiplication by constants and the Clifford multiplication, respectively), andtake � to C ⊕ C

9 (the charge conjugation plus the natural map assigning a vector to a pair of spinors). Let R ∈ R(f4).Decompose it as the sum R = S + T , where S and T take values in so(9,C) and �, respectively. Let λ,μ ∈ C, x, y, z ∈so(9,C), and X, Y , Z ∈ �. Applying the Bianchi identity to X , Y and Z , we get that S|�2�⊗� ∈ R(spin(9,C)). On the otherhand, R(spin(9,C)) = 0, consequently, S(X, Y ) = 0. Applying the Bianchi identity to λ, x and X , we get T (λ, x) = 0 andS(X, λ) = 0. Applying the Bianchi identity to λ, X and Y , we get T (X, Y ) = 0. Applying the Bianchi identity to X , Y andx, we get S(X, x)Y + S(Y , x)X = 0. This means that for each fixed x the map S(·, x) : � → so(9,C) lies in the first skew-symmetric prolongation for the representation spin(9,C), which is trivial, as we already know. Consequently, S(x, X) = 0.Writing down the Bianchi identity for other vectors, we conclude that R = 0. Thus, R(f4) = 0. Since this representation isorthogonal, we have R(f4 ⊕ C) = 0.

Suppose now that the semi-simple part gs is not simple, then it can be written as a direct sum gs = g1 ⊕ g2, whereg1 ⊂ sl(n1,C), g2 ⊂ sl(n2,C), and V = C

n1 ⊗ Cn2 .

First suppose that n1,n2 � 3. By the same arguments as in [19], each gi must be either sl(ni,C), or so(ni,C), orsp(ni,C). Suppose that g1 = so(n1,C). Consider the subalgebra so(n1,C) ⊕ sl(n2,C) ⊕ C ⊂ sl(n1,C) ⊕ sl(n2,C) ⊕ C. LetRτ ∈ R(sl(n1,C) ⊕ sl(n2,C) ⊕ C) be as in Section 5, where τ ∈ V ∗ ⊗ V ∗ , V = C

n1 ⊗ Cn2 . Suppose that Rτ ∈ R(so(n1,C) ⊕

sl(n2,C) ⊕ C). Then we get A(x1 ⊗ x2, u1 ⊗ u2) − tr(A(x1 ⊗ x2, u1 ⊗ u2)) idCn1 ∈ so(n1,C). Hence for all v1, z1 ∈ Cn1 it holds


0 = τ (x1, x2, v1, u2)g(u1, z1) + τ (u1, u2, v1, x2)g(x1, z1) + τ (x1, x2, z1, u2)g(v1, u1)

+ τ (u1, u2, z1, x2)g(v1, x1) − τ (x1, x2, u1, u2)g(v1, z1) − τ (u1, u2, x1, x2)g(v1, z1).

Taking u1 = z1, x1 and v1 mutually orthogonal, we get that τ (x1, x2, v1, u2) = 0 whenever g(x1, v1) = 0. Taking v1 = x1 or-thogonal to u1 = z1 such that g(x1, x1) = g(u1, u1) = 1, we get τ (x1, x2, x1, u2) = −τ (z1, u2, z1, x2) whenever g(x1, z1) = 0.In particular, τ (z1, u2, z1, x2) does not depend on z1 under the condition g(x1, z1) = 0 and g(z1, z1) = 1. Considering anorthonormal basis of C

n1 , we conclude that τ (x1, x2, u1, u2) = g(x1, u1)w(x2, u2) for some skew-symmetric (not necessarynon-degenerate) bilinear form on C

n2 . Thus, R(so(n1,C)⊕ sl(n2,C)⊕ C) = R(so(n1,C)⊕ sl(n2,C)) Λ2(Cn2 )∗ . Using this,it is easy to get that R∇(so(n1,C) ⊕ sl(n2,C)) = 0. Thus so(n1,C) ⊕ sl(n2,C) is a symmetric Berger algebra, but it doesnot admit a curvature tensor that is annihilated by this algebra and such that its image coincides with this Lie algebra (thisrepresentation does not appear in Table 2). Consequently, any absolutely irreducible real representation h ⊂ sl(n1n2,R) withsuch complexification cannot appear as the holonomy algebra of a linear torsion-free connection on a purely odd super-manifold (if h ⊂ sl(n1n2,R) appears as the holonomy algebra of an odd supermanifold, then the curvature tensor of thismanifold is parallel; hence its value at the point must be annihilated by h and the image of this curvature tensor mustcoincide with h).

Similarly, if n1 is even, then R(sp(n1,C) ⊕ sl(n2,C) ⊕ C) = R(sp(n1,C) ⊕ sl(n2,C)) �2(Cn1 )∗ . We also get thatR(so(n,C) ⊕ sp(2m,C)) is one-dimensional and it is spanned by the curvature tensor Rτ with τ = g ⊗ w .

By the same arguments as in [19] it can be proved that if n1 = 2 and g is a non-symmetric Berger algebra, then g1 =sl(2,C) and g2 is one of gl(n2,C), sl(n2,C), so(n2,C), or sp(n2,C).

The theorem is proved. �7. An outlook to the general case

Consider the identity representation of the orthosymplectic Lie superalgebra osp(n|2m,C) ⊂ gl(n|2m,C) on the vectorsuperspace C

n ⊕ Π(C2m). Recall that osp(n|2m,C) is the supersubalgebra of gl(n|2m,C) preserving the form g + Ω , whereg is the standard non-degenerate symmetric bilinear form on C

n and Ω is the standard non-degenerate skew-symmetricbilinear form on C

2m . For the even part of osp(n|2m,C) we have osp(n|2m,C)0 = so(n,C) ⊕ sp(2m,C). Note that so(n,C)

annihilates Π(C2m) and acts on Cn in the natural way. Similarly, sp(2m,C) annihilates C

n and acts on Π(C2m) in thenatural way. It is easy to describe the space R(osp(n|2m,C)) using the method of [9]. In particular, the following curvaturetensors take values in osp(n|2m,C)0: R0|Cn∧Cn , R0|Π(C2m)∧Π(C2m) , and R1|Cn⊗Π(C2m) , where R0 ∈ R(osp(n|2m,C))0 andR1 ∈ R(osp(n|2m,C))1 .

For a subalgebra h ⊂ so(n,C) define the space of weak curvature tensors of type h,

P g(h) = {P ∈ (

Cn)∗ ⊗ h

∣∣ g(

P (x)y, z) + g

(P (y)z, x

) + g(

P (z)x, y) = 0 for all x, y, z ∈ C

n}.This space was introduced in [9,11] and it appears if one considers the space of curvature tensors for the holonomy algebrasof Lorentzian manifolds. Note that if R ∈ R(h), then for any fixed x ∈ C

n it holds R(·, x) ∈ P g(h). A subalgebra h ⊂ so(n,C)

is called a weak-Berger algebra if it is spanned by the images of the elements of P g(h).The following important theorem is proved by T. Leistner in [11].

Theorem 7.1. Let h ⊂ so(n,C) be an irreducible subalgebra. Then h is a weak-Berger algebra if and only if it is a Berger algebra.

Similarly, for a subalgebra h ⊂ sp(2m,C) define the space of weak skew-curvature tensors of type h,

PΩ(h) = {P ∈ (

C2m)∗ ⊗ h

∣∣ Ω(

P (x)y, z) + Ω

(P (y)z, x

) + Ω(

P (z)x, y) = 0 for all x, y, z ∈ C

2m}.

Note that if R ∈ R(h), then for any fixed x ∈ C2m it holds R(·, x) ∈ PΩ(h). We call a subalgebra h ⊂ sp(2m,C) a weak-skew-

Berger algebra if it is spanned by the images of the elements of PΩ(h).In view of Theorem 7.1, we get the following.

Hypothesis 7.1. Let h ⊂ sp(2m,C) be an irreducible subalgebra. Then h is a weak-skew-Berger algebra if and only if it is a skew-Bergeralgebra.

Let now R0 and R1 be as above, then prso(n,C) ◦R0|Cn∧Cn ∈ R(so(n,C)), prsp(2m,C) ◦R0|Π(C2m)∧Π(C2n) ∈ R(sp(2m,C)

acting on Π(C2m)), and for any fixed x0 ∈ Cn and x1 ∈ Π(C2m) it holds prso(n,C) ◦R1(·, x1)|Cn ∈ P g(so(n,C)) and

prsp(2m,C) ◦R1(·, x0)|Π(C2m) ∈ PΩ(sp(2m,C)). On the other hand, there is no such obvious restrictions on prsp(2m,C) ◦R0|Cn∧Cn

and prso(n,C) ◦R0|Π(C2m)∧Π(C2n) .Suppose now that we have a simple supersubalgebra g ⊂ osp(n|2m,C) such that the representations of g0 in the both

Cn and Π(C2m) are faithful. This is not the case for the identity representation of osp(n|2m,C), but this is the case for

the adjoint representations of the simple Lie superalgebras of classical type and it seems to be the case for the most of or-thosymplectic representations of simple Lie superalgebras. In this case prsp(2m,C) ◦R0|Cn∧Cn and prso(n,C) ◦R0|Π(C2m)∧Π(C2n)

are determined, respectively, by prso(n,C) ◦R0|Cn∧Cn and prsp(2m,C) ◦R0|Π(C2m)∧Π(C2n) , and therefore are strongly restricted.


Module Hypothesis 7.1 we get.

Theorem 7.2. Let g ⊂ osp(n|2m,C) be a simple Berger supersubalgebra such that the representations of g0 in both Cn and Π(C2m)

are faithful. Then g0 ⊂ so(n,C) is a Berger algebra and g0 ⊂ sp(2n,C) is a skew-Berger algebra.

Similarly, for a simple Berger supersubalgebra g ⊂ gl(n|m,C) such that the representations of g0 in both Cn and Π(Cm)

are faithful, we expect that there are two ideals g1,g2 ∈ g0 such that g1 + g2 = g0 and g1 ⊂ gl(n,C) is a Berger algebra, andg2 ⊂ gl(m,C) is a skew-Berger algebra.

In another paper we will discuss the ideas of this section in details.

Acknowledgements

I would like to thank L. Schwachhöfer for useful discussions. I am grateful to D.A. Leites for the communications con-cerning the skew-symmetric prolongations. I thank the Department of Mathematics and Statistics of the Masaryk Universityfor the excellent atmosphere for work.

References

[1] J.F. Adams, Lectures on Exceptional Lie Groups, The University of Chicago Press, 1996, edited by Zafer Mahmud and Mamoru Mimura.[2] D.V. Alekseevsky, Riemannian manifolds with exceptional holonomy groups, Funksional Anal. Prilozhen. 2 (2) (1968) 1–10.[3] E.M. Andreev, E.B. Vinberg, A.G. Elashvili, Orbits of highest dimension of semisimple linear Lie groups, Functional Anal. Appl. 1 (1967) 257–261.[4] M. Berger, Sur les groupers d’holonomie des variétés àconnexion affine et des variétés riemanniennes, Bull. Soc. Math. France 83 (1955) 279–330.[5] R. Bryant, Classical, exceptional, and exotic holonomies: A status report, in: Actes de la Table Ronde de Géométrie Différentielle en l’Honneur de Marcel

ee e Berger, Soc. Math. France, 1996, pp. 93–166.[6] R. Bryant, Recent advances in the theory of holonomy, Séminaire Bourbaki 51 ème année 861 (1998–1999).[7] V. Cortés, Odd Riemannian symmetric spaces associated to four-forms, Math. Scand. 98 (2) (2006) 201–216.[8] A.S. Galaev, Holonomy of supermanifolds, Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg 79 (1) (2009) 47–78.[9] A.S. Galaev, The spaces of curvature tensors for holonomy algebras of Lorentzian manifolds, Differential Geom. Appl. 22 (2005) 1–18.

[10] V.G. Kac, Lie superalgebras, Adv. Math. 26 (1977) 8–96.[11] T. Leistner, On the classification of Lorentzian holonomy groups, J. Differential Geom. 76 (3) (2007) 423–484.[12] D.A. Leites, E. Poletaeva, V. Serganova, On Einstein equations on manifolds and supermanifolds, J. Nonlinear Math. Phys. 9 (4) (2002) 394–425.[13] S. Merkulov, L. Schwachhöfer, Classification of irreducible holonomies of torsion-free affine connections, Ann. Math. 150 (1999) 77–149.[14] P.-A. Nagy, Prolongations of Lie algebras and applications, arXiv:0712.1398.[15] C. Olmos, A geometric proof of the Berger holonomy theorem, Ann. of Math. (2) 161 (1) (2005) 579–588.[16] E. Poletaeva, Analogues of Riemann tensors for the odd metric on supermanifolds, Acta Appl. Math. 31 (2) (1993) 137–169.[17] E. Poletaeva, The analogs of Riemann and Penrose tensors on supermanifolds, arXiv:math/0510165.[18] J. Simens, On the transitivity of holonomy systems, Annals of Math. 76 (2) (1962) 213–234.[19] L.J. Schwachhöfer, Connections with irreducible holonomy representations, Adv. Math. 160 (1) (2001) 1–80.[20] V.V. Serganova, Classification of simple real Lie superalgebras and symmetric superspaces, Funktsional Anal. Appl. 17 (3) (1983) 200–207 (in Russian).

ISSN 1995-0802, Lobachevskii Journal of Mathematics, 2011, Vol. 32, No. 2, pp. 163–173. c© Pleiades Publishing, Ltd., 2011.

Irreducible Holonomy Algebras of Odd Riemannian Supermanifolds

A. S. Galaev

(Submitted by V. Lychagin)Received November 15, 2010

Abstract—Possible irreducible holonomy algebras g ⊂ sp(2m, R) of odd Riemannian supermani-folds and irreducible subalgebras g ⊂ gl(n, R) with non-trivial first skew-symmetric prolongationsare classified. An approach to the classification of some classes of the holonomy algebras ofRiemannian supermanifolds is discussed.

DOI: 10.1134/S1995080211020041

Keywords and phrases: Riemannian supermanifold, Levi-Civita superconnection, holonomyalgebra, Berger superalgebra.

1. INTRODUCTION

Berger’s classification of holonomy algebras of Riemannian manifolds is an important result thathas found applications both in geometry and theoretical physics [1–4, 10, 12]. For pseudo-Riemannianmanifolds the corresponding problem is solved only in the irreducible case by Berger and in several partialcases, e.g. in the Lorentzian signature [7].

Since theoretical physicists discovered supersymmetry, supermanifolds began to play an importantrole both in geometry and physics [6, 15–17, 23]. In [8] the holonomy algebras of linear connections onsupermanifolds are defined. In particular, if (M, g) is a Riemannian supermanifold, then its holonomyalgebra g may be identified with a subalgebra of the orthosymplectic Lie superalgebra osp(p, q|2m),where p + q|2m is the dimension of M and (p, q) is the signature of the metric g restricted to theunderlying smooth manifold of M. It is natural to pose the problem of classification of the holonomyalgebras g ⊂ osp(p, q|2m) of Riemannian supermanifolds. Since for m = 0 this is the unsolved problemof the differential geometry, one should consider some restrictions on g. The first natural restriction is theirreducibility of g ⊂ osp(p, q|2m). Let us suppose also that

g = (⊕igi) ⊕ z, (1)

where gi are simple Lie superalgebras of classical type and z is a trivial or one-dimensional center (ifm = 0 or p = q = 0, then this assumption follows automatically from the irreducibility of g).

In the present paper we obtain a classification of possible irreducible holonomy algebras of oddRiemannian supermanifolds, in this case

g ⊂ osp(0|2m) � sp(2m, R)

is a usual Lie algebra. This result is the mirror analog of the Berger classification. Moreover, the aimof this paper is to collect some facts that will be needed for the classification of irreducible holonomyalgebras g ⊂ osp(p, q|2m) of the form (1). More precisely, any holonomy algebra g ⊂ sp(2m, R) of anodd Riemannian supermanifold is a skew-Berger algebra, i.e. g is spanned by the images of the spaceR(g) that consists of symmetric bilinear forms on R

2m with values in g satisfying the first Bianchiidentity. These algebras are the analogs of the Berger algebras that are defined in a similar way [4,18, 21]. All previously known irreducible Berger algebras were realized as the holonomy algebras [4,

Supported by the grant 201/09/P039 of the Grant Agency of Czech Republic and by the grant MSM 0021622409 of theCzech Ministry of Education.

163

164 GALAEV

21], hence skew-Berger algebras may be considered as the candidates to the holonomy algebras of oddsupermanifolds. Complex irreducible skew-Berger subalgebras of gl(n, C) are classified recently in [9].

Suppose now that g ⊂ osp(p, q|2m) is of the form (1) and irreducible. Its even part

g0 ⊂ so(p, q) ⊕ sp(2m, R)

preserves the decomposition Rp,q ⊕R

2m into the even and odd parts. For the most of the representations,g0 acts diagonally in R

p,q ⊕ R2m, i.e. its representations in the both subspaces are faithful. In [9] it is

explained that in this case prso(p,q) g0 ⊂ so(p, q) is a Berger algebra and prsp(2m,R) g0 ⊂ sp(2m, R) is askew-Berger algebra. Since g0 is reductive, prso(p,q) g0 is known. Below we will show that to know allpossible prsp(2m,R) g0, it is enough to classify irreducible skew-Berger subalgebras h ⊂ sp(2k, R) and

to classify skew-Berger subalgebras h ⊂ sp(2m, R) that preserve a decomposition R2m = W ⊕W1 into

the direct sum of two Lagrangian subspaces and act diagonally in W ⊕ W1. The last classification canbe reduced to the classification of irreducible subalgebras h ⊂ gl(n, R) with the non-trivial first skew-symmetric prolongation

h[1] = {ϕ ∈ (Rn)∗ ⊗ h|ϕ(x)y = −ϕ(y)x for all x, y ∈ R

n}.For subalgebras h ⊂ so(n, R) the first skew-symmetric prolongation is studied in [19], where someapplications are obtained. Thus, knowing prso(p,q) g0 and prsp(2m,R) g0, and using the theory of repre-sentations of simple Lie superalgebras [14], it is possible to find g ⊂ osp(p, q|2m). These ideas and theresults of this paper will allow to classify irreducible subalgebras g ⊂ osp(p, q|2m) of the form (1).

The paper has the following structure. In Section 2 we give necessary preliminaries. Section 3 dealswith odd Riemannian symmetric superspaces. In Section 4 the classification of irreducible subalgebrasg ⊂ gl(n, R) with non-trivial first skew-symmetric prolongation is obtained. In Section 5 irreducible notsymmetric skew-Berger subalgebras g ⊂ sp(2m, R), i.e. possible irreducible holonomy algebras of notsymmetric odd Riemannian supermanifolds, are classified.

2. PRELIMINARIES

Odd Riemannian supermanifolds, connections, holonomy algebras. First we rewrite somegeneral definitions from the theory of supermanifolds [15, 16, 17] for the case of odd supermanifolds.A connected supermanifold M of dimension 0|k is a pair

({x},Λ(k))

where x is the only point of the manifold and Λ(k) is a Grassman superalgebra of k generators,which is considered as the superalgebra of functions on M. Such supermanifolds M are called oddsupermanifolds. If ξ1, . . . , ξk are generators of Λ(k), then

Λ(k) = ⊕ki=0Λ

iR

k = Λ(k)0 ⊕ Λ(k)1,

where Rk = span{ξ1, . . . , ξk}, and Λ(k)0 and Λ(k)1 are spanned by the elements of even and odd degree,

respectively. The elements of Λ(k)0 and Λ(k)1 are called homogeneous. For a homogeneous element f ,the parity |f | ∈ Z2 = {0, 1} is defined to be |0| or |1| if f ∈ Λ(k)0 or f ∈ Λ(k)1\{0}, respectively. It holds|ξ1| = · · · = |ξk| = 1 and

fh = (−1)|f ||h|hf

for all homogenous f, h ∈ Λ(k). In particular, ξiξj = −ξjξi. Any function f ∈ Λ(k) can be written in theform

f =m∑

r=0

∑

α1<···<αr

fα1...αrξα1 · · · ξαr ,

where fα1...αr ∈ R. By definition, the value of f at the point x is f(x) = f∅. The functions ξ1, . . . , ξk arecalled coordinates on M. The Λ(k)-supermodule

TM = (TM)0 ⊕ (TM)1

LOBACHEVSKII JOURNAL OF MATHEMATICS Vol. 32 No. 2 2011

IRREDUCIBLE HOLONOMY ALGEBRAS 165

of vector fields on M consists of R-linear maps X : Λ(k) → Λ(k) such that the homogeneous X satisfy

X(fh) = (Xf)h + (−1)|X||f |f(Xh)

for all homogeneous functions f, g. The vector fields ∂ξ1, . . . , ∂ξk are defined in the obvious way.These vector fields are odd. The tangent space TxM to M at the point x can be identified withspan

R{∂ξ1 , . . . , ∂ξk} and it is an odd vector superspace. It holds

TM = Λ(k) ⊗ TxM.

The value of a vector field X = Xα∂ξα at the point x is defined as Xx = Xα(x)(∂ξα)x ∈ TxM.

A connection on M is an even R-linear map

∇ : TM ⊗R TM → TMof R-supermodules such that

∇fY X = f∇Y X and ∇Y fX = (Y f)X + (−1)|Y ||f |f∇Y X

for all homogeneous functions f and vector fields X, Y on M. The curvature tensor R of ∇ and itscovariant derivatives ∇rR and their values at the point x are defined in the usual way. From [8] it followsthat the holonomy algebra g of the connection ∇ at the point x can be defined in the following way:

g = span

⎧⎨

⎩∇r∂ξαr ,...,∂ξα1

Rx(∂ξβ , ∂ξγ )

∣∣∣∣∣∣0 ≤ r ≤ k, 1 ≤ β, γ ≤ k,

1 ≤ α1 < · · · < αr ≤ k

⎫⎬

⎭ ⊂ gl(TxM) � gl(0|k, R).

Thus g is a usual Lie algebra acting in the odd superspace TxM. Considering the isomorphismΠTxM � R

k, we get g ⊂ gl(k, R). Here Π is the parity changing functor. The holonomy group of theconnection ∇ at the point x is defined as the corresponding connected Lie subgroup of Gl(k, R).

A Riemannian supermetric on M is an even linear map

g : 2TM → Λ(k)

such that its value

ω = gx ∈ 2T ∗xM

is non-degenerate. Since TxM is an odd vector superspace, ω is a symplectic form on ΠTxM � Rk.

Hence in this case k must be even, k = 2m. On such Riemannian supermanifold there exists the Levi-Civita superconnection ∇. For its holonomy algebra it holds

g ⊂ osp(0|2m, R) � sp(2m, R).

Skew-Berger algebras. The main task of this paper is to classify possible irreducible holonomyalgebras g ⊂ sp(2m, R) of odd Riemannian supermanifolds. This can be done using the followingalgebraic properties of the representation g ⊂ sp(2m, R).

Let V be a real or complex vector space and g ⊂ gl(V ) a subalgebra. The space of skew-symmetriccurvature tensors of type g is defined as follows:

R(g) =

⎧⎨

⎩R ∈ 2V ∗ ⊗ g

∣∣∣∣∣∣R(X,Y )Z + R(Y,Z)X + R(Z,X)Y = 0

for all X,Y,Z ∈ V

⎫⎬

⎭ .

The subalgebra g ⊂ gl(V ) is called a skew-Berger subalgebra if it is spanned by the images of theelements R ∈ R(g). The space R(g) and the notion of the Berger algebra (or more generally of a Bergersuperalgebra) is defined in the same way, the only difference is that R is a (super) skew-symmetricbilinear form with values in g. Obviously R(g) = R(g ⊂ gl(ΠV )) and g ⊂ gl(V ) is a skew-Bergeralgebra if and only if g ⊂ gl(ΠV ) is a Berger superalgebra.


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Let g ⊂ gl(k, R) Consider the space of linear maps from Rk to R(g) satisfying the second Bianchi

identity,

R∇(g) =

⎧⎨

⎩S ∈ (Rk)∗ ⊗ R(g)

∣∣∣∣∣∣SX(Y,Z) + SY (Z,X) + SZ(X,Y ) = 0

for all X,Y,Z ∈ Rk

⎫⎬

⎭ .

A skew-Berger subalgebra g ⊂ gl(n, R) is called symmetric if R∇(g) = 0. Note that if an odd su-permanifold M is endowed with a torsion-free connection ∇ and g is its holonomy algebra, then∇r

∂ξαr ,...,∂ξα1Rx ∈ R(g) and ∇r

∂ξαr ,...,∂ξα2 ,·Rx ∈ R∇(g). In particular, g ⊂ gl(k, R) is a skew-Berger

algebra.Let ω be the standard symplectic form on R

2m. A subalgebra g ⊂ sp(2m, R) is called weakly-irreducible if it does not preserve any proper non-degenerate subspace of R

2m. The next theorem isthe partial case of the Wu Theorem for Riemannian supermanifolds proved in [8].

Theorem 1. Let g ⊂ sp(2m, R) = sp(V ) be a skew-Berger subalgebra, then there is a decompo-sition

V = V0 ⊕ V1 ⊕ · · · ⊕ Vr

into a direct sum of symplectic subspaces and a decomposition

g = g1 ⊕ · · · ⊕ gr

into a direct sum of ideals such that gi annihilates Vj if i �= j and gi ⊂ sp(Vi) is a weakly-irreducible Berger subalgebra.

If g ⊂ gl(k, R) is the holonomy algebra of an odd Riemannian supermanifold (M, g), then theabove decompositions define a decomposition of (M, g) into the product of a flat odd Riemanniansupermanifold and of odd Riemannian supermanifolds with the weakly-irreducible holonomyalgebras gi ⊂ sp(Vi).

Let g ⊂ sp(2m, R) = sp(V ) be a skew-Berger subalgebra. By the above theorem, we may assumethat it is weakly-irreducible. Suppose that it is not irreducible. Suppose also that g is a reductiveLie algebra. Since g is not irreducible, it preserves a degenerate subspace W ⊂ V . Consequently, g

preserves the isotropic subspace L = W ∩ W⊥ (W⊥ is defined using ω). Since g is totally reducible,there exists a complementary invariant subspace L′ ⊂ V . Since g is weakly-irreducible, the subspace L′

is degenerate. If L′ is not isotropic, then g preserves the kernel of the restriction of ω to L′ and g preservesa complementary subspace in L′ to this kernel, which is non-degenerate. Hence L′ is isotropic andV = L⊕L′ is the direct of two Lagrangian subspaces. The form ω on V allows to identify L′ with the dualspace L∗ and the representations of g on L and L′ are dual. Since g ⊂ sp(V ) is weakly-irreducible, therepresentation g ⊂ gl(L) is irreducible. Let R ∈ R(g). From the Bianchi identity it follows that R(x, y) =0 and R(ϕ,ψ) = 0 for all x, y ∈ L and ϕ,ψ ∈ L∗. Moreover, for each fixed ϕ ∈ L∗ it holds R(·, ϕ) ∈ (g ⊂gl(L))[1], where (g ⊂ gl(L))[1] is the first skew-symmetric prolongation for the representation g ⊂ gl(L)(similarly, for each fixed x ∈ L it holds R(·, x) ∈ (g ⊂ gl(L∗))[1]). Consequently, (g ⊂ gl(L))[1] �= 0 andsuch algebras are classified in Section 4. Thus we will get classification of all reductive skew-Bergersubalgebras g ⊂ sp(2m, R).

3. ODD SYMMETRIC SUPERSPACES AND SIMPLE LIE SUPERALGEBRAS

Symmetric superspaces are studied in [22, 16, 11]. A class of odd symmetric superspaces isconsidered in [5].

An odd supermanifold (M,∇) is called symmetric if ∇R = 0. In this case for the holonomy algebrag we have

g = span{Rx(∂ξβ , ∂ξγ )|1 ≤ β, γ ≤ k} = Rx(TxM, TxM).

Moreover, g annihilates Rx ∈ R(g) [8].Similarly as in [21] it can be shown that if g ⊂ gl(k, R) is irreducible skew-Berger algebra such

that the representation of g in R(g) is trivial, then g is a symmetric skew-Berger algebra.



Table 1. Simple complex Lie superalgebras of type I

g g0 g1 Restriction

osp(n|2m, C) so(n, C) ⊕ sp(2m, C) Cn ⊗ C

2m n �= 2

osp(4|2, α, C) sl(2, C) ⊕ sl(2, C) ⊕ sl(2, C) C2 ⊗ C

2 ⊗ C2 α ∈ C\{0,−1}

F (4) so(7, C) ⊕ sl(2, C) C8 ⊗ C

2

G(3) G2 ⊕ sl(2, C) C7 ⊗ C

2

pq(n, C) sl(n, C) sl(n, C) n ≥ 3

From [8] it follows that if the holonomy algebra g of a torsion-free connection ∇ on an oddsupermanifold M is a symmetric skew-Berger algebra, then (M,∇) is symmetric.

Let (M,∇) be an odd symmetric supermanifold. Define the Lie superalgebra

k = g ⊕ ΠRk

with the Lie superbrackets

[ΠX,ΠY ] = R(X,Y ), [A,ΠX] = Π(AX), [A,B] = [A,B]g,

where A,B ∈ g and X,Y ∈ Rk. Conversely, let k be a Lie superalgebra with the even part g and the

odd part ΠRk. Let G be the connected Lie group with the Lie algebra g and K be the connected Lie

supergroup with the Lie superalgebra k. The Lie supergroup K can be given by the Harish-Chandra pair(G, k) [11]. The factor superspace M = K/G is an odd supermanifold and it admits a unique symmetricsuperconnection [16].

Thus we obtain a one to one correspondence between connected odd symmetric superspaces (M,∇)and Lie superalgebras k = g ⊕ ΠR

k. Moreover, ΠRk is the tangent space to M and g ⊂ gl(k, R) is the

holonomy algebra. The space (M,∇) is Riemannian if and only if g ⊂ sp(k, R).

Let k = k0 ⊕ k1 be a (real or complex) simple Lie superalgebra. It is of classical type if therepresentation of k0 on k1 is totally reducible. In this case, k is of type I if the representation of k0 onk1 is irreducible; k is of type II if k0 preserves a decomposition

k1 = k−1 ⊕ k1

such that the representations of k0 in k−1 and k1 are faithful and irreducible.

Let g be a reductive Lie algebra. Suppose that g ⊂ gl(k, R) is irreducible, or there exists a g-invariantdecomposition

ΠRk = k−1 ⊕ k1

such that the representations of g on k−1 and k1 are faithful and irreducible. Suppose that there existsR ∈ R(g) such that g annihilates R and R(Rk, Rk) = g, then the Lie superalgebra k = g ⊕ ΠR

k definedas above is simple, this follows from Propositions 1.2.7 and 1.2.8 from [13].

Thus we have reduced the classification of weakly-irreducible reductive subalgebras g ⊂ gl(k, R)admitting elements R ∈ R(g) such that g annihilates R and R(Rk, Rk) = g to the classification of realsimple Lie superalgebras of classical type. Remark that we are interested only in the case g ⊂ sp(k, R).If g ⊂ gl(k, R) is not irreducible, then g ⊂ sp(k, R) if and only if k−1 � k∗1.

Real simple Lie superalgebras of classical type are exhausted by simple complex Lie superalgebrasof classical type considered as real Lie superalgebras and by real forms of complex Lie superalgebras ofclassical type. The real forms are found in [20]. To make the exposition complete we list these algebrasin Tables 1–4.


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Table 2. Simple complex Lie superalgebras of type II

g g0 g1 g−1 Restriction

sl(n|m, C) sl(n, C) ⊕ sl(m, C) ⊕ C Cn ⊗ C

m∗C

n∗ ⊗ Cm n �= m

psl(n|n, C) sl(n, C) ⊕ sl(n, C) Cn ⊗ C

n∗C

n∗ ⊗ Cn

osp(2|2m, C) so(2, C) ⊕ sp(2m, C) C2m

C2m

pe(n, C) sl(n, C) 2sl(n, C) Λ2sl(n, C)∗ n ≥ 3

Table 3. Simple real Lie superalgebras of type I

g ⊗ C g g0 g1

osp(n|2m, C) osp(r, n − r|2m, R) so(r, n − r) ⊕ sp(2m, R) Rr,n−r ⊗ R

2m

osp(2n|2m, C) hosp(r, m − r|n) so(n, H) ⊕ sp(r, m − r) Hn ⊗H H

r,m−r

osp(1|2m, C) osp(1|2m, R) sp(2m, R) R2m

osp(4|2, α, C) sl(2, R) ⊕ sl(2, R) ⊕ sl(2, R) R2 ⊗ R

2 ⊗ R2

su(2) ⊕ su(2) ⊕ sl(2, R) R4 ⊗ R

2

sl(2, C) ⊕ sl(2, R) R1,3 ⊗ R

2

F (4) sl(2, R) ⊕ so(7) R2 ⊗ Δ7

sl(2, R) ⊕ so(3, 4) R2 ⊗ Δ3,4

su(2) ⊕ so(2, 5) C2 ⊗ Δ2,5

su(2) ⊕ so(1, 6) C2 ⊗ Δ1,6

G(3) sl(2, R) ⊕ G2 R2 ⊗ R

7

sl(2, R) ⊕ G∗2(2) R

2 ⊗ R3,4

pq(n, C) pq(n, R) sl(n, R) sl(n, R)

su(p, n − p) su(p, n − p)

pq(n, H) sl(n2 , H) sl(n

2 , H)

sl(n|m, C) su(s, n − s|r, m − r) su(s, n − s) ⊕ su(r, m − r) ⊕ Ri Cs,n−s ⊗ C

r,m−r

psl(n|n, C) psu(s, n − s|r, n − r) su(s, n − s) ⊕ su(r, n − r) Cs,n−s ⊗ C

r,n−r

sl(n, C) Cn ⊗ Cn

osp(2|2m, C) hosp(r, m − r|1) Ri ⊕ sp(r, m − r) Hr,m−r

Table 4. Simple real Lie superalgebras of type II

g ⊗ C g g0 g1

sl(n|m, C) sl(n|m, R) sl(n, R) ⊕ sl(m, R) ⊕ R Rn ⊗ R

m∗ ⊕ Rn∗ ⊗ R

m

sl(n2 |

m2 , H) sl(n

2 , H) ⊕ sl(m2 , H) ⊕ R H

n2 ⊗H H

m2 ∗ ⊕ H

n2 ∗ ⊗H H

m2

osp(2|2m, C) osp(2|2m, R) R ⊕ sp(2m, R) R2m ⊕ R

2m∗

pe(n, C) pe(n, R) sl(n, R) 2sl(n, R) ⊕ Λ2sl(n, R)∗

pe(n2 , R) sl(n

2 , H) 2sl(n2 , H) ⊕ Λ2sl(n

2 , H)∗



4. SKEW-SYMMETRIC PROLONGATIONS OF LIE ALGEBRASIrreducible subalgebras g ⊂ gl(n, F) (F = R or C) with non-trivial prolongations

g(1) = {ϕ ∈ (Fn)∗ ⊗ g|ϕ(x)y = ϕ(y)x for all x, y ∈ Fn}

are well known, see e.g. [4]. Here we classify irreducible subalgebras g ⊂ gl(n, F) such that the skew-symmetric prolongation

g[1] = {ϕ ∈ (Fn)∗ ⊗ g|ϕ(x)y = −ϕ(y)x for all x, y ∈ F

n}of g is non-zero.

Irreducible subalgebras g ⊂ so(n, R) with non-zero skew-symmetric prolongations are classifiedin [19]. These subalgebras are exhausted by the whole orthogonal Lie algebra so(n, R) and by the adjointrepresentations of compact simple Lie algebras.

Irreducible subalgebras g ⊂ gl(n, C) with g[1] �= 0 are classified in [9]. We give this list in Table 5. Toget this result it was used that g[1] coincides with Π(g ⊂ gl(0|n, C))(1) and the fact that the whole Cartanprolong

g∗ = ΠV ⊕ g ⊕ (g ⊂ gl(0|n, C))(1) ⊕ (g ⊂ gl(0|n, C))(2) ⊕ · · ·is an irreducible transitive Lie superalgebra with the consistent Z-grading and g1 �= 0. All such Z-gradedLie superalgebras are classified in [13]. The second prolongation g[2] is defined in the obvious way.

Now we classify irreducible subalgebras g ⊂ gl(n, R) with g[1] �= 0. Let g ⊂ gl(n, R) be such subalge-bra. If this representation is absolutely irreducible, i.e. Rn does not admit a complex structure commutingwith the elements of g, then g⊗C ⊂ gl(n, C) is an irreducible subalgebra, and (g⊗C)[1] = g[1] ⊗C �= 0.Note that if in this case the representation g ⊂ gl(n, R) is different from the adjoint one and from thestandard representation of so(n, R), then (g ⊗ C)(1) �= 0 or (g ⊗ C ⊕ C)(1) �= 0 (then g(1) �= 0). Henceabsolutely irreducible subalgebras g ⊂ gl(n, R) with g[1] �= 0 are exhausted by absolutely irreduciblesubalgebras g ⊂ gl(n, R) (up to the center of g) with g[1] �= 0, by the adjoint representations of real formsof complex simple Lie algebras, and by so(p, n − p) ⊂ gl(n, R). The result is given in Table 6, where weuse the following notation from [4]:

Hn(C) = {A ∈ Matn(C)|A∗ = A}, Sn(H) = {A ∈ Matn(H)|A∗ = −A},An(H) = {A ∈ Matn(H)|A∗ = A}.

The first and the second skew-symmetric prolongations can be found from the relation (g ⊗ C)[k] =g[k] ⊗ C.

Suppose that the representation g ⊂ gl(n, R) is non-absolutely irreducible, i.e. E = Rn admits a

complex structure J commuting with the elements of g. In this case the complexificated space E ⊗ C

admits the decomposition E ⊗ C = V ⊕ V , where V and V are the eigenspaces of the extension ofJ to E ⊗ C corresponding to the eigenvalues i and −i, respectively. The Lie algebra g ⊗ C preservesthis decomposition. Consider the ideal g1 = g ∩ Jg ⊂ g. Since g is reductive, there is an ideal g2 ⊂ g

such that g = g1 ⊕ g2. The Lie algebra g1 ⊗ C admits the decomposition g1 ⊗ C = g′1 ⊕ g′′1 into theeigenspaces of the extension of J to g1 ⊗ C corresponding to the eigenvalues i and −i, respectively. It iseasy to see that g′1 annihilates V , g′′1 annihilates V , and g2 ⊗C acts diagonally in V ⊕ V . We immediatelyconclude that

(g ⊗ C)[1] = (g′1 ⊂ gl(V ))[1] ⊕ (g′′1 ⊂ gl(V ))[1].

It is clear that the representation of g′1 ⊕ (g2 ⊗ C) in V is irreducible. If dim g2 ≥ 2, then this represen-tation is of the form of the tensor product of irreducible representations of g′1 and g2 ⊗ C. Obviously,in this case (g′1 ⊂ gl(V ))[1] = 0, similarly (g′′1 ⊂ gl(V ))[1] = 0. We conclude that dim g2 ≤ 1, and g1 ⊂gl(n

2 , C) ⊂ gl(n, R) is a complex subalgebra considered as the real one.

Thus irreducible subalgebras g ⊂ gl(n, R) with g[1] �= 0 are exhausted by the subalgebras fromTable 5 considered as the real ones, by the subalgebras from Table 6, and subalgebras of the formg1 ⊕ z ⊂ gl(n, R), where z ⊂ C is a center of real dimension 1 or 2, and g1 is an subalgebra from Table 5considered as the real one and with the trivial center.


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Table 5. Complex irreducible subalgebras g ⊂ gl(V ) with g[1] �= 0

g V g[1] g[2]

sl(n, C) Cn, n ≥ 3 (Cn ⊗ Λ2(Cn)∗)0 (Cn ⊗ Λ3(Cn)∗)0

gl(n, C) Cn, n ≥ 2 C

n ⊗ Λ2(Cn)∗ Cn ⊗ Λ3(Cn)∗

sl(n, C) 2C

n, n ≥ 3 Λ2(Cn)∗ 0

gl(n, C) 2C

n, n ≥ 3 Λ2(Cn)∗ 0

sl(n, C) Λ2C

n, n ≥ 5 2(Cn)∗ 0

gl(n, C) Λ2C

n, n ≥ 5 2(Cn)∗ 0

sl(n, C) ⊕ sl(m, C) ⊕ C Cn ⊗ C

m, n, m ≥ 2 V ∗ 0

n �= m

sl(n, C) ⊕ sl(n, C) Cn ⊗ C

n, n ≥ 3 V ∗ 0

sl(n, C) ⊕ sl(n, C) ⊕ C Cn ⊗ C

n, n ≥ 3 V ∗ 0

so(n, C) Cn, n ≥ 4 Λ3V ∗ Λ4V ∗

so(n, C) ⊕ C Cn, n ≥ 4 Λ3V ∗ Λ4V ∗

sp(2n, C) ⊕ C C2n n ≥ 2 V ∗ 0

g is simple g C id 0

g ⊕ C, gis simple g C id 0

Table 6. Absolutely irreducible subalgebras g ⊂ gl(n, R) with g[1] �= 0 (z denotes either 0 or R)

g V

sl(n, R) Rn, n ≥ 3

gl(n, R) Rn, n ≥ 2

sl(n, R) ⊕ z 2R

n, n ≥ 3

sl(n, H) ⊕ z Sn(H), n ≥ 2

sl(n, R) ⊕ z Λ2R

n, n ≥ 5

sl(n, H) ⊕ z An(H), n ≥ 3

sl(n, R) ⊕ sl(m, R) ⊕ R Rn ⊗ R

m, n > m ≥ 2

sl(n, R) ⊕ sl(n, R) ⊕ z Rn ⊗ R

n, n ≥ 3

sl(n, H) ⊕ sl(m, H) ⊕ R Hn ⊗ H

m, n > m ≥ 1

sl(n, H) ⊕ sl(n, H) ⊕ z Hn ⊗ H

n, n ≥ 2

sl(n, C) ⊕ R Hn(C), n ≥ 3

so(p, q) ⊕ z Rp+q , p + q ≥ 4

sp(2n, R) ⊕ R R2n, n ≥ 2

g ⊕ z, g is a real form of a g

simple complex Lie algebra



Table 7. Irreducible skew-Berger subalgebras g ⊂ sp(2m, C) = sp(V )

g V Restriction

sp(2m, C) C2m n ≥ 1

sl(2, C) ⊕ so(m, C) C2 ⊗ C

m m ≥ 3

spin(12, C) Δ+12 = C

32

sl(6, C) Λ3C

6 = C20

sp(6, C) Vπ3 = C14

so(n, C) ⊕ sp(2q, C) Cn ⊗ C

2q n ≥ 3, q ≥ 2

GC

2 ⊕ sl(2, C) C7 ⊗ C

2

so(7, C) ⊕ sl(2, C) C8 ⊗ C

2

5. IRREDUCIBLE HOLONOMY ALGEBRAS OF NOT SYMMETRIC ODD RIEMANNIANSUPERMANIFOLDS

Section 3 provides the classification of irreducible symmetric skew-Berger algebras g ⊂ sp(2m, R),hence it is left to classify irreducible non-symmetric skew-Berger algebras g ⊂ sp(2m, R).

Irreducible skew-Berger subalgebras g ⊂ gl(n, C) are classified in [9]. In Table 7 we list irreducibleskew-Berger subalgebras g ⊂ sp(2m, C).

Note that for the last three subalgebras from Table 7 it holds R(g) � C and this space is annihilatedby g, i.e. those algebras are symmetric skew-Berger algebras.

We get now the list of irreducible not symmetric skew-Berger subalgebras g ⊂ sp(2m, R). Let V be areal vector space and g ⊂ gl(V ) an irreducible subalgebra. Consider the complexifications VC = V ⊗R C

and gC = g ⊗R C ⊂ gl(VC). It is easy to see that R(gC) = R(g) ⊗R C. Suppose that g ⊂ gl(V ) isabsolutely irreducible, i.e. there exists a complex structure J on V commuting with the elements of g.Then V can be considered as a complex vector space. Consider the natural representation i : gC → gl(V )in the complex vector space V . The following proposition is the analog of Proposition 3.1 from [21].

Proposition 1. Let V be a real vector space and g ⊂ gl(V ) an irreducible subalgebra.(1) If the subalgebra g ⊂ gl(V ) is absolutely irreducible, then g ⊂ gl(V ) is a skew-Berger

algebra if and only if gC ⊂ gl(VC) is a skew-Berger algebra.

(2) If the subalgebra g ⊂ gl(V ) is not absolutely irreducible and if (i(gC))[1] = 0, then g ⊂ gl(V )is a skew-Berger algebra if and only if Jg = g and g ⊂ gl(V ) is a complex irreducible skew-Bergeralgebra.

First of all, the algebras of Table 7 exhaust the second possibility of the proposition with (i(gC))[1] = 0,and only the first 5 algebras of Table 7 are not symmetric skew-Berger algebras.

Suppose that the subalgebra g ⊂ sp(V ) is an absolutely irreducible not symmetric skew-Bergeralgebra. Then gC ⊂ sp(VC) is one of the first 5 algebras of Table 7. Note that each of these algebras isalso a Berger algebra [21]. From Proposition 3.1 and Proposition 3.1 from [21] it follows that g ⊂ sp(V )is a Berger algebra, hence we may deduce all absolutely irreducible not symmetric skew-Berger algebrag ⊂ sp(V ) from [21].

We are left with the not absolutely irreducible subalgebras g ⊂ sp(V ) such that (i(gC))[1] �= 0.Consider any such g. Let as in Section 4, g1 = g ∩ Jg. Then g = g1 ⊕ g2 and gC = g′1 ⊕ g′′1 ⊕ (g2 ⊗C). More over, g′1 and g′′1 are isomorphic. Table 5 implies that the only possible i(gC) is sl(n, C) ⊕sl(n, C) ⊕ C. Then V = C

n ⊗ Cn, g1 = sl(n, C) and g2 = R. The Lie algebra gC acting in VC preserves

the decomposition VC = W ⊕ W , where W and W are the eigenspaces of the extension of J to VC

corresponding to the eigenvalues i and −i, respectively. Moreover, g′1 � sl(n, C) annihilates W , g′′1 �sl(n, C) annihilates W , and C acts diagonally in W ⊕ W . This shows that

R(gC) ⊂ R(gl(n, C) ⊂ gl(W )

)⊕ R

(gl(n, C) ⊂ gl(W )

).


172 GALAEV

Table 8. Possible irreducible holonomy algebras g ⊂ sp(2m, R) = sp(V ) of not symmetric odd Riemanniansupermanifolds

g V Restriction

sp(2m, R) R2m m ≥ 1

u(p, q) Cp,q p + q ≥ 2

su(p, q) Cp,q p + q ≥ 2

so(n, H) Hn n ≥ 2

sp(1) ⊕ so(n, H) Hn n ≥ 2

sl(2, R) ⊕ so(p, q) R2 ⊗ R

p,q p + q ≥ 3

spin(2, 10) Δ+2,10 = R

32

spin(6, 6) Δ+6,6 = R

32

so(6, H) ΔH

6 = H8

sl(6, R) Λ3R

6 = R20

su(1, 5) {ω ∈ Λ3C

6| ∗ w = w}su(3, 3) {ω ∈ Λ3

C6| ∗ w = w}

sp(6, R) R14 ⊂ Λ3

R6

sp(2m, C) C2m m ≥ 1

sl(2, C) ⊕ so(m, C) C2 ⊗ C

m m ≥ 3

spin(12, C) Δ+12 = C

32

sl(6, C) Λ3C

6 = C20

sp(6, C) Vπ3 = C14

Note that dimC W = 2n. From [9] it follows that the both sl(n, C) and gl(n, C) do not appear as theskew-Berger subalgebras of gl(W ) for W of such dimension, hence R(gC) = 0. This shows that g1 = 0,hence g ∩ Jg = 0, i.e. g ⊂ gC is a Real form. We have to consider the real forms g of the algebrash appearing in Table 5 such that the restriction to g of the corresponding representation h ⊂ gl(E)is irreducible and take V = E considered as the real vector space. Now gC = h acts diagonally inVC = W ⊕ W . Let R ∈ R(gC ⊂ VC) and S ∈ ∇R(gC ⊂ VC). From the first Bianchi identity it followsthat R(X,Y ) = 0 whenever both X and Y belong either to W or to W . Next, for each X1 ∈ W andX ∈ W , it holds

R(X1, ·|W )|W ∈ ((gC)W ⊂ gl(W ))[1], R(X, ·|W )|W ∈ ((gC)W ⊂ gl(W ))[1].

From this and the second Bianchi identity it follows that

S·|W (X1, ·|W )|W ∈ ((gC)W ⊂ gl(W ))[2], S·|W (X, ·|W )|W ∈ ((gC)W ⊂ gl(W ))[2].

Hence, if g ⊂ sp(V ) is a skew-Berger algebra and ((gC)W ⊂ gl(W ))[2] = 0, then ∇R(g) = 0, i.e. g ⊂sp(V ) is a symmetric skew-Berger algebra. Thus we get only the following 4 possibilities for h ⊂ gl(E):

gl(n, C), sl(n, C), so(n, C), so(n, C) ⊕ C ⊂ gl(n, C).

The corresponding g ⊂ sp(2m, R) are the following:

u(n), su(n) ⊂ sp(2n, R), so(n, H), so(n, H) ⊕ Ri ⊂ sp(4n, R).

Let us find the spaces R(g)⊗C = R(gC ⊂ gl(VC)). As we have seen above, any R ∈ R(gC ⊂ gl(VC)) isuniquely defined by the values R(X,X1), where X ∈ W and X1 ∈ W . Let e1, . . . , em be a basis in W ande1, . . . , em the dual basis in W . We may write R(ei, e

j) = Aji for some Aj

i ∈ gC|W . Define the numbers



Ajlik such that Aj

iek =∑

l Ajlikel. Then Aj

i el =

∑k Ajl

ikek. Let g = u(n), then Aj

i ∈ gl(n, C) and m = n.

From above it follows that R ∈ R(gC) if and only if Ajlik = −Ajl

ki and Ajlik = −Alj

ik. Hence, R(u(n)) ⊗ C

is isomorphic (W ∧ W )⊗ (W ∗ ∧ W ∗). For g = su(n) we get the additional condition∑

k Ajkik = 0. This

shows that u(n), su(n) ⊂ sp(2n, R) are skew-Berger subalgebras. Since

(so(2n, C) ⊂ gl(2n, C))[1] = (so(2n, C) ⊕ C ⊂ gl(2n, C))[1],

so(n, H)⊕Ri ⊂ sp(4n, R) is not a skew-Berger subalgebra. Finally consider g = so(n, H) ⊂ sp(4n, R).Any R ∈ R(gC ⊂ VC) is defined as above by the numbers Aj

i with the additional condition Ajlik = −Ajk

il .This shows that R(gC ⊂ VC) � ∧4

C2n, i.e. so(n, H) ⊂ sp(4n, R) is a skew-Berger subalgebra. We

obtain the following classification theoremTheorem 2. Possible irreducible holonomy algebras g ⊂ sp(2m, R) of not symmetric odd

Riemannian supermanifolds are listed in Table 8.

REFERENCES1. D. V. Alekseevsky, Funct. Anal. Appl. 2 (2), 1 (1968).2. M. Berger, Bull. Soc. Math. France 83, 279 (1955).3. A. L. Besse, Einstein manifolds (Springer-Verlag, Berlin-Heidelberg-New York, 1987).4. R. Bryant, in Actes de la Table Ronde de Geometrie Differentielle (Luminy, 1992), Semin. Congr. 1, 93 (1996).5. V. Cortes, Math. Scand. 98, (2), 201 (2006).6. P. Deligne and J. W. Morgan, in Notes on supersymmetry (following Joseph Bernstein), Quantum Fields

and Strings: A Course for Mathematicians 1/2, 41 (1999).7. A. Galaev and T. Leistner, Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys.,

Eur. Math. Soc. Zurich 53 (2008).8. A. S. Galaev, Abhandlungen aus dem Mathematischen Seminar der Universitat Hamburg 79, 47 (2009).9. A. S. Galaev, Differential Geom. Appl. 27, (6), 743 (2009).

10. G. W. Gibbons, Progress of Theoretical Physics Supplement No. 177, 33 (2009).11. O. Goertsches, Mat. Z. 260 (3), 557 (2008).12. D. Joyce, Riemannian holonomy groups and calibrated geometry, (Oxford University Press, Oxford,

2007).13. V. G. Kac, Adv. Math. 26, 8 (1977).14. V. G. Kac, Representations of classical Lie superalgebras, Lectures Notes in Mathematics 676,

(Springer-Verlag, Berlin, 1978).15. D. A. Leites, Russian Math. Surveys, 35, 1 (1980).16. D. A. Leites, Theory of supermanifolds, (Petrozavodsk, 1983) [in Russian].17. Yu. I. Manin, Gauge Field Theory and Complex Geometry, (Nauka, Moscow, 1984; Springer Verlag,

Berlin, 1988).18. S. Merkulov, L. Schwachhofer, Ann. Math. 150, 77 (1999).19. P.-A. Nagy, Prolongations of Lie algebras and applications, arXiv:0712.1398.20. M. Parker, J. Math. Phys. 21, 689 (1980).21. L. J. Schwachhofer, Adv. Math. 160, 1 (2001).22. V. V. Serganova, Funct. Anal. Appl. 17, 200 (1983).23. V. S. Varadarajan, American Mathematical Society, Courant ecture notes 11, 2004.24. H. Wu, Pacific Journal of Math. 20, 351 (1967).


Ann Glob Anal Geom (2012) 42:1–27DOI 10.1007/s10455-011-9299-4

Irreducible holonomy algebras of Riemanniansupermanifolds

Anton S. Galaev

Received: 15 March 2011 / Accepted: 4 October 2011 / Published online: 18 October 2011© Springer Science+Business Media B.V. 2011

Abstract Possible irreducible holonomy algebras g ⊂ osp(p, q|2m) of Riemanniansupermanifolds under the assumption that g is a direct sum of simple Lie superalgebrasof classical type and possibly of a 1-dimensional center are classified. This generalizes theclassical result of Marcel Berger about the classification of irreducible holonomy algebrasof pseudo-Riemannian manifolds.

Keywords Riemannian supermanifold · Levi-Civita superconnection · Holonomy algebra ·Berger superalgebra

Mathematics Subject Classification (2000) 58A50 · 53C29

1 Introduction

Possible irreducible holonomy algebras (or, equivalently, connected irreducible holon-omy groups) of non-locally symmetric pseudo-Riemannian manifolds are classified byBerger in [4]. These algebras are the following: so(p, q), so(p,C), u(r, s), su(r, s), sp(r, s),sp(r, s)⊕sp(1), sp(r,R)⊕sl(2,R), sp(r,C)⊕sl(2,C), spin(7) ⊂ so(8), spin(4, 3) ⊂ (4, 4),spin(7,C) ⊂ so(8, 8), G2 ⊂ so(7), G∗

2(2) ⊂ so(4, 3) and GC

2 ⊂ so(7, 7). This result, espe-cially in the case of Riemannian manifolds, has a lot of consequences and applications bothin geometry and physics, see [2, 6, 8, 22] and the references therein.

Since theoretical physicists discovered supersymmetry, supermanifolds began to playan important role both in mathematics and physics [11, 26, 28, 39]. There is also aninterest to Riemannian supermanifolds, see e.g. [3, 7, 9, 10, 19]. In particular, Calabi-Yau

Communicated by N. Hitchin (Oxford).

A. S. Galaev (B)Department of Mathematics and Statistics, Faculty of Science, Masaryk University in Brno,Kotlárská 2, 611 37 Brno, Czech Republice-mail: [email protected]

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2 Ann Glob Anal Geom (2012) 42:1–27

supermanifolds (i.e. Riemannian supermanifolds with the holonomy algebras contained insu(p0, q0|p1, q1)) were considered recently in several physical papers, e.g. [1, 33, 41].

Holonomy groups and holonomy algebras for superconnections on supermanifolds areintroduced recently in [13]. In this article, we generalize the result of M. Berger for the caseof Riemannian supermanifolds.

For a supersubalgebra g ⊂ osp(p, q|2m), we define the vector superspace R(g) of curva-ture tensors of type g that consists of linear maps from�2

Rp,q|2m to g satisfying the Bianchi

super identity. We call g a Berger superalgebra if g is spanned by the images of the elementsof R(g). The holonomy algebra of a Riemannian supermanifold is a Berger superalgebra.Consequently, Berger superalgebras may be considered as the candidates to the holonomyalgebras. A Berger superalgebra g ⊂ osp(p, q|2m) is called symmetric if the superspaceR∇(g), consisting of linear maps from R

p,q|2m to R(g) satisfying the second Bianchi superidentity, is trivial. If such g is the holonomy algebra of a Riemannian supermanifold, thenthis supermanifold is locally symmetric. The classification of irreducible symmetric Bergersuperalgebras can be deduced from [37]. Irreducible Berger supersubalgebras g ⊂ osp(0|2m)are classified in [16]. In this article, we classify irreducible non-symmetric Berger supersub-algebras g ⊂ osp(p, q|2m) (p + q > 0) of the form

g = (⊕i gi )⊕ z, (1)

where gi are simple Lie superalgebras of classical type, and z is a trivial or one-dimensionalcenter. The obtained list is the following:

osp(p, q|2m), osp(r |2k,C), u(p0, q0|p1, q1), su(p0, q0|p1, q1),

hosp(r, s|k), hosp(r, s|k)⊕ sp(1), ospsk(2k|r, s)⊕ sl(2,R),

ospsk(2k|r,C)⊕ sl(2,C).

This list generalizes the list of irreducible holonomy algebras of non-locally symmetricpseudo-Riemannian manifolds, in the same time, we do not get the analogs of the importantholonomy algebras G2 and spin(7). We discuss the geometries of the supermanifolds witheach of the obtained holonomy algebras.

Remark that in general a semi-simple Lie superalgebra g has the form g = ⊕i (gi ⊗�(ni )),where gi is a simple Lie superalgebra (either of classical or of Cartan type), and�(ni ) is theGrassmann superalgebra of ni variables [23]. Moreover, there exist representations of solv-able Lie superalgebras in vector superspaces of dimensions greater then one [23]. Thus, weconsider only a part of irreducible subalgebras g ⊂ osp(p, q|2m). For irreducible subalgebrasg ⊂ so(p, q) the property (1) holds automatically, i.e., our assumption is quit natural.

In Sect. 2, we recall some facts about holonomy algebras of pseudo-Riemannian manifolds.In Sects. 3 and 4, we provide some preliminaries on Lie superalgebras and supermanifolds.In Sect. 6, we formulate the main result of this article. The remaining sections are dedicatedto the proof. In Sect. 9, we describe the superspaces R(g) for subalgebras g ⊂ osp(p, q|2m).Let V = R

p,q|2m . We show that any element R ∈ R(g)0 satisfies

prso(p,q) ◦R|�2V0∈ R(prso(p,q) g0), prsp(2m,R) ◦R|�2V1

∈ R(prsp(2m,R) g0),

where R(prso(p,q) g0) is the space of curvature tensors for the subalgebra prso(p,q) g0 ⊂so(p, q) and R(prsp(2m,R) g0) is a similar space for the subalgebra prsp(2m,R) g0 ⊂ sp(2m,R),this space is defined in Sect. 8. Next, any R ∈ R(g)1 satisfies

prso(p,q) ◦R(·, ξ)|V0∈ Pη(prso(p,q) g0), prsp(2m,R) ◦R(·, x)|V1

∈ Pω(prsp(2m,R) g0),

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Ann Glob Anal Geom (2012) 42:1–27 3

where x ∈ V0 and ξ ∈ V1 are fixed, Pη(prso(p,q) g0) is the space of weak curvature ten-sors for the subalgebra prso(p,q) g0 ⊂ so(p, q) and Pω(prsp(2m,R) g0) is a similar spacefor the subalgebra prsp(2m,R) g0 ⊂ sp(2m,R). We discuss these spaces in Sects. 7 and 8.Elements of the space Pη(so(n)) appear as a part of the curvature tensor of a Lorentzianmanifold [25, 15]. These properties of the space R(g) give strong conditions on the rep-resentation g0 ⊂ so(p, q) ⊕ sp(2m,R) of the even part of g. We prove that under someassumption prso(p,q) g0 ⊂ so(p, q) is the holonomy algebra of a pseudo-Riemannian mani-fold and prsp(2m,R) g0 ⊂ sp(2m,R) is in the list of possible reductive holonomy algebras ofRiemannian odd supermanifolds. These facts allow us to prove the classification theorem.Many facts and tables that we use are collected in [16].

2 Holonomy algebras of pseudo-Riemannian manifolds

The theory of holonomy algebras of pseudo-Riemannian manifolds can be found e.g. in[6, 8, 22]. Here we collect some facts that motivate the topic of this article and will be usedbelow.

Let (M, g) be a connected pseudo-Riemannian manifold of signature (p, q). The holon-omy group of (M, g) at a point x ∈ M is a Lie group that consists of the pseudo-orthogonaltransformations given by the parallel displacements along piece-wise smooth loops at thepoint x , and it can be identified with a Lie subgroup of the pseudo-orthogonal Lie groupO(p, q). The corresponding Lie subalgebra of so(p, q) is called the holonomy algebra. If themanifold M is simply connected, then the holonomy group is connected and it is uniquelydefined by the holonomy algebra.

A subalgebra g ⊂ so(p, q) is called weakly-irreducible if it does not preserve any propernon-degenerate subspace of the pseudo-Euclidean space R

p,q . By the Wu Theorem [40] anypseudo-Riemannian manifold can be decomposed (at least locally) in the product of a flatpseudo-Riemannian manifold and of pseudo-Riemannian manifolds with irreducible holon-omy algebra. In particular, for any holonomy algebra g ⊂ so(V ) there exists an orthogonaldecomposition

Rp,q = V0 ⊕ V1 ⊕ · · · ⊕ Vr

into a direct sum of pseudo-Euclidean subspaces and a decomposition

g = g1 ⊕ · · · ⊕ gr

into a direct sum of ideals such that gi annihilates Vj if i = j and gi ⊂ so(Vi ) is theweakly-irreducible holonomy algebra of a pseudo-Riemannian manifold.

Possible connected irreducible holonomy groups (i.e. possible irreducible holonomyalgebras) of not locally symmetric pseudo-Riemannian manifolds classified Marcel Bergerin [4]. Later it was proved that all these algebras can be realized as the holonomy algebras ofpseudo-Riemannian manifolds [8]. Here is the list and the description of the correspondinggeometries (see e.g. [8]):

so(p, q): generic pseudo-Riemannian manifolds;so(p,C) ⊂ so(p, p): generic holomorphic pseudo-Riemannian manifolds;u(r, s) ⊂ so(2r, 2s): pseudo-Kählerian manifolds;su(r, s) ⊂ so(2r, 2s): special pseudo-Kählerian manifolds or Calabi-Yau manifolds;sp(r, s) ⊂ so(4r, 4s): pseudo-hyper-Kählerian manifolds;sp(r, s)⊕ sp(1) ⊂ so(4r, 4s): pseudo-quaternionic-Kählerian manifolds;

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sp(r,R)⊕ sl(2,R) ⊂ so(2r, 2r): pseudo-para-Kählerian manifolds;sp(r,C)⊕ sl(2,C) ⊂ so(4r, 4r): complex pseudo-para-Kählerian manifolds;spin(7) ⊂ so(8), spin(4, 3) ⊂ so(4, 4), spin(7,C) ⊂ so(8, 8): 8-dimensional pseudo-Riemannian manifolds with a parallel 4-form and their complex analog;G2 ⊂ so(7), G∗

2(2) ⊂ so(4, 3), GC

2 ⊂ so(7, 7): 7-dimensional pseudo-Riemannian mani-folds with a parallel 3-form and their complex analog.

Compact Riemannian manifolds with the holonomy groups SU(2), SU(3), G2 and Spin(7)are extremely useful in theoretical physics, see references in [22].

The list of weakly-irreducible (and irreducible) reductive holonomy algebras of locallysymmetric pseud-Riemannian manifolds can be found in [5].

In the case of Riemannian manifolds weakly-irreducible subalgebras g ⊂ so(n) are thesame as the irreducible ones. Weakly-irreducible not irreducible Berger subalgebras g ⊂so(p, q) are classified only if p = 1 (p is the number of minuses of the metric), i.e. in thecase of Lorentzian manifolds, if g ⊂ u(1, q) ⊂ so(2, 2q) and there are some partial resultsin the neural signature (p, p), see the review [18].

Let us shortly explain how a classification of holonomy algebras can be obtained.Let g ⊂ so(p, q) = so(V ) be a subalgebra, where V = R

p,q is the pseudo-Euclideanspace. The space of curvature tensors of type g is defined as follows

R(g) ={

R ∈ �2V ∗ ⊗ g

∣∣∣∣ R(X, Y )Z + R(Y, Z)X + R(Z , X)Y = 0for allX, Y, Z ∈ V

}.

The above identity is called the first Bianchi identity. Let L(R(g)) ⊂ g be the ideal spannedby the images of the elements form R(g). From the Ambrose–Singer Theorem it follows thatfor the holonomy algebra g of a pseudo-Riemannian manifold it holds L(R(g)) = g.

Consider the vector space

R∇(g) ={

S ∈ V ∗ ⊗ R(g)∣∣∣∣SX (Y, Z)+ SY (Z , X)+ SZ (X, Y ) = 0

for all X, Y, Z ∈ V

}.

If g satisfies R∇(g) = 0, then any pseudo-Riemannian manifold with the holonomy algebrag is automatically locally flat. M. Berger classified irreducible subalgebras g ⊂ so(p, q)that satisfy L(R(g)) = g and R∇(g) = 0. By this reason subalgebra g ⊂ so(p, q) satisfy-ing L(R(g)) = g are called Berger algebras [29, 34]. If g ⊂ so(V ) is a Berger subalgebraand R∇(g) = 0, then g is called a symmetric Berger algebra, otherwise g is called a non-symmetric Berger algebra. It is known that if g is a reductive Lie algebra and the g-moduleR(g) is trivial, then R∇(g) = 0 [34].

Let g ⊂ so(V ) be a reductive weakly-irreducible Berger subalgebra. If g is not irreduc-ible, then it preserves a degenerate subspace W ⊂ V . Consequently, g preserves the isotropicsubspace L = W ∩ W ⊥. Since g is totally reducible, there exists a complementary invariantsubspace L ′ ⊂ V . Since g is weakly-irreducible, the subspace L ′ is degenerate. If L ′ is notisotropic, then g preserves the kernel of the restriction of the metric to L ′ and g preserves acomplementary subspace in L ′ to this kernel, which is non-degenerate. Hence L ′ is isotropicand V = L ⊕ L ′ is the direct sum of isotropic subspaces. This can happen only if p = q . Themetric on V allows to identify L ′ with the dual space L∗ and the representations of g on Land L ′ are dual. This shows that the representation g ⊂ gl(L) is irreducible. Let R ∈ R(g).From the Bianchi identity it follows that R(x, y) = 0 and R(ϕ, ψ) = 0 for all x, y ∈ Land ϕ,ψ ∈ L∗. Moreover, for each fixed ϕ ∈ L∗ it holds R(·, ϕ) ∈ (g ⊂ gl(L))(1), where(g ⊂ gl(L))(1) is the first prolongation for the representation g ⊂ gl(L) (similarly, for each

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fixed x ∈ L it holds R(·, x) ∈ (g ⊂ gl(L∗))(1)). Consequently, (g ⊂ gl(L))(1) = 0. The listof irreducible subalgebras and g ⊂ gl(L) with (g ⊂ gl(L))(1) = 0 can be found in [8].

3 Lie superalgebras

Here we define some Lie superalgebras that we will use. Further information about Lie su-peralgebras can be found e.g. in [11, 23, 26]. For the purposes of this article, in [16] we listsimple Lie superalgebras of classical type.

A vector superspace V is a Z2-graded vector space V = V0 ⊕ V1, Z2 = {0, 1}. Theelements x ∈ V0 ∪ V1 are called homogeneous. Elements x ∈ V0 are called even, and wedefine the parity |x | of x by putting |x | = 0, while elements x ∈ V1\{0} are called odd,we write |x | = 1. All notions and constructions defined for the usual vector spaces can beextended to the vector superspaces, i.g. if V and W are vector superspaces, then their tensorproduct V ⊗ W is a vector superspace with the grading

(V ⊗ W )0 = (V0 ⊗ W0)⊕ (V1 ⊗ W1), (V ⊗ W )1 = (V0 ⊗ W1)⊕ (V1 ⊗ W0).

Linear maps F : V → W that preserve the parity are called morphisms.A superalgebra is a vector superspace A = A0 ⊕ A1 with a morphism · : A ⊗ A → A, i.e.

for homogeneous vectors it holds |x · y| = |x |+ |y|. An important example of a superalgebrais a Grassmann superalgebra

�(m) = �Rm = ⊕m

k=1�kR

m

with the obvious Z2-grading. The Grassmann superalgebra is super commutative, this meansthat its homogeneous elements satisfy xy = (−1)|x ||y|yx .

A Lie superalgebra is a superalgebra g = g0 ⊕g1 with the multiplication [·, ·] : g⊗g → g

such that the homogeneous elements satisfy

[x, y] = −(−1)|x ||y|[y, x],[[x, y], z] + (−1)|x |(|y|+|z|)[[y, z], x] + (−1)|z|(|x |+|y|)[[z, x], y] = 0.

In particular, g0 is a Lie algebra and g1 is a g0-module.A simple Lie superalgebra g is of classical type if the representation of g0 on g1 is totally

reducible, otherwise it is of the Cartan type. A simple Lie superalgebra g of classical type isof type I if the representation of g0 in g1 is irreducible and it is of type II if g1 is a direct sumof two irreducible g0-modules.

Consider several examples.Consider the vector superspace V = V0 ⊕ V1. The Lie algebra of all endomorphisms of

V is denoted by gl(V ). Its even part gl(V )0 = gl(V0) ⊕ gl(V1) consists of even endomor-phisms, i.e. of morphisms; the odd part of gl(V ) consists of odd endomorphisms, i.e. of theendomorphisms changing the parity and it holds gl(V )1 = V ∗

0⊗ V1 ⊕ V ∗

1⊗ V0.

Consider the vector superspace V = Rn|m = R

n ⊕�Rm . Here � is the parity changing

functor, in particular it is used to show that the vector space �Rm is purely odd. In this case

gl(V ) is denote by gl(n|m,R) (if V is a real vector superspace). In the matrix form we have

gl(n|m,R) ={(

A BC D

)},

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where

gl(n|m,R)0 ={(

A 00 D

)}, gl(n|m,R)1 =

{(0 BC 0

)}.

Consider the Lie superalgebra sl(n|m,R) = {ξ ∈ gl(n|m,R)| str ξ = 0}, where

str

(A BC D

)= tr A − tr D.

If n = m, then the Lie superalgebra sl(n|m) is simple of the classical type II.Let g be an even non-degenerate supersymmetric form on the vector superspace R

n|2m =R

n ⊕ �R2m , i.e. g(Rn,�R

2m) = g(�R2m,Rn) = 0, the restriction η of g to R

n is non-degenerate and symmetric (with some signature (p, q), p + q = n), and the restriction ωof g to R

2m is non-degenerate and skew-symmetric. Having such form, we denote Rn|2m

also by Rp,q|2m . The orthosymplectic Lie superalgebra is defined as the supersubalgebra of

gl(n|2m,R) preserving g,

osp(p, q|2m)i = {ξ ∈ gl(n|2m,R)i | g(ξ x, y)+ (−1)|x |i g(x, ξ y) = 0}, i ∈ Z2.

If p + q = 2, then the Lie superalgebra osp(p, q|2m) is simple of classical type I. In partic-ular, if the restriction of g to R

n is positive definite, choose a basis with respect to which the

matrix of g has the form

⎛⎝ 1n 0 0

0 0 1m

0 −1m 0

⎞⎠. Then,

osp(n|2m) =⎧⎨⎩

⎛⎝ A B1 B2

Bt2 C1 C2

−Bt1 C3 −Ct

1

⎞⎠

∣∣∣∣∣∣ At = −A,Ct2 = C2,Ct

3 = C3

⎫⎬⎭.

Similarly, the Lie superalgebra ospsk(2k|p, q), p + q = m, is defined as the supersubalge-bra of gl(2k|m,R) preserving an even non-degenerate supersymmetric form on the vectorsuperspace R

2k ⊕�Rm , in this case the restriction of g to R

2k is non-degenerate and skew-symmetric, and the restriction of g to R

m is non-degenerate and symmetric (with the signature(p, q), p + q = m). For example,

osp(2k|m) =⎧⎨⎩

⎛⎝ C1 C2 B1

C3 −Ct1 B2

Bt2 −Bt

1 A

⎞⎠

∣∣∣∣∣∣ At = −A,Ct2 = C2,Ct

3 = C3

⎫⎬⎭.

Suppose that n is even and suppose that we have an even non-degenerate supersymmetricform on the vector superspace R

n ⊕�R2m such that the restriction of g to R

n has signature(2p0, 2q0), 2p0 + 2q0 = n. Suppose that we have also an even complex structure J onR

n ⊕�R2m such that g(J x, J y) = g(x, y) for all x, y ∈ R

n ⊕�R2m . Note that the restric-

tion of g(J ·, ·) to R2m is symmetric and non-degenerate and let this form have the signature

(2p1, 2q1), p1 + q1 = m. By definition,

u(p0, q0|p1, q1) = {ξ ∈ osp(2p0, 2q0|2m)|[ξ, J ] = 0},su(p0, q0|p1, q1) = {ξ ∈ u(p0, q0|p1, q1)| str(J ◦ ξ) = 0}.

Similarly, suppose that n and 2m are integers divided by 4, m = 2k, and suppose that wehave an even non-degenerate supersymmetric form on the vector superspace R

n ⊕ �R2m

such that the restriction of g to Rn has the signature (4r, 4s), 4r + 4s = n. Suppose that we

have a quaternionic structure I, J, K on Rn ⊕�R

2m (i.e. I, J, K are even complex structures

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and they generate the Lie algebra isomorphic to sp(1)) such that g(I x, I y) = g(J x, J y) =g(K x, K y) = g(x, y) for all x, y ∈ R

n ⊕�R2m . By definition,

hosp(r, s|k) = {ξ ∈ osp(4r, 4s|2m)|[ξ, I ] = [ξ, J ] = [ξ, K ] = 0}.Note that the normalizer of sp(1) in osp(4r, 4s|2m) coincides with sp(1) ⊕ hosp(r, s|k).Remark also that hosp(r, s|k)0 ∩ sp(2m,R) = so∗(k), where so∗(k) is the subalgebra ofgl(k,H) acting on R

4k = Hk and preserving the skew-quaternionic-Hermitian form

ω(·, ·)+ iω(I ·, ·)+ jω(J ·, ·)+ kω(K ·, ·),here ω is the restriction of g to R

4k considered as a skew-symmetric bilinear form on R4k .

4 Supermanifolds and their holonomy groups

The introduction to the theory of supermanifolds can be found in [11, 26, 28, 39]. Riemanniansupermanifolds are considered e.g. in [3, 7, 9, 10, 19]. The holonomy theory is introduced in[13].

A real smooth supermanifold M of dimension n|m is a pair (M,OM), where M is aHausdorff topological space and OM is a sheaf of commutative superalgebras with unityover R locally isomorphic to

Rn;m = (Rn,O

Rn;m = ORn ⊗�(m)),

where ORn is the sheaf of smooth functions on Rn , and�(m) is the Grassmann superalgebra

of m generators. The sections of the sheaf OM are called superfunctions (or just functions)on M. The ideal (OM)1 ⊕ ((OM)1)

2 consists of the nilpotent elements of OM, and thesheaf OM defined as the quotient OM/((OM)1 ⊕ ((OM)1)

2) furnish M with the structureof a real smooth manifold. We get the canonical projection ∼: OM → OM , f �→ f . Thevalue of a superfunction f at a point x ∈ M is by definition f (x). If m = 0, then M = Mis a smooth manifold.

Denote by TM the tangent sheaf, i.e. the sheaf of superderivatives of the sheaf OM. Fora point x ∈ M define the tangent space

TxM = TM(U )/(OM(U )xTM(U )),

where U ⊂ M is any open subset containing x and OM(U )x is the ideal in OM(U )consisting of functions vanishing at the point x . The space TxM is a vector superspaceof the same dimension as M and its even part coincides with Tx M . Consider the tangentbundle T M = ∪x∈M TxM as a bundle of vector superspaces over M , its even part is thetangent bundle of M . We may consider also T M as a usual vector bundle over M . For asection X ∈ TM(U ), its value Xx ∈ TxM is defined in the obvious way. We get the naturalprojection ∼: TM(U ) → (T M,U ).

A connection on M is an even morphism ∇ : TM ⊗R TM → TM of sheaves of super-modules over R such that

∇ f Y X = f ∇Y X and ∇Y f X = (Y f )X + (−1)|Y || f | f ∇Y X

for all homogeneous functions f , vector fields X, Y on M, here | · | ∈ Z2 = {0, 1} denotesthe parity. In particular, |∇Y X | = |Y | + |X |. The curvature tensor of the connection ∇ isgiven by

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R(Y, Z) = [∇Y ,∇Z ] − ∇[Y,Z ], (2)

where Y and Z are vector fields on M.Obviously, the restriction

∇ = (∇|(T M)⊗(T M))∼ : (T M)⊗ (T M) → (T M)

is a connection on the vector bundle T M. Since ∇ is even, the subbundles T M =(T M)0, (T M)1 ⊂ T M are parallel. We obtain also a linear connection on M . Letγ : [a, b] ⊂ R → M be a curve and τγ : Tγ (a)M → Tγ (b)M the parallel displacementalong γ defined by the connection ∇.

The holonomy algebra hol(∇)x of the connection ∇ at a point x ∈ M is the supersubal-gebra of the Lie superalgebra gl(TxM) generated by the operators of the form

τ−1γ ◦ ∇r

Yr ,...,Y1Ry(Y, Z) ◦ τγ : TxM → TxM,

where γ is any piecewise smooth curve in M beginning at the point x ; y ∈ M is the end-pointof the curve γ , r ≥ 0, and Y, Z , Y1, . . . , Yr ∈ TyM.

Now we define the holonomy group. Recall that a Lie supergroup G = (G,OG) is a groupobject in the category of supermanifolds. The underlying smooth manifold G is a Lie group.The Lie superalgebra g of G can be identified with the tangent space to G at the identitye ∈ G. The Lie algebra of the Lie group G is the even part g0 of the Lie superalgebra g. AnyLie supergroup G is uniquely given by a pair (G, g) (Harish-Chandra pair), where G is aLie group, g = g0 ⊕ g1 is a Lie superalgebra such that g0 is the Lie algebra of the Lie groupG and a certain conditions hold [11, 19].

Denote by Hol(∇)0x the connected Lie subgroup of the Lie group GL((TxM)0) ×GL((TxM)1) corresponding to the Lie subalgebra (hol(∇)x )0 ⊂ gl((TxM)0)⊕gl((TxM)1)⊂ gl(TxM). Let Hol(∇)x be the Lie subgroup of the Lie group GL((TxM)0)×GL((TxM)1)

generated by the Lie groups Hol(∇)0x and Hol(∇)x . Clearly, the Lie algebra of the Lie groupHol(∇)x is (hol(∇)x )0. The Lie supergroup Hol(∇)x given by the Harish-Chandra pair(Hol(∇)x , hol(∇)x ) is called the holonomy group of the connection ∇ at the point x . TheLie supergroup Hol(∇)0x given by the Harish-Chandra pair (Hol(∇)0x , hol(∇)x ) is called therestricted holonomy group of the connection ∇ at the point x .

If M = M is a usual smooth manifold and ∇ is a connection on M , than Hol(∇)x andhol(∇)x coincide with the holonomy group and the holonomy algebra of (M,∇), respectively.

If the manifold M is simply connected, then the group Hol(∇)x is connected and thewhole information about Hol(∇)x gives us the holonomy algebra hol(∇)x .

A Riemannian supermanifold (M, g) is a supermanifold M of dimension n|2m endowedwith an even non-degenerate supersymmetric metric

g : TM ⊗OM TM → OM,

see e.g. [10]. In particular, the value gx of g at a point x ∈ M satisfies: gx ((TxM)0,

(TxM)1) = 0, gx |(Tx M)0×(Tx M)0is non-degenerate, symmetric and gx |(Tx M)1×(Tx M)1

isnon-degenerate, skew-symmetric. The metric g defines a pseudo-Riemannian metric g onthe manifold M . Note that g is not assumed to be positively defined. The supermanifold(M, g) has a unique linear connection ∇ such that ∇ is torsion-free and ∇g = 0. Thisconnection is called the Levi-Civita connection. We denote the holonomy algebra of theconnection ∇ by hol(M, g). Since g is parallel, hol(M, g) ⊂ osp(p, q|2m) and Hol(∇) ⊂O(p, q)× Sp(2m,R), where (p, q) is the signature of the pseudo-Riemannian metric g.

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We call a supersubalgebra g ⊂ osp(p, q|2m)weakly-irreducible if it does not preserve anynon-degenerate vector supersubspace of R

p,q ⊕�R2m . The following theorem generalizes

the Wu theorem [40].

Theorem 1 [13] Let (M, g) be a Riemannian supermanifold such that the pseudo-Riemannian manifold (M, g) is simply connected and geodesically complete. Then thereexist Riemannian supermanifolds (M0, g0), (M1, g1), . . . , (Mr , gr ) such that

(M, g) = (M0 × M1 × · · · × Mr , g0 + g1 + · · · + gr ), (3)

the supermanifold (M0, g0) is flat and the holonomy algebras of the supermanifolds(M1, g1), . . . , (Mr , gr ) are weakly irreducible. In particular,

hol(M, g) = hol(M1, g1)⊕ · · · ⊕ hol(Mr , gr ). (4)


5 Berger superalgebras

Let V = V0⊕V1 be a real or complex vector superspace and g ⊂ gl(V ) a supersubalgebra. Thespace of algebraic curvature tensors of type g is the vector superspace R(g) = R(g)0⊕R(g)1,where

R(g) =⎧⎨⎩R ∈ �2V ∗ ⊗ g

∣∣∣∣∣∣R(X, Y )Z + (−1)|X |(|Y |+|Z |)R(Y, Z)X

+(−1)|Z |(|X |+|Y |)R(Z , X)Y = 0for all homogeneous X, Y, Z ∈ V

⎫⎬⎭.

Here | · | ∈ Z2 denotes the parity. The identity that satisfy the elements R ∈ R(g) is calledthe first Bianchi super identity. Obviously, R(g) is a g-module with respect to the actionA · R = RA,

RA(X, Y ) = [A, R(X, Y )] − (−1)|A||R| R(AX, Y )− (−1)|A|(|R|+|X |)R(X, AY ), (5)

where A ∈ g, R ∈ R(g) and X, Y ∈ V are homogeneous.If M is a supermanifold, and ∇ is a linear torsion-free connection on the tangent sheaf

TM with the holonomy algebra hol(∇)x at some point x , then for the covariant derivatives ofthe curvature tensor, we have (∇r

Yr ,...,Y1R)x ∈ R(hol(∇)x ) for all r ≥ 0 and tangent vectors

Y1, . . . , Yr ∈ Tx M . Moreover, |(∇rYr ,...,Y1

R)x | = |Y1| + · · · + |Yr |, whenever Y1, . . . , Yr arehomogeneous.


L(R(g)) = span{R(X, Y )|R ∈ R(g), X, Y ∈ V } ⊂ g.

From (5) it follows that L(R(g)) is an ideal in g. We call a supersubalgebra g ⊂ gl(V ) aBerger superalgebra if L(R(g)) = g.

If V is a vector space, which can be considered as a vector superspace with the trivial oddpart, then g ⊂ gl(V ) is a usual Lie algebra, which can be considered as a Lie superalgebrawith the trivial odd part. Berger superalgebras in this case are the same as the usual Bergeralgebras.

Proposition 1 [13] Let M be a supermanifold of dimension n|m with a linear torsion-freeconnection ∇. Then its holonomy algebra hol(∇) ⊂ gl(n|m,R) is a Berger superalgebra.

123



R∇(g) =⎧⎨⎩S ∈ V ∗ ⊗ R(g)

∣∣∣∣∣∣SX (Y, Z)+ (−1)|X |(|Y |+|Z |)SY (Z , X)

+(−1)|Z |(|X |+|Y |)SZ (X, Y ) = 0for all homogeneous X, Y, Z ∈ V

⎫⎬⎭.

If M is a supermanifold and ∇ is a linear torsion-free connection on TM, then(∇r

Yr ,...,Y2,· R)x ∈ R∇(hol(∇)x ) for all r ≥ 1 and Y2, . . . , Yr ∈ Tx M . Moreover,|(∇r

Yr ,...,Y2,· R)x | = |Y2| + · · · + |Yr |, whenever Y2, . . . , Yr are homogeneous.

A Berger superalgebra g is called symmetric if R∇(g) = 0. This is a generalization of theusual symmetric Berger algebras, see e.g. [34], and the following is a generalization of thewell-known fact about smooth manifolds.

Proposition 2 [13] Let M be a supermanifold with a torsion free connection ∇. If hol(∇) is asymmetric Berger superalgebra, then (M,∇) is locally symmetric (i.e. ∇ R = 0). If (M,∇)is a locally symmetric superspace, then its curvature tensor at any point is annihilated bythe holonomy algebra at this point and its image coincides with the holonomy algebra.

A geometric theory of Riemannian symmetric superspaces is developed recently in [19].The proof of the following proposition is as in [34].

Proposition 3 Let g ⊂ gl(V ) be an irreducible Berger superalgebra of the form (1). If g

annihilates the module R(g), then g is a symmetric Berger superalgebra.

In [37] simply connected symmetric superspaces of simple Lie supergroups of isome-tries are classified. In particular this implies the classification of the holonomy algebras ofRiemannian symmetric superspaces and of irreducible Berger superalgebras g ⊂ osp

(p, q|2m) of the form (1). Hence we assume that the Riemannian supermanifolds underthe consideration are not locally symmetric.

6 The main results

Here is the main theorem of this article.

Theorem 2 Let (M, g) be a not locally symmetric Riemannian supermanifold of dimensionp + q|2m (p + q > 0) with an irreducible holonomy algebra hol(M, g) ⊂ osp(p, q|2m)that is of the form (1), then hol(M, g) ⊂ osp(p, q|2m) coincides with one of the Lie super-algebras from Table 1.

The Ricci tensor of a supermanifold (M, g) is defined by the formula

Ric(Y, Z) = str(X �→ (−1)|X ||Z | R(Y, X)Z), (6)

where U ⊂ M is open and X, Y, Z ∈ TM(U ) are homogeneous.The definitions of the supermanifolds considered below are similar to the usual ones, see

e.g. [10, 13]. Foe example, a Riemannian supermanifold (M, g) is called a Kählerian super-manifold if it admits an even parallel g-orthogonal complex structure. Since the holonomyalgebra annihilates the values of the parallel tensors [13], in this case it must be contained inu(p0, q0|p1, q1). By definition, a special Kählerian supermanifold or a Calabi-Yau super-manifold is a Ricci-flat Kählerian supermanifold.

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Table 1 Irreducible non-symmetric Berger supersubalgebras g ⊂ osp(p, q|2m)(p + q > 0) of the from (1)and the connected Lie subgroups G ⊂ SO(p, q)× Sp(2m,R) corresponding to g0 ⊂ so(p, q)⊕ sp(2m,R)

g G (p, q|2m)

osp(p, q|2m) SO(p, q)× Sp(2m,R) (p, q|2m)osp(p|2k,C) SO(p,C)× Sp(2k,R) (p, p|4k)u(p0, q0|p1, q1) U(p0, q0)× U(p1, q1) (2p0, 2q0|2p1 + 2q1)su(p0, q0|p1, q1) U(1)(SU(p0, q0)× SU(p1, q1)) (2p0, 2q0|2p1 + 2q1)hosp(r, s|k) Sp(p0, q0)× SO∗(k) (4r, 4s|4k)hosp(r, s|k)⊕ sp(1) Sp(1)(Sp(p0, q0)× SO∗(k)) (4r, 4s|4k)ospsk (2k|r, s)⊕ sl(2,R) Sp(2k,R)× SO(r, s)× SL(2,R) (2k, 2k|2r + 2s)ospsk (2k|r,C)⊕ sl(2,C) Sp(2k,C)× SO(r,C)× SL(2,C) (4k, 4k|4r)

Proposition 4 [13] Let (M, g) be a Kählerian supermanifold, then Ric = 0 if and only ifhol(M, g) ⊂ su(p0, q0|p1, q1).

Riemannian supermanifolds with the holonomy algebras osp(p, q|2m) are generic. Herewe give the geometric characterization of simply connected supermanifolds with the holon-omy algebras g different from osp(p, q|2m) (the conditions on the holonomy algebra andthe corresponding geometries are equivalent for simply connected supermanifolds):

g ⊂ osp(p|2k,C): holomorphic Riemannian supermanifolds;g ⊂ u(p0, q0|p1, q1): Kählerian supermanifolds;g ⊂ su(p0, q0|p1, q1): special Kählerian supermanifolds or Calabi-Yau supermanifolds;g ⊂ hosp(r, s|k): hyper-Kählerian supermanifolds;g ⊂ hosp(r, s|k)⊕ sp(1): quaternionic-Kählerian supermanifolds;g ⊂ ospsk(2k|r, s)⊕ sl(2,R): para-Kählerian supermanifolds;g ⊂ ospsk(2k|r,C)⊕ sl(2,C): holomorphic para-Kählerian supermanifolds.

Proposition 5 [13] Let (M, g) be a quaternionic-Kählerian supermanifold, then Ric = 0if and only if hol(M, g) ⊂ hosp(p0, q0|p1, q1). In particular, if (M, g) is hyper-Kählerian,then Ric = 0; if M is simply connected, (M, g) is quaternionic-Kählerian and Ric = 0,then (M, g) is hyper-Kählerian.

Now the natural problem is to construct examples of supermanifolds with each of theobtained possible holonomy algebras. Note that examples of special Kählerian manifolds(i.e. Calabi-Yau manifolds) delivers the Calabi-Yau Theorem. In [33] it is shown that theCalabi-Yau Theorem does not hold for Kählerian supermanifolds of real odd dimension two.In [41] it is shown that the arguments of [33] work only for the odd dimension two andit is conjectured that Calabi-Yau Theorem is true for manifolds of odd dimensions biggerthen two. In [1] some examples of Calabi-Yau supermanifolds are constructed. Examples ofquaternionic-Kählerian supermanifolds are constructed in [10].

The rest sections are dedicated to the proof of Theorem 2.

7 Weak-Berger algebras

Let g ⊂ so(p, q) = so(V ) be a subalgebra. Denote by η the pseudo-Euclidian metric onV = R

p,q . The vector space

Pη(g) ={

P ∈ V ∗ ⊗ g

∣∣∣∣ η(P(X)Y, Z)+ η(P(Y )Z , X)+ η(P(Z)X, Y ) = 0for all X, Y, Z ∈ V

}

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is called the space of weak-curvature tensors of type g. A subalgebra g ⊂ so(V ) is calleda weak-Berger algebra if h is spanned by the images of the elements P ∈ P(h). It is nothard to see that if R ∈ R(g) and x ∈ V is fixed, then R(·, x) ∈ Pη(g). In particular, anyBerger algebra is a weak-Berger algebra. The converse statement is not obvious, it is provedrecently in [25].

Theorem 3 Let g ⊂ so(p, q) be an irreducible weak-Berger subalgebra, then it is a Bergersubalgebra.

Remark that in the origin theorem g is a subalgebra of so(n). The above result immediatelyfollows from the complexification process described in [25].

The spaces Pη(g) for irreducible Berger subalgebras g ⊂ so(n) are found in [15]. Thisresult can be easily extended to the case of subalgebras g ⊂ so(p, q). In particular, it isproved that if g = g1 ⊕ g2 ⊂ so(V 1 ⊗ V 2) = so(V ), where g1 ⊂ gl(V 1) and g2 ⊂ gl(V 2)

are irreducible, then

Pη(g1 ⊂ so(V )) = Pη(g2 ⊂ so(V )) = 0,

unless the complexification of g ⊂ so(V ) coincides with

sp(2m,C)⊕ sl(2,C) ⊂ so(4m,C).

For example, for sp(m)⊕ sp(1) ⊂ so(4m) it holds

Pη(sp(1) ⊂ so(V )) = 0, Pη(sp(m) ⊂ so(V )) = (sp(2m,C) ⊂ sl(2m,C))(1).

In [15] it is shown that if g ⊂ so(n) is an irreducible subalgebra and Pη(g) = 0 orR(g) = 0, then either g is a Berger subalgebra, or g = sp( n

4 )⊕ RJ . A similar result holdsfor irreducible subalgebras g ⊂ so(p, q).

Let g ⊂ gl(L) be an irreducible subalgebra. Then g is a weakly-irreducible subalgebra ofso(L ⊕ L∗) = so(p, p), where p = dim L . Let η be the natural metric on L ⊕ L∗. It is easyto see that P ∈ Pη(g) if and only if

prgl(L) ◦P|L ∈ (g ⊂ gl(L))(1), prgl(L∗) ◦P|L ∈ (g ⊂ gl(L∗))(1).

Thus if g ⊂ so(L ⊕ L∗) is a weak-Berger subalgebra, then (g ⊂ gl(L))(1) = {0} and allg ⊂ gl(L) are known [8].

8 The case of Riemannian odd supermanifolds

Let (M, g) be a Riemannian supermanifold of dimension 0|2m, such supermanifolds arecalled odd. In this case for the holonomy algebra we get

g ⊂ osp(0|2m) � sp(2m,R),

i.e. g is a usual Lie algebra acting in a purely odd vector superspace. The possible irreducibleholonomy algebras of such supermanifolds are classified in [16].

Let V be a real or complex vector space and g ⊂ gl(V ) a subalgebra. The space ofskew-symmetric curvature tensors of type g is defined as follows

R(g) ={

R ∈ �2V ∗ ⊗ g

∣∣∣∣ R(X, Y )Z + R(Y, Z)X + R(Z , X)Y = 0for all X, Y, Z ∈ V

}.

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The subalgebra g ⊂ gl(V ) is called a skew-Berger subalgebra if it is spanned by the images ofthe elements R ∈ R(g). Obviously R(g) = R(g ⊂ gl(�V )) and g ⊂ gl(V ) is a skew-Bergeralgebra if and only if g ⊂ gl(�V ) is a Berger superalgebra.

Letω be the standard symplectic form on V = R2m . A subalgebra g ⊂ sp(2m,R) is called

weakly-irreducible if it does not preserve any proper non-degenerate subspace of R2m . The

Wu Theorem for supermanifolds implies the following statement. Let g ⊂ sp(2m,R) be anirreducible skew-Berger subalgebra, then there is a decomposition

V = V0 ⊕ V1 ⊕ · · · ⊕ Vr

into a direct sum of symplectic subspaces and a decomposition

g = g1 ⊕ · · · ⊕ gr

into a direct sum of ideals such that gi annihilates Vj if i = j and gi ⊂ sp(Vi ) is a weakly-irreducible skew-Berger subalgebra.

Irreducible skew-Berger subalgebras g ⊂ gl(n,C) are classified in [14]. Irreducible skew-Berger subalgebras g ⊂ sp(2m,R) are classified in [16].

Let g ⊂ sp(2m,R) = sp(V ) be a reductive weakly-irreducible subalgebra. Suppose thatg is not irreducible. As in Sect. 2, we may show that V is of the form V = L ⊕ L∗, whereg ⊂ gl(L) is irreducible. If g ⊂ sp(V ) is a skew-Berger subalgebra, then (g ⊂ gl(L))[1] ={0}, where

g[1] = {ϕ ∈ L∗ ⊗ g|ϕ(x)y = −ϕ(y)x for all x, y ∈ L}is the skew-symmetric prolongation of the subalgebra g ⊂ gl(L). Irreducible subalgebrasg ⊂ gl(L) with g[1] = 0 are classified in [16].

Let V be a complex or real vector space with a symplectic form ω. Let g ⊂ sp(V ) be asubalgebra. The vector space

Pω(g) ={

P ∈ V ∗ ⊗ g

∣∣∣∣ ω(P(X)Y, Z)+ ω(P(Y )Z , X)+ ω(P(Z)X, Y ) = 0for all X, Y, Z ∈ V

}

is called the space of skew-symmetric weak-curvature tensors of type g. A subalgebra g ⊂sp(V ) is called a skew-symmetric weak-Berger algebra if g is spanned by the images ofthe elements P ∈ Pω(g). It is not hard to see that if R ∈ R(g) and X ∈ V is fixed, thenR(·, X) ∈ Pω(g). In particular, any skew-Berger algebra is a skew-symmetric weak-Bergeralgebra. The converse statement gives the following theorem.

Theorem 4 Let g ⊂ sp(2m,R) be an irreducible skew-symmetric weak-Berger subalgebra,then it is a skew-Berger subalgebra.

The proof of this theorem is a modified copy of the proof from [25] of Theorem 3.If the representation g ⊂ sp(2m,R) is not absolutely irreducible, then Pω(g) is isomorphic

to (gC ⊂ gl(m,C))[1] and the proof follows from [14, 16].If the representation g ⊂ sp(2m,R) is absolutely irreducible, then we need a classification

of irreducible skew-symmetric weak-Berger subalgebras g ⊂ sp(2m,C). It can be achieve inthe same way as the classification of irreducible weak-Berger subalgebras g ⊂ so(n,C). Infact, in [25] it is obtained a necessary condition for an irreducible subalgebra g ⊂ so(n,C)to be a weak-Berger subalgebra, then there were classified all subalgebras g ⊂ gl(n,C)satisfying this condition and the subalgebras g ⊂ sp(2m,C) were noted. It is easy to seethat a skew-symmetric weak-Berger subalgebras g ⊂ sp(2m,C) satisfy the same necessarycondition. Thus the proof follows immediately. ��

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The spaces Pω(g) for irreducible weak-Berger subalgebras g ⊂ sp(2n,R) can be foundby methods of [15]. In particular, it is can be proved that if

g = g1 ⊕ g2 ⊂ sp(V 1 ⊗ V 2) = sp(V ),

where g1 ⊂ gl(V 1) and g2 ⊂ gl(V 2) are irreducible, then

Pω(g1 ⊂ sp(V )) = Pω(g2 ⊂ sp(V )) = 0,

unless the complexification of g ⊂ sp(V ) coincides with

so(n,C)⊕ sl(2,C) ⊂ sp(2n,C).

For example, for so(n,R)⊕ sl(2,R) ⊂ sp(2n,R) it holds

Pω(sl(2,R) ⊂ sp(V )) = 0, Pω(so(n,R) ⊂ so(V )) = (so(n,C) ⊂ sl(n,C))[1].

Let g ⊂ gl(L) be an irreducible subalgebra. Then g is a weakly-irreducible subalgebra ofsp(L ⊕ L∗) = sp(2m,R), where m = dim L . Let ω be the natural symplectic form L ⊕ L∗.It is easy to see that P ∈ Pω(g) if and only if

prgl(L) ◦P|L ∈ (g ⊂ gl(L))[1], prgl(L∗) ◦P|L ∈ (g ⊂ gl(L∗))[1].

Thus if g ⊂ sp(L ⊕ L∗) is a skew-symmetric weak-Berger subalgebra, then (g ⊂ gl(L))[1] ={0} and g ⊂ gl(L) is given in [16].

9 Structure of the spaces R(g)

Consider a subalgebra g ⊂ osp(p, q|2m) = osp(V ) and describe the space R(g). Denotethe supersymmetric metric on V by g. It can be represented as the sum g = η + ω, where ηis a pseudo-Euclidean metric of signature (p, q) on V0 = R

p,q and ω is a symplectic formon �V1 = R

2m . We identify osp(V ) with �2V , the element X ∧ Y ∈ osp(V ) is defined by

(X ∧ Y )Z = (−1)|Y ||Z |g(X, Z)Y − (−1)(|Y |+|Z |)|X |g(Y, Z)X,

where X, Y, Z ∈ V are homogeneous. Note that so(p, q) � �2V0 and sp(2m,C) � �2V1 =�2�V1. From the Bianchi super identity it follows that any R ∈ R(g) satisfies

g(R(X, Y )Z ,W ) = (−1)(|X |+|Y |)(|Z |+|W |)g(R(Z ,W )X, Y ) (7)

for all homogeneous X, Y, Z ,W ∈ V . This means that R : �2V → g ⊂ �2V is a super-symmetric map. In particular, R is zero on the orthogonal complement g⊥ ⊂ �2V .

First consider that space R(g)0. Let R ∈ (�2V ∗ ⊗ g)0. Define the following maps:

A = prso(p,q) g0 ◦ R|�2V0⊕�2V1: �2V0 ⊕�2V1 → prso(p,q) g0,

B = prR2m∗⊗Rp+q g1 ◦ R|V0⊗V1: V0 ⊗ V1 → prR2m∗⊗Rp+q g1,

C = prRp+q ∗⊗R2m g1 ◦ R|V0⊗V1: V0 ⊗ V1 → prRp+q ∗⊗R2m g1,

D = prsp(2m,R) g0 ◦ R|�2V0⊕�2V1: �2V0 ⊕�2V1 → prsp(2m,R) g0.

In the definition of B and C , we used the inclusion

osp(p, q|2m) ⊂ gl(p + q|2m,R)

= gl(p + q,R)⊕ gl(2m,R)⊕ R2m∗ ⊗ R

p+q ⊕ Rp+q ∗ ⊗ R

2m .

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Since R takes values in g ⊂ osp(p, q|2m), we obtain

ω(C(x, ξ)y, δ) = −η(y, B(x, ξ)δ) (8)

for all x, y ∈ V0 and ξ, δ ∈ V1, i.e. the maps B and C define each other. Extend the mapsA, B,C, D to �2V mapping the natural complements to zero. Then R = A + B + C + D.In the matrix form we may write

R(x, y) = −R(y, x) =(

A(x, y) 00 D(x, y)

), R(ξ, δ) = R(δ, ξ) =

(A(ξ, δ) 0

0 D(ξ, δ)

),

R(x, ξ) = −R(ξ, x) =(

0 B(x, ξ)C(x, ξ) 0

),

where x, y ∈ V0 and ξ, δ ∈ V1.Writing down the Bianchi identity, we get that R ∈ R(g)0 if and only if the following

conditions hold: A|�2V0∈ R (prso(p,q) g0), D|�2V1

∈ R(prsp(2m,R) g0),

D(x, y)ξ + C(y, ξ)x + C(ξ, x)y = 0, (9)

A(ξ, δ)x − B(δ, x)ξ + B(x, ξ)δ = 0 (10)

for all x, y ∈ V0 and ξ, δ ∈ V1.Suppose that R ∈ R(g)0. Using (7), we get

ω(D(x, y)ξ, δ) = η(A(ξ, δ)x, y), (11)

ω(C(x, ξ)y, δ) = −ω(C(y, δ)x, ξ), η(B(x, ξ)δ, y) = −η(B(y, δ)ξ, x) (12)

for all x, y ∈ V0 and ξ, δ ∈ V1. In particular, we see that A|�2V1and D|�2V0

define eachother. The meanings of the restrictions (9) and (10) on A|�2V1

and D|�2V0are not so clear.

On the other hand, if representation of g0 is diagonal in V0 ⊕ V1 (by this we mean that theboth representations of g0 on V0 and V1 are faithful), then A|�2V1

and D|�2V0are given by

D|�2V1and A|�2V0

, respectively. We will use this in many situations.

We turn now to the space R(g)1. Let R ∈ (�2V ∗ ⊗ g)1. Define the following maps:

A = prso(p,q) g0 ◦ R|V0⊗V1: V0 ⊗ V1 → prso(p,q) g0,

B = prR2m∗⊗Rp+q g1 ◦ R|�2V0⊕�2V1: �2V0 ⊕�2V1 → prR2m∗⊗Rp+q g1,

C = prRp+q ∗⊗R2m g1 ◦ R|�2V0⊕�2V1: �2V0 ⊕�2V1 → prRp+q ∗⊗R2m g1,

D = prsp(2m,R) g0 ◦ R|V0⊗V1: V0 ⊗ V1 → prsp(2m,R) g0.

Since R takes values in g ⊂ osp(p, q|2m), we obtain

ω(C(x, y)z, ξ) = −η(z, B(x, y)ξ), ω(C(ξ, δ)z, θ) = −η(z, B(ξ, δ)θ) (13)

for all x, y, z ∈ V0 and ξ, δ, θ ∈ V1. Thus the maps B and C define each other. Extend themaps A, B,C, D to�2V mapping the natural complements to zero. Then R = A+B+C+D.In the matrix form, we may write

R(x, y) = −R(y, x) =(

0 B(x, y)C(x, y) 0

), R(ξ, δ) = R(δ, ξ) =

(0 B(ξ, δ)

C(ξ, δ) 0

),

R(x, ξ) = −R(ξ, x) =(

A(x, ξ) 00 D(x, ξ)

),

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where x, y ∈ V0 and ξ, δ ∈ V1. Writing down the Bianchi identity, we get that R ∈ R(g)1 ifand only if the following conditions hold:

B(x, y)z + B(y, z)x + B(z, x)y = 0, (14)

C(ξ, δ)θ + C(δ, θ)ξ + C(θ, ξ)δ = 0, (15)

B(x, y)ξ + A(y, ξ)x + A(ξ, x)y = 0, (16)

C(ξ, δ)x − D(δ, x)ξ + D(x, ξ)δ = 0 (17)

for all x, y, z ∈ V0 and ξ, δ, θ ∈ V1. Let us fix ξ ∈ V1. Using (14), we get

η(B(x, y)ξ, z)+ η(B(y, z)ξ, x)+ η(B(z, x)ξ, y) = 0

for all x, y, z ∈ V0. Using this and (16), we conclude that for each fixed ξ ∈ V1 it holdsR(·, ξ) ∈ Pη(prso(p,q) g0). Similarly, for each x ∈ V0 it holds R(·, x) ∈ Pω(prsp(2m,R) g0).This will be extremely useful especially in the case when the representation of g0 or of someideal of g0 is diagonal in V0 ⊕ V1.

In [27] it is shown that

R(osp(p, q|2m)) � �2(�2V )/�4V . (18)

The osp(p, q|2m)-supermodule R(osp(p, q|2m)) is decomposed into the direct sum ofthree irreducible components. This generalizes the well-known decomposition of theso(p, q)-module R(so(p, q)) [2] and defines the decomposition of the elements R ∈R(osp(p, q|2m)) into the Weyl tensor, the trace-free part of the Ricci tensor and the scalarcurvature.

Let us compare (18) with the above description. For that we consider R(osp(p, q|2m))as an so(p, q)⊕ sp(2m,R)-module. It holds

�2V = �2V0 ⊕ (V0 ⊗ V1)⊕�2V1,

�2 (�2V ) = �2(�2V0)⊕

�2(V0 ⊗ V1)⊕

�2(�2V1)⊕

�2V0 ⊗ (V0 ⊗ V1)⊕�2V0 ⊗�2V1

⊕(V0 ⊗ V1)⊗�2V1,

�4V = �4V0

⊕�3V0 ⊗ V1

⊕�2V0 ⊗�2V1

⊕V0 ⊗�3V1

⊕�4V1.

This implies

R(osp(p, q|2m))0 � (�2(�2V )/�4V )0 = �2(�2V0)/�4V0⊕

�2(�2V1)/�4V1

⊕�2(V0 ⊗ V1).

Note that R(so(p, q)) � �2(�2V0)/�4V0 [2]. Similarly,

R(sp(2m,R)) � �2(�2�V1)/�4 �V1 = �2(�2V1)/�4V1.

We conclude that

R(osp(p, q|2m))0 � R(so(p, q))⊕ R(sp(2m,R))⊕ �2(V0 ⊗ V1).

Let us describe this isomorphism. The inclusions R(so(p, q)), R(sp(2m,R)) ⊂ R(osp(p,q|2m))0 are obvious. Let B ∈ �2(V0 ⊗ V1) = �2(V0 ⊗�V1), i.e. B is a skew-symmetricendomorphism of V0 ⊗�V1 with respect to the skew-symmetric form η⊗ω, and B satisfies

η ⊗ ω(B(x, ξ), y ⊗ δ) = −η ⊗ ω(B(y, δ), x ⊗ ξ).

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Note that this corresponds to (11). Equation 8 defines the element C and the Eqs. 9 and 10defines the restrictions D|�2V0

and A|�2V1. Put in addition A|�2V0

= 0 and D|�2V1= 0.

We obtain and element R ∈ R(osp(p, q|2m))0. This defines the inclusion �2(V0 ⊗ V1) ⊂R(osp(p, q|2m))0.

Next,

R(osp(p, q|2m))1 � V1 ⊗ (�2V0 ⊗ V0)/�3V0

⊕V0 ⊗ (�2V1 ⊗ V1)/�

3V1.

It holds Pη(so(p, q)) � (�2V0 ⊗ V0)/�3V0 [15]. Similarly, Pω(sp(2m,R)) � (�2�V1 ⊗

�V1)/�3 �V1. We obtain

R(osp(p, q|2m))1 � Pη(so(p, q))⊗ V1

⊕�Pω(sp(2m,R))⊗ V0.

Let P ∈ Pη(so(p, q)) and ζ ∈ V0 be fixed. Define R ∈ R(osp(p, q|2m))1 by putting

A(x, δ) = ω(δ, ζ )P(x), D(x, δ) = 0, B(x, y)ξ = ω(ζ, ξ)(P(x)y − P(y)x), B(ξ, δ) = 0.

In the same way any elements P⊗x ∈ Pω(sp(2m,R))⊗V0 define an R ∈ R(osp(p, q|2m))1.This gives the exact form of the obtained isomorphism.

Information about the spaces R(g) for some Lie superalgebras g ⊂ gl(n|k) not containedin osp(p, q|2m) can be found in [27, 32].

10 Adjoint representations of simple Lie superalgebras

Proposition 6 Let g be a simple (real or complex) Lie superalgebra admitting an even non-degenerate g-invariant bilinear supersymmetric form, i.e. such that the adjoint representationof g is orthosymplectic. Then R(g) = R(g)0 is one-dimensional and it is spanned by the Liesuperbrackets of g.

Proof First of all, from the Jacobi super identity it follows that [·, ·] ∈ R(g)0 for each simpleLie superalgebra. Note that the representation of g0 is diagonal in g0 ⊕ g1 (up to the centerof g0, which does not play a role).

First we prove that R(g)1 = 0. Suppose that g0 contains at least two simple ideals h1 andh2. Let R ∈ R(g)1, then for each fixed ξ ∈ g1 and any x ∈ h1 we have R(x, ξ) ∈ h1. On theother hand, for each fixed x ∈ h1 we have prg0

◦R(x, ·) ∈ Pω(prsp(g1)g0), but the g0-module

g1 is a tensor product of irreducible representations of simple ideals in g0 (if g is of type I)and it is a direct sum of two such representations (if g is of type II). This and Sect. 8 show thatprg0

◦R(x, ·) can not take values in a one simple ideal of g0 (unless g0 or its complexificationcoincides with so(n,C) ⊕ sl(2,C), i.e. if g = osp(p, q|2) or g = osp(p|2,C), these casescan be considered in the same way as g = osp(1|2m,R) below and we get R(g)1 = 0).We have prg0

◦R(x, ·) = 0 for all x ∈ h1. Similarly, prg0◦R(x, ·) = 0 for all x ∈ h2 and

prg0◦R(x, ·) = 0 for all x ∈ g0, i.e. R = 0.

Suppose thatthe semi-simple part of g0 is simple, then g = osp(1|2m,F) or g =osp(2|2m,F), F = R or C (for other simple g such that the semi-simple part of g0 is simple,the adjoint representation of g is not orthosymplectic).

Consider the case g = osp(1|2m,F), the case g = osp(2|2m,F) is similar. Since the com-plexification of the adjoint representation of g = osp(1|2m,R) is irreducible, it is enoughto consider the adjoint representation of g = osp(1|2m,C). Let R ∈ R(g)1. Then for eachx ∈ g0 = sp(2m,C) it holds prg0

◦R(x, ·) ∈ Pω(sp(2m,C)), and for each ξ ∈ g1 = C2m it

123


holds prg0◦R(·, ξ) ∈ Pη(adsp(2m,C)). That is R(g)1 is contained in the diagonal form in the

sp(2m,C)-module

(C2m ⊗ Pη(adsp(2m,C)))⊕ (sp(2m,C)⊗ Pω(sp(2m,C))).

Suppose that m ≥ 2. In [15] we prove that Pη(adsp(2m,C)) � sp(2m,C) and any P ∈Pη(adsp(2m,C)) is of the form P(·) = [x, ·], where x ∈ sp(2m,C). Note that Pω(sp(2m,C))contains a submodule isomorphic to C

2m and any element P of this module is of the formP(·) = ξ � · for some ξ ∈ C

2m , here for ξ, δ ∈ C2m the element ξ � δ ∈ sp(2m,C) is

defined by

(ξ � δ)θ = ω(ξ, θ)δ + ω(δ, θ)ξ.

We conclude that R(g)1 is contained in the diagonal form in the sp(2m,C)-module

(C2m ⊗ sp(2m,C))⊕ (sp(2m,C)⊗ C2m).

Moreover for each R ∈ R(g)1 there exist linear maps ϕ : C2m → sp(2m,C) and ψ :

sp(2m,C) → C2m such that R(x, ξ) = [ϕ(ξ), x] = ψ(x) � ξ for all ξ ∈ C

2m and x ∈sp(2m,C). Since R(g)1 is an sp(2m,C)-module, ϕ and ψ are proportional as the elementsof C

2m ⊗ sp(2m,C). To show that the equation [ϕ(ξ), x] = ψ(x)� ξ for all ξ ∈ C2m and

x ∈ sp(2m,C) has only the trivial solution it is enough to decompose the sp(2m,C)-moduleC

2m ⊗ sp(2m,C) into the direct some of irreducible components and to check this equationfor a non-zero representative of each component. We have

C2m ⊗ sp(2m,C) = V3π1 ⊕ Vπ1+π2 ⊕ C

2m .

Let (ξα)α=−m,...,−1,1,...,m be the standard basis of C2m , i.e. ω(ξα, ξβ) = δα,−β . Then

sp(2m,C) is spanned be the elements of the form ξα � ξβ . Let

ϕ = cψ = ξ1 ⊗ ξ1 � ξ1 ∈ V3π1 .

Substituting to the equation ξ = ξ−1 and x = ξ−1 � ξ1, we get

0 = [ξ1 � ξ1, ξ−1 � ξ1].On the other hand,

[ξ1 � ξ1, ξ−1 � ξ1] = 2ξ1 � ξ1 = 0.

Hence ϕ = cψ = ξ1 ⊗ ξ1 � ξ1 is not a solution of the equation. Similarly, taking

ϕ = cψ =∑α

ξα ⊗ ξ−α � ξ1 ∈ C2m, ξ = ξ1, x = ξ1 � ξ1,

we get that ϕ = cψ = ∑α ξα ⊗ ξ−α � ξ1 is not a solution of the equation. Finally, the

sp(2m,C)-module C2m ⊗ sp(2m,C) contains the weight space of the weight π1 + π2 of

dimension 2 and this space consists of the vectors

c1ξ1 ⊗ ξ1 � ξ2 + c2ξ2 ⊗ ξ1 � ξ1, c1, c2 ∈ C.

Let ϕ = cψ be equal to such vector. Taking ξ = ξ−1 and x = ξ−2 � ξ1, we get c1 = 0;taking ξ = ξ−2 and x = ξ−1 � ξ2, we get c2 = 0. Thus we may conclude that R(g)1 = 0. Ifm = 1, then C

2 ⊗ sp(2,C) = V3π1 ⊕ C2. It is not hard to see that Pω(sp(2,C)) = C

2. Andthe further proof is the same.

Next we prove that R(g)0 is one-dimensional.

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Let R ∈ R(g)0 be given as above by the linear maps A, B,C, D. We have seen thatA|�2V0

(or D|�2V1) defines uniquely A and D. We claim that it defines the whole R. Indeed,

suppose that A = 0 and D = 0. Let ξ, δ ∈ g1 and x ∈ g0. We have ξ · R(x, δ) = [ξ, R(x, δ)].If R(x, δ) = 0, then since g is simple, there exists a ξ such that ξ · R(x, δ) = 0. On the otherhand, ξ · R ∈ R(g)1 = 0 and we get a contradiction, this proves the claim.

If the semi-simple part of g0 is simple, then A|�2V0being an element in R(adg0

) is pro-portional to the Lie brackets in g0 [34]. Since A|�2V0

defines uniquely R ∈ R(g)0, we getthat R(g)0 is one-dimensional.

Suppose that g is of type I and the semi-simple part of g0 is not simple, then R(g0 ⊂sp(g1)) is one-dimensional (Sect. 8). Hence D|�2V1

belongs to a one-dimensional space.Since D|�2V1

defines uniquely R ∈ R(g)0, we get that R(g)0 is one-dimensional.Finally suppose that g is of type II and the semi-simple part of g0 is not simple, i.e.

g0 = h1 ⊕h2 ⊕ z. Then obviously A|h1⊗h2 = 0, A|�2h1= c1[·, ·]h1 , and A|�2h2

= c2[·, ·]h2

are annihilated by g0. Then D|�2V1belongs to a two-dimensional space annihilated by g0. On

the other hand, g0 may annihilate only a one-dimensional subspace in R(g0 ⊂ sp(g1)). Hencethere exists a c ∈ C such that for each R it holds c1 = c c2. Thus R(g)0 is one-dimensional.The proposition is proved. ��

11 Proof of Theorem 2

Lemma 1 Let g ⊂ osp(p, q|2m) be an irreducible subalgebra of the form (1). If R(g)1 = 0,then R(g) is a trivial g-module.

Proof We have (gi )1 · R(g)0 ⊂ R(g)1 = 0. Since each gi is simple of classical type, itholds (gi )0 = [(gi )1, (gi )1]. Consequently, (gi )0 · R(g)0 = 0. Suppose that z = 0. Sinceosp(p, q|2m)0 ∩ q(2m,R) = 0, by the Schur Lemma z = RJ , where J is an even complexstructure on V . It is not hard to see that J · R(g) = 0. ��First we consider case by case simple real Lie superalgebras g of classical type (they areclassified in [31] and we list them for the convenience in [16]). For each g we find all irre-ducible representations g ⊂ osp(p, q|2m) = osp(V ) such that g is a non-symmetric Bergersupersubalgebra. We explain the way of the considerations and then give several examplesdemonstrating this proof.

We begin with the case when g0 is of the form h1 ⊕h2 ⊕ z, where h1 and h2 are simple andz is trivial or one-dimensional. If g is of type I, i.e. the g0-module g1 is irreducible, then g1 isof the form W1 ⊗ W2, h1 ⊂ so(W1) and h2 ⊂ sp(W2) are irreducible. If g is of type II, i.e.the g0-module g1 is of the form g−1 ⊕ g1, where g−1 and g1 are irreducible g0-modules, thenthere are two vector spaces U1 and U2 such that h1 ⊂ gl(U1), h2 ⊂ gl(U2) are irreducible,g−1 = U∗

1 ⊗ U2, and g1 = U∗2 ⊗ U1.

Consider several cases:Case a. h1 annihilates V1.Suppose that g is of type I. Since the inclusion i : g ↪→ osp(p, q|2m) is a Lie superalgebra

homomorphism, the restriction

i |g1: g1 = W1 ⊗ W2 → osp(V )1 = V0 ⊗ V1

is g0-equivariant. In particular, it is h1-equivariant. Since h1 annihilates W2 and V1, weconclude that V0 is a direct sum of h1-submodules isomorphic to W1 and of an h1-trivialsubmodule. Similarly, if g is of type II, then V0 is a direct sum of h1-submodules isomorphicto U1 ⊕ U∗

1 and of an h1-trivial submodule.

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Under the current assumption we have three cases:

Case a.1. h2 annihilates V0.Suppose that g is of type I. By the above arguments, V1 is a direct sum of h2-submodules isomorphic to W2 and of an h2-trivial submodule. From Sects. 7and 8, we read that if V0 contains more then one h1-submodule isomorphic toW1 and if V1 contains more then one h2-submodule isomorphic to W2, thenPη(h1 ⊂ so(V0)) = 0 and Pω(h2 ⊂ so(V1)) = 0. Consequently, R(g)1 = 0 andg is symmetric. Thus either V0 contains exactly one h1-submodule isomorphic toW1, or V1 contains exactly one h2-submodule isomorphic to W2. Similarly, if g

is of type II, then V0 contains exactly one h1-submodule isomorphic to U1 ⊕ U∗1

or V1 contains exactly one h2-submodule isomorphic to U2 ⊕U∗2 . Next we check

when such representation of g exists. For this we may pass to the complexificationof g and of its representation. If the resulting representation is not irreducible, wetake one of its irreducible components. Note that the type of g may change.Let g be of type I. Suppose, for instance, that V1 contains exactly one h2-submod-ule isomorphic to W2. Since g1 ⊗W2 contains only one submodule annihilated byh2 and isomorphic to W1 as the h1-module and since the representation of g1 onV is g0-equivariant, g preserves the vector supersubspace (g1 · V1)⊕ V1 ⊂ V andits even part contains only one h1-submodule isomorphic to W1. From the irre-ducibility of V it follows that V0 contains exactly one h1-submodule isomorphicto W1 and V1 contains exactly one h2-submodule isomorphic to W2. Supposethat V contains a non-trivial vector v annihilated by g0. Then the homogeneouscomponents of v are also annihilated by g0 and we may assume that v is homo-geneous. Suppose that v ∈ V0. Consider the map from g1 to V1 sending ϕ ∈ g1to ϕv. This map is g0-equivariant and since the g0-modules g1 and V1 are notisomorphic, this map is zero. This shows that Rv ⊂ V is an invariant subspaceand it must be trivial. Thus we get that V0 = W1 and V1 = W2. Such a represen-tation of g is either the identity one, or it does not exist. Let g be of type II. Bythe similar arguments we get that V0 = U1 and V1 = U2. This happens e.g. forthe complexification of the identity representation of su(p0, q0|p1, q1).

Case a.2. h2 annihilates V1.In this case h1 ⊕h2 annihilates V1. Let U ⊂ V0 be an irreducible h1 ⊕h2-module.Since U is not g-invariant, g1 · U = 0. On the other hand, g1 · U ⊂ V1 is annihi-lated by h1 ⊕ h2, i.e. g1 ⊗U contains a one-dimensional subspace annihilated byh1 ⊕h2. This may happens only if U � g1 (if g is of type I), or if U is isomorphiceither to g−1, or to g1 (if g is of type II). We show that such representations donot exist.

Case a.3. The representation of h2 is diagonal in V0 ⊕ V1.We may decompose V0 as the direct sum V0 = L1 ⊕ L2 ⊕ L3 ⊕ L4 such thath1 ⊕ h2 annihilates L4, h1 annihilates L2, h2 annihilates L1, L1 is a direct sumof h1-submodules isomorphic to W1 (resp. U1 ⊕ U∗

1 ), L2 is an h2-submodule,and L3 is an h1 ⊕ h2-submodule (such that each irreducible component of L3

is faithful for both h1 and h2). If L3 = 0, then from Sect. 7 it follows thatPη(prso(L1⊕L2)

g0) = 0 and this implies R(g)1 = 0. Thus, L3 = 0. Since g is aBerger algebra, one of the following holds:

1. There exists an R ∈ R(g)0 such that for some x, y ∈ L2 it holds 0 = R(x, y) ∈h2. Expressing R in terms of the maps A, B,C, D as above, we get A|�2 L2

= 0 and

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D|�2 L2= 0. Hence A|�2V1

= 0 and D|�2V1= 0. This implies that h2 ⊂ so(L2) is a

Berger subalgebra and h2 ⊂ sp(V1) is a skew-Berger subalgebra.2. There exists an R ∈ R(g)0 such that for some ξ, δ ∈ V1 it holds 0 = R(ξ, δ) ∈ h2. This

case is similar to Case 1 and we get the same conclusion.3. There exists an R ∈ R(g)1 such that for some x ∈ L2 and ξ ∈ V1 it holds 0 = R(x, ξ) ∈

h2. This shows that h2 ⊂ so(L2) is a Berger subalgebra and h2 ⊂ sp(V1) is a skew-Bergersubalgebra.Thus L2 is an irreducible h2-module or L2 = L ⊕ L∗, where h2 ⊂ gl(L) is irreducible.The same holds for V1. Next, as in Case a.1, we show that from the irreducibility ofg ⊂ osp(V ) it follows that L2 = 0, and we get a contradiction.

Case b. h1 annihilates V0.This case is analogous to Case a. Note that if g is of type I, then in this caseh1 ⊂ sp(V1) and V1 is a direct sum of h1-submodules isomorphic to W1 and of ah1-trivial submodule. Since h1 ⊂ so(W1), we get that h1 ⊂ su(W1).As above we may consider Cases b.1, b.2, b.3.

We are left with the following case:Case c. The representations of h1 and h2 are diagonal in V0 ⊕ V1.

We may decompose V0 as the direct sum V0 = L1⊕L2⊕L3⊕L4 such that h1⊕h2 annihilatesL4, h1 annihilates L2, h2 annihilates L1, L1 is an h1-submodule, L2 is an h2-submodule, andL3 is an h1 ⊕ h2-submodule. Let V0 = L ′

1 ⊕ L ′2 ⊕ L ′

3 ⊕ L ′4 be the similar decomposition.

Since R(g)1 = 0, we get that Pη(prso(V0)g0) = 0 and Pω(prsp(V1)

g0) = 0. This showsthat if L1 = 0, then L3 = 0. Moreover L2 = 0, since the representation of h1 is diagonalin V0 ⊕ V1. Thus either L3 = 0, or L1 = 0 and L2 = 0. Furthermore, if L3 = 0, thenh1 ⊕ h2 ⊂ so(L3) is a Berger subalgebra (which is irreducible or L3 is of the form U ⊕ U∗and h1 ⊕ h2 ⊂ gl(U ) is irreducible). Similarly, if L1 = 0 and L2 = 0, then h1 ⊂ so(L1) andh2 ⊂ so(L2) are Berger subalgebras. The same statements we get for V1 (instead of Bergersubalgebras we get skew-Berger subalgebras). If L1 = 0, L2 = 0, L ′

1 = 0, and L ′2 = 0 then

we show in the same way as in Case a.1 that the representation is not irreducible. If L1 = 0,L2 = 0, and L ′

3 = 0, then we show as for the adjoint representations that R(g)1 = 0. IfL3 = 0, and L ′

3 = 0, then either h1 ⊕h2 does not appear as a Berger or skew-Berger algebra,or h1 ⊕ h2 appears only as a reducible Berger and a reducible skew-Berger algebra. In thelast case, the representation of g is not irreducible.

Note that in each case we get a decomposition of V into irreducible g0-modules. Thenwe ask if such representation of g exists, in other words, we should check if the obtainedrepresentation of g0 can be extended to an irreducible representation of g. This can be done bypassing to the complex case. Then we may use the theory of representations of the complexsimple Lie superalgebras [12, 20, 21, 23, 24, 30, 35, 36, 38]. Any irreducible representationg ⊂ gl(V ) is the highest weight representation V� and the weight� is given by its labels onthe Kac-Dynkin diagram of g. There is a way to decompose the g0-module V� into irreduciblecomponents. One g0-module V� is obtained directly from �. Then V� must coincide withone of the irreducible g0-modules obtained by us. The weight � defines uniquely � and weneed only to check that V� consists exactly of the irreducible g0-modules obtained by us.We will demonstrate this technics in the examples below.

Example 1 Let g be the real form of the complex simple Lie superalgebra F(4) with g0 =sl(2,R)⊕so(7,R) and g1 = R

2 ⊗�, where� � R8 is the spinor representation of so(7,R).

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Case a.1. We have V0 = � and V1 = R2. Note that V0 ⊗ V1 = osp(8|2,R)1, hence

[V0, V1] = sl(2,R) ⊕ so(8,R). This shows that the representation of g on R8|2

does not exist.

Case a.2. We consider it after Case a.3.Case a.3. We have V0 = � ⊕ L , where sl(2,R) ⊂ so(L) is an irreducible Berger subal-

gebra, and V1 = R2. As in Case a.1, such representation does not exist. We may

prove it also in another way. First since sl(2,R) ⊂ so(L) is an irreducible Bergeralgebra, the only possible L are R

3 and R5. Passing to the complexification we

get that as g0 = sl(2,C) ⊕ so(7,C)-modules, V0 = C8 ⊕ L , where C

8 is thespinor representation of so(7,C), L is either C

3, or C5, and V1 = �C

2. Notethat neither g1 ⊗ C

8, nor g1 ⊗ C2 contain any of the g0-modules C

3 and C5.

This means that g1 · (C8 ⊕ �C2) ⊂ C

8 ⊕ �C2, i.e. the vector supersubspace

C8 ⊕�C

2 ⊂ V is g-invariant, hence L = 0. In fact, the method of the decompo-sition of a g0-module V into irreducible components discussed above is foundedon the fact that if U ⊂ V is an irreducible g0-module, then g1 takes it into someirreducible components of the tensor product g1 ⊗ U .Let us show how the above discussed method can be applied to our representa-tions. The information about the irreducible representations of F(4) can be foundin [35]. Any irreducible representation (with the highest weight �) of F(4) isgiven by the labels (a1, a2, a3, a4) on the Kac-Dynkin diagram. Define the fol-lowing number b = 1

3 (2a1 −3a2 −4a3 −2a4). The labels must satisfy the condi-tions: b, a2, a3, a4 are non-negative integers; if b = 0, then a1 = · · · = a4 = 0;b = 1; if b = 2, then a2 = a4 = 0; if b = 3, then a2 = 2a4 + 1. The weight� is given by the labels (b, a2, a3, a4) on the Dynkin diagram of the Lie alge-bra sl(2,C) ⊕ so(7,C). In our case � must be one of (0, 0, 0, 1), (1, 0, 0, 0),(2, 0, 0, 0), (4, 0, 0, 0). The first two cases do not satisfy the conditions on thelabels. The third case corresponds to the adjoint representation, which is differentfrom our ones. In the second case V contains a g0-module with the highest weight(3, 1, 0, 0), while our representations do not contain such submodule.

Coming back to Case a.2, we get that in this case the representation is given by b = 1,a2 = 0, a3 = 0, and a4 = 1. But the representations of F(4) with b = 1 do not exist.

Case b. Does not appear, since the representation of so(7,R) in � is not unitary.

Case c. Section 2 and the papers [5, 16] show that the only representation of sl(2,R) ⊕so(7,R) as a skew-Berger algebra is in the space R

2⊗�, and Lie algebra sl(2,R)⊕so(7,R) does not appear as the Berger subalgebra of so(p, q). Thus, as we haveseen, in this case R(g)1 = 0.

Example 2 Let g be the real form of the complex simple Lie superalgebra osp(4|2, α,C)withg0 = sl(2,R)⊕sl(2,R)⊕sl(2,R) and g1 = R

2⊗R2⊗R

2. Suppose that g ⊂ osp(p, q|2m) =osp(V ) is an irreducible Berger supersubalgebra. We may consider several cases.

First suppose that the representation of none of the Lie algebras sl(2,R) is diagonalin V0 ⊕ V1. Using the fact that the representation of sl(2,R) in R

2 is symplectic and thearguments of Case a.1, we get that V1 = R

2 as the sl(2,R)-module, and V0 = R2 ⊗ R

2

as the sl(2,R) ⊕ sl(2,R)-module. To analyze such representations we turn to the complexcase. Then we may use the theory of representations of the complex simple Lie superal-gebras osp(4|2, α,C) [38]. Any representation of osp(4|2, α,C) is given by the numbers

123


(a2, a1, a3) (the labels on the Kac-Dynkin diagram of osp(4|2, α,C)) such that a2, a3 andthe number b = 1

1+α (2a1 − a2 − αa3) are non-negative integers. Furthermore, if b = 0,then a1 = a2 = a3 = 0; if b = 1, then α(a3 + 1) = ±(a2 + 1). In [38] it is shown that Vcontains the following sl(2,C)⊕ sl(2,C)⊕ sl(2,C)-submodules: (b, a2, a3), (b − 1, a2 ±1, a3 ± 1), (b − 1, a2 ± 1, a3 ∓ 1), (b − 2, a2 ± 2, a3), (b − 2, a2, a3 ± 2), (b − 2, a2, a3),(b − 3, a2 ± 1, a3 ± 1), (b − 3, a2 ± 1, a3 ∓ 1), (b − 4, a2, a3) (the representations are givenby the labels on the Dynkin diagram of sl(2,C)⊕ sl(2,C)⊕ sl(2,C)). In our case (b, a2, a3)

is one of (1, 0, 0), (0, 1, 0), (0, 0, 1), (0, 1, 1), (1, 1, 0), (1, 0, 1). In the first case α = 1 andwe get the identity representation of osp(4|2,R); the second, the third and the forth cases arenot possible; in the last two cases V contains the representations (0, 2, 1) and (0, 1, 2) thatgive the contradiction. Thus the only possible representation is the identity representation ofosp(4|2,R).

Next we suppose that a number of the representation of the Lie algebras sl(2,R) are diag-onal in V0 ⊕ V1. By the same arguments as in Cases a.3 and b, we show that if g is a Bergersuperalgebra, then it is symmetric.

The representations of the simple Lie superalgebras g such that the semi-simple partof g0 is simple can be considered in the same way. The situation becomes simpler, sincethe representation of g0 is diagonal in V0 ⊕ V1 (except for the identity representations ofosp(1|2m,R) and osp(2|2m,R)). We immediately conclude that prso(V0)

g0 ⊂ so(V0) is aBerger subalgebra and prsp(V1)

g0 ⊂ sp(V1) is a skew-Berger subalgebra.

Example 3 Consider the Lie superalgebra g = pe(n,R). Recall that g0 = sl(n,R) and g1 =�2

Rn ⊕�2

Rn∗. Suppose that g ⊂ osp(p, q|2m) = osp(V ) is an irreducible non-symmetric

Berger algebra. Then prso(V0)g0 ⊂ so(V0) is a Berger algebra and prsp(V1)

g0 ⊂ sp(V1) is askew-Berger algebra. Section 2 and the results from [5, 16] show that V0 and V1 should becontained in the following list:sl(n,R), R

n ⊕ Rn∗, �2

Rn ⊕ �2

Rn∗, �2

Rn ⊕�2

Rn∗,

�3R

6 (n = 6), and �4R

8 (n = 8).To study these representations we turn to the complexification. Suppose, for example, thatV0 = C

n ⊕ Cn∗. The g0-submodule g1 · C

n must coincide with an irreducible componentfrom the list obtaining by the complexification of the above one. We have

g1 ⊗ Cn = V3π1 ⊕ Vπ1+π2 ⊕ Vπ1+πn−2 ⊕ C

n∗.

Hence, g1 · Cn = C

n∗ ⊂ V1. The tensor product g1 ⊗ Cn∗ does not contain C

n∗. This meansthat the vector supersubspace C

n ⊕ Cn∗ (where C

n∗ ⊂ V1) of V is g-invariant and we get acontradiction. All the other representatives from the above list can be considered in a similarway.

Thus we conclude that if g is a simple real Lie superalgebra and there exists an irreduciblerepresentation g ⊂ osp(p, q|2m) such that g is a non-symmetric Berger subalgebra, then thisrepresentation is the identity one of g and g is one of the following Lie algebras with theiridentity representations: osp(p, q|2m), osp(p|2m,C), su(p0, q0|p1, q1) and hosp(p, q|m).These Lie superalgebras with their identity representations are non-symmetric Berger super-algebras, since they contain, respectively, the subalgebras so(p, q), so(p,C), su(p0, q0) andhosp(p, q), which are non-symmetric Berger algebras.

Suppose that g is a simple real Lie superalgebra and there exists an irreducible repre-sentation g ⊂ osp(p, q|2m) such that g ⊕ z is a non-symmetric Berger supersubalgebra,where z is a supersubalgebra of osp(p, q|2m) commuting with g. Since g is not containedin q(n,R), i.e. it does not commute with an odd complex structure, by the Schur Lemma for

123


representations of Lie superalgebras, z is either RJ , where J is an even complex structure,or z = sp(1), i.e. z is spanned by an even quaternionic structure J1, J2, J3. From Sect. 2 itfollows that if h ⊂ so(p, q) is an irreducible subalgebra, then if h ⊕ z is a non-symmetricBerger algebra, then h is a non-symmetric Berger algebra. Similarly, from [16] it followsthat if h ⊂ sp(2m,R) is an irreducible subalgebra then if h ⊕ z is a skew-Berger algebra,then h is a skew-Berger algebra. The same note holds for representations in U ⊕ U∗, whereh ⊂ sl(U ) is irreducible. This shows that the above method can be applied also to irreduciblesubalgebras g ⊕ z ⊂ osp(p, q|2m), where g is simple. We obtain the identity representationsof u(p0, q0|p1, q1), hosp(p, q|m)⊕ RJ and hosp(p, q|m)⊕ sp(1). Since u(p0, q0|p1, q1)

contains u(p0, q0), it is a non-symmetric Berger superalgebra. Since

R(sp(p, q)⊕ RJ ) = R(sp(p, q)), R(so(m,H)⊕ RJ ) = R(so(m,H)),

and J acts diagonally in V0 ⊕ V1, we get that

R(hosp(p, q|m)⊕ RJ ) = R(hosp(p, q|m)),i.e. hosp(p, q|m) ⊕ RJ is not a Berger superalgebra. Generalizing the curvature tensor ofthe quaternionic projective space (with indefinite metric) [2], we define the curvature tensorR ∈ R(hosp(p, q|m)⊕ sp(1)) by

R(X, Y ) = 1

2

3∑α=1

g(JαX, Y )Jα − 1

4

(X ∧ Y +

3∑α=1

JαX ∧ JαY),

where X, Y ∈ V . The restriction of R to �2R

4p,4q coincides with the curvature tensor ofthe quaternionic projective space, its image is not contained in sp(p, q). This shows thatR ∈ R(hosp(p, q|m)). Thus,

R(hosp(p, q|m)) = R(hosp(p, q|m)⊕ sp(1))

and hosp(p, q|m)⊕ sp(1) is a non-symmetric Berger superalgebra.Let now g1 ⊂ gl(V 1) and g2 ⊂ gl(V 2) be two irreducible real supersubalgebras. Con-

sider the tensor product of these representations g = g1 ⊕ g2 ⊂ gl(V 1 ⊗ V 2) = gl(V ).Suppose that g ⊂ osp(V ). Note that if the real vector superspaces V 1 and V 2 admit complexstructures commuting, respectively, with the elements of g1 and g2, then the representationof g = g1 ⊕ g2 in V 1 ⊗ V 2 is reducible, and we consider its representation in V 1 ⊗C V 2.

For the even and odd parts of V 1 ⊗ V 2, we have

(V 1 ⊗ V 2)0 = V 10

⊗ V 20

⊕ V 11

⊗ V 21, (V 1 ⊗ V 2)1 = V 1

0⊗ V 2

1⊕ V 1

1⊗ V 2

0.

This shows that if the even and odd parts of both V 1 and V 2 are non-trivial, then the repre-sentation of g0 is diagonal in (V 1 ⊗ V 2)0 ⊕ (V 1 ⊗ V 2)1. Consequently, prso((V 1⊗V 2)0)

g0is a Berger algebra and prsp((V 1⊗V 2)1)

g0 is a skew-Berger algebra. Using the arguments asabove, it is easy to see that if g is a Berger superalgebra, then it is symmetric. The same worksfor the tensor product of several representations (we may assume that g2 is a direct some ofsimple Lie superalgebras and V 2 is a tensor products of irreducible representations of theseLie superalgebras).

Next, we assume that the even and odd parts of V 1 are non-trivial and V 2 is either purelyeven or purely odd. If V 2 is purely odd, then V 1 ⊗ V 2 = �V 1 ⊗ �V 2, where �V 2 ispurely even. Thus we may assume that V 2 is purely even, i.e. V 2 is a usual vector space.Since g = g1 ⊕ g2 ⊂ osp(V 1 ⊗ V 2), we get that either g1 ⊂ osp(V 1) and g2 ⊂ so(V 2), org1 ⊂ ospsk(V 1) and g2 ⊂ sp(V 2).

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Let V 1 and V 2 be a complex vector superspace and a complex vector space, respectively,and let V = V 1⊗V 2. Let g1 be a supersymmetric bilinear forms on V 1 and g2 be a symmetricbilinear form on V 2 From the results of [27] it follows that

R(sl(V 1)⊕ sl(V 2)⊕ C) � V ∗ ⊗ V ∗.

Any τ ∈ V ∗ ⊗ V ∗ defines the curvature tensor Rτ by

Rτ (x1 ⊗ x2, u1 ⊗ u2) = A(x1 ⊗ x2, u1 ⊗ u2)+ B(x1 ⊗ x2, u1 ⊗ u2),

where A(x1 ⊗ x2, u1 ⊗ u2) ∈ sl(V 1) ⊕ C, B(x1 ⊗ x2, u1 ⊗ u2) ∈ sl(V 2) ⊕ C, and forv1 ∈ V 1 and v2 ∈ V 2 we have

A(x1 ⊗ x2, u1 ⊗ u2)v1 = (−1)|v1||u1|τ(x1, x2, v1, u2)u1

−(−1)(|v1|+|u1|)|x1|τ(u1, u2, v1, x2)x1,

B(x1 ⊗ x2, u1 ⊗ u2)v2 = τ(x1, x2, u1, v2)u2 − (−1)|u1||x1|τ(u1, u2, x1, v2)x2.

In particular,

tr(B(x1 ⊗ x2, u1 ⊗ u2)) = (τ (x1, x2, u1, u2)− (−1)|u1||x1|τ(u1, u2, x1, x2)).

If q > 1, then by the same arguments as in [14, 29] it can be shown that if Rτ ∈ R(osp(V 1)⊕so(V 2)) ⊂ R(sl(V 1)⊕ sl(V 2)⊕ C) then it is given by τ ∈ �2V ∗ such that

τ(x1, x2, u1, u2) = cg1(x1, u1)g2(x2, u2),

where c ∈ C. Hence,

R(osp(V 1)⊕ so(V 2)) = R(osp(V 1)⊕ so(V 2))0

is one-dimensional. Thus osp(V 1)⊕so(V 2) ⊂ osp(V 1 ⊗V 2) is a symmetric Berger superal-gebra and R(g) = 0 for any proper supersubalgebra g ⊂ osp(V 1)⊕so(V 2) ⊂ osp(V 1⊗V 2).Similarly, if g1 is a super-skew-symmetric bilinear forms on V 1 and g2 is a symplectic formon V 2, then

R(ospsk(V 1)⊕ sp(V 2)) = R(ospsk(V 1)⊕ sp(V 2))0 = CRτ ,

where

τ(x1, x2, u1, u2) = g1(x1, u1)g2(x2, u2).

The same holds if V 1 and V 2 are real.Thus we are left with the cases g2 = sl(2,R) and g2 = sl(2,C) (in the last case g1 ⊂

ospsk(V 1) admits a complex structure and we consider the representation of g1 ⊕ g2 inV 1 ⊗C V 2). Suppose that g2 = sl(2,R). We have g1 ⊂ ospsk(V 1). Since there are no reduc-tive Lie algebras h such that h ⊕ sl(2,R) appears both as a Berger subalgebra of so(p, q)and as a skew-Berger subalgebra of sp(2m,C), there are no ideal in g1

0that acts diagonally in

V 10

⊕V 11

. Hence g1 ⊂ ospsk(V 1) is the identity representation of the Lie superalgebra g1 (thisrepresentation must exist). Section 2, the results of [16] and the condition R(g)1 = 0 implyg1 = ospsk(V 1) = ospsk(2m|r, s). Similarly, if g2 = sl(2,C), then g1 = ospsk(2m|r,C).We get the following two algebras:

ospsk(2m|r, s)⊕ sl(2,R)⊂ osp(R2m|r,s ⊗ R2), ospsk(2m|r,C)⊕sl(2,C)⊂osp(C2m|r ⊗C

2).

123


The second representation is the complexification of the identity representation ofhosp(r, r |m) ⊕ sp(1), hence the second representation gives us a non-symmetric Bergersuperalgebra. The complexification of the first representation is the second one, hence thefirst representation gives us a non-symmetric Berger superalgebra.

The theorem is proved. ��Acknowledgements I am grateful to D. V. Alekseevsky for useful discussions on the topic of this article.This work was supported by the grant 201/09/P039 of the Grant Agency of Czech Republic and by the grantMSM 0021622409 of the Czech Ministry of Education

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