groups and algebras for theoretical physics · groups and algebras for theoretical physics masters...

Groups and Algebrasfor

Theoretical Physics

Masters course in theoretical physics at

The University of BernSpring Term 2016

R

SUSANNE REFFERT

Contents

Contents

1 Complex semi-simple Lie Algebras 21.1 Basic notions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 The Cartan–Weyl basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 The Killing form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.5 Simple roots and the Cartan matrix . . . . . . . . . . . . . . . . . . . . . . . 71.6 The Chevalley basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.7 Dynkin diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.8 The Cartan classification for finite-dimensional simple Lie algebras . . . . . 101.9 Fundamental weights and Dynkin labels . . . . . . . . . . . . . . . . . . . . 121.10 The Weyl group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.11 Normalization convention . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.12 Examples: rank 2 root systems and their symmetries . . . . . . . . . . . . . 171.13 Visualizing the root system of higher rank simple Lie algebras . . . . . . . . 191.14 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.15 Highest weight representations . . . . . . . . . . . . . . . . . . . . . . . . . 221.16 Conjugate representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.17 Remark about real Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . 271.18 Characteristic numbers of simple Lie algebras . . . . . . . . . . . . . . . . . 271.19 Relevance for theoretical physics . . . . . . . . . . . . . . . . . . . . . . . . 27

2 Generalizations and extensions: Affine Lie algebras 302.1 From simple to affine Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . 302.2 The Killing form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3 Simple roots, the Cartan matrix and Dynkin diagrams . . . . . . . . . . . . . 342.4 Classification of the affine Lie algebras . . . . . . . . . . . . . . . . . . . . . 352.5 A remark on twisted affine Lie algebras . . . . . . . . . . . . . . . . . . . . . 382.6 The Chevalley basis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 382.7 Fundamental weights . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.8 The affine Weyl group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.9 Outer automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.10 Visualizing the root systems of affine Lie algebras . . . . . . . . . . . . . . . 472.11 Highest weight representations . . . . . . . . . . . . . . . . . . . . . . . . . 50

3 Advanced topics: Beyond affine Lie algebras 553.1 The Virasoro algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.2 Lie superalgebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583.3 Quantum groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

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Part 1

Complex semi-simple Lie Algebras

Symmetries, and with them, groups and algebras are of paramount importance in theo-retical physics. The basic concepts have already been introduced in the course AdvancedConcepts in Theoretical Physics. This course will build on the material treated there, witha special emphasis on techniques that prove to be useful to the theorist. In order to beself-contained, a certain amount of repetition is however inevitable.

This course consists of three parts. The first part is dedicated to simple Lie algebras,which are basically a theorist’s daily bread. The second part treats affine Lie algebras andthe third generalizations beyond the affine case which keep appearing in various contextsin theoretical physics.1

While the topic is certainly mathematical, treating the structure theory of Lie algebras,this course is aimed at physicists. Proofs are generally not given and I do not work at thehighest possible level of generality (I do not e.g. work over a general field F).

1.1 Basic notions

In the following, we will be working over the field C, which simplifies many things as it isalgebraically closed.

A Lie algebra g is a vector space equipped with an antisymmetric binary operation

[ , ] : g× g→ g (1.1)

which satisfies the Jacobi identity

[X, [Y, Z]] + [Z, [X, Y]] + [Y, [Z, X]] = 0, X, Y, Z ∈ g. (1.2)

This binary operation is usually referred to as either a commutator or the Lie bracket.As you know from ACTP, a Lie algebra g describes the Lie group G in the vicinity of theidentity via the exponential map

eiaX ∈ G for X ∈ g, (1.3)

where a is a parameter.A representation associates to every element of g a linear operator on a vector space V

which respects the commutation relations of the algebra. The maximal number of linearlyindependent states that generate V is the dimension of the representation. Relative to a

1In retrospect, I would have chosen a different title for the course, since it ended up being centered aroundsimple Lie algebras and their generalizations while groups have not made an important appearance.

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Part 1. Complex semi-simple Lie Algebras

given basis, each element of g can be represented as a square matrix, and the basis vectorsare represented as column matrices.

A representation is irreducible if the matrices representing the elements of g cannotall be brought into a block-diagonal form by a change of basis.

A Lie algebra can be specified by a set of generators Ja and their commutationrelations

[Ja, Jb] = ∑c

i f abc Jc. (1.4)

The numbers f abc are the so-called structure constants of the Lie algebra. The number of

generators is the dimension of the Lie algebra.A simple Lie algebra is a Lie algebra that contains no proper ideal (no proper subset of

generators La such that [La, Jb] ∈ La ∀Jb). A semi-simple Lie algebra is a direct sumof simple Lie algebras. In the following, we will focus only on the simple and semi-simplecases.

1.2 The Cartan–Weyl basis

In many cases, we want to work independently from a specific basis. In the case of Liealgebras, however, choosing a particular basis is most convenient. In the following, wewant to construct the generators of g in the standard Cartan–Weyl basis. While this choiceis canonical in the case of finite-dimensional semi-simple Lie algebras, for more generalcases, no fully canonical form exists. First we need to find the maximal set of commutingHermitian generators Hi, i = 1, . . . , r, where r is the rank of the algebra g:

[Hi, H j] = 0. (1.5)

This set of generators Hi forms the Cartan subalgebra h. The generators in h can besimultaneously diagonalized.

The remaining generators Eα of g are chosen such that they satisfy

[Hi, Eα] = αiEα. (1.6)

The vector α = (α1, . . . , αr) is called a root2. The Eα are ladder operators. A basissatisfying both (1.5) and (1.6) is called a Cartan–Weyl basis (also standard or canonicalbasis).

As h is the maximal Abelian subalgebra of g, the roots are non-degenerate. The root αnaturally maps an element Hi ∈ h to the number αi:

α(Hi) = αi. (1.7)

The roots are therefore elements of the dual of the Cartan subalgebra,

α ∈ h∗. (1.8)

With (Eα)† = E−α, we see from hermitian conjugation of Eq. (1.6) that whenever α is aroot, so is −α. We use the notation

∆ = set of all roots, (1.9)

also called the root system. The root components αi can be regarded as the non-zeroeigenvalues of the Hi in the adjoint representation, in which the Lie algebra g itself is

2The name is chosen as the αi are roots of the characteristic equation for Hi.

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the vector space on which the generators act. A matrix representation of the adjointrepresentation in the basis Ja is given by

(Ja)bc = −i fabc. (1.10)

In the adjoint representation, we have

Eα 7→ |Eα〉 ≡ |α〉, (1.11)

Hi 7→ |Hi〉, (1.12)

identifying the generators and the states of the representation. The action of a generator Xin the adjoint representation is

ad(X)Y = [X, Y] (1.13)

so thatad(Hi)Eα = αiEα 7→ Hi|α〉 = αi|α〉. (1.14)

The one-to-one correspondence between the |α〉 and the Eα reflects the fact that the rootsare non-degenerate. In the adjoint representation, the zero eigenvalue has degeneracy r(associated to the |Hi〉).

dim(adj. rep.) = dim(algebra) = r + # roots. (1.15)

We now need to specify the remaining commutation relation of the algebra g in theCartan–Weyl basis Hi, Eα. From the Jacobi identity, we find

[Hi, [Eα, Eβ]] = (αi + βi)[Eα, Eβ]. (1.16)

Therefore, [Eα, Eβ] ∝ Eα+β if α + β ∈ ∆,[Eα, Eβ] = 0 if α + β /∈ ∆,[Eα, Eβ] is a linear comb. of Hi if α = −β.

(1.17)

We will use in the following the expressions

α · H =r

∑i=1

αi Hi, |α|2 =r

∑i=1

αiαi. (1.18)

The full set of commutation relations of g in the Cartan–Weyl basis is given by

[Hi, H j] = 0, (1.19)

[Hi, Eα] = αiEα (1.20)

[Eα, Eβ] =

Nα,βEα+β if α + β ∈ ∆

2|α2|α · H if α = −β

0 otherwise,

(1.21)

where Nα,β = const.

Example: The Cartan–Weyl basis for sl(2, C). The basis

E =

(0 10 0

), H =

(1 00 −1

), F =

(0 01 0

)(1.22)

of sl(2, C) has commutation relations

[H, E] = 2E, [H, F] = −2F, [E, F] = H. (1.23)

This fulfills the commutation relations of a Cartan–Weyl basis.

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1.3 The Killing form

In order to define a scalar product on g, we will in the following define the Killing form.This will allow us, among other things, to fix normalizations. It is defined as follows:

K(X, Y) = Tr(adX adY), X, Y ∈ g. (1.24)

The Killing form obeys ∀ X, Y ∈ g

K(X, Y) = K(Y, X) symmetry, (1.25)

K([X, Y], Z) = K(X, [Y, Z]) invariance. (1.26)

For semi-simple Lie algebras, the Killing form is non-degenerate:

K(X, Y) = 0 ∀Y ⇒ X = 0. (1.27)

This is an alternative way of defining semi-simplicity.

Example: The Killing form of sl(2, C). Reusing the basis elements E, H, F in this order-ing and their commutation relation from the last example, we first express them in theadjoint representation:

adE =

0 −2 00 0 10 0 0

, adH =

2 0 00 0 00 0 −2

, adF =

0 0 0−1 0 00 2 0

. (1.28)

Using these matrices, we can now directly calculate the Killing form on this basis:

K(E, E) = 0, K(E, H) = 0, K(E, F) = 4, (1.29)

K(H, H) = 8, K(H, F) = 0, K(F, F) = 0. (1.30)

In matrix form,

K(X, Y) =

0 0 40 8 04 0 0

. (1.31)

F

Often, a normalized version of the Killing form is used instead:

K(X, Y) =1

2gTr(adX adY), (1.32)

where g is the dual Coxeter number. The standard basis Ja is understood to be orthonor-mal with respect to K,

K(Ja, Jb) = δa,b. (1.33)

The same is true for the generators of the Cartan sub-algebra:

K(Hi, H j) = δi,j. (1.34)

As the Killing form acts as a scalar product, it can be used to raise and lower indices:

f abc = ∑

df ad

c[K(Jd, Jb)]−1. (1.35)

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fabc is anti-symmetric in all three indices. In the orthonormal basis Ja, the position of theindices is irrelevant.

From the cyclic property of the trace, it follows that

K([Z, X], Y) + K(X, [Z, Y]) = 0. (1.36)

The Killing form is uniquely characterized by this property. It also follows that

[Eα, E−α] = K(Eα, E−α)α · H, (1.37)

K(Eα, E−α) =2|α|2 . (1.38)

The fundamental role of the Killing form is to establish an isomorphism between the Cartansubalgebra h and its dual h∗:

K(Hi, · ) : h→ R, (1.39)

for fixed Hi. To every element γ ∈ h∗, there corresponds a Hγ ∈ h:

γ(Hi) = K(Hi, Hγ). (1.40)

For a root α, we have in particular:

Hα = α · H = ∑i

αi Hi. (1.41)

With this isomorphism, we finally also have a positive definite scalar product on h∗:

(γ, β) = K(Hβ, Hγ). (1.42)

Since roots are elements of h∗, this defines a scalar product on root space. In particular,

|α|2 = (α, α). (1.43)

1.4 Weights

So far, we have studied the algebra g from the point of view of the adjoint representationwhich encodes the essential structure of the algebra. We have seen that in the adjoint,the eigenvalues of the Cartan generators are called the roots and that the Killing forminduces a scalar product between them. We now study the more general context of a finitedimensional representation.

For an arbitrary representation, we can always find a basis |λ〉 such that

Hi|λ〉 = λi|λ〉. (1.44)

The eigenvalues λi make up the vector

λ = (λ1, . . . , λr) (1.45)

which is called a weight.λ(Hi) = λi. (1.46)

The scalar product between weights is fixed by the Killing form. In the adjoint repre-sentation, weights are called roots. From [Hi, Eα] = αiEα, we learn that Eα changes theeigenvalue of a state by α:

HiEα|λ〉 = [Hi, Eα]|λ〉+ EαHi|λ〉 = (λi + αi)Eα|λ〉. (1.47)

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If Eα is non-zero, it must be proportional to a state |λ + α〉, therefore the name ladder orstep operator for Eα.

We are mostly interested in finite dimensional representations. For any |λ〉 in a finite-dimensional representation, there are p, q ∈ Z+ such that

(Eα)p+1|λ〉 ∝ Eα|λ + pα〉 = 0, (1.48)

(E−α)q+1|λ〉 ∝ E−α|λ− qα〉 = 0, (1.49)

for any root α.Note that the triplet of generators Eα, E−α, α · H/|α|2 form an su(2) subalgebra analo-

gous toJ+, J−, J3 : [J+, J−] = 2J3, [J3, J±] = ±J±. (1.50)

Due to this fact, many of the properties of simple Lie algebras can be analyzed to a largeextent by making judicious use of the properties of su(2).

If |λ〉 is in a finite-dimensional representation, also its projection onto su(2) is finite-dimensional.

Let the dimension of the subalgebra su(2) be 2j + 1. From the state |λ〉, the state withhighest J3 projection m = j can be reached by a finite number, p, of applications of J+. qapplications of J− on the other hand lead to the state with m = −j:

j =(α, λ)

|α|2 + p, −j =(α, λ)

|α|2 − q. (1.51)

Eliminating j leads to

2(α, λ)

|α|2 = −(p− q). (1.52)

Therefore, any weight λ in a finite-dimensional representation is such that (α,λ)|α|2 is an integer.

This is true in particular for λ = β, where β is a root.

1.5 Simple roots and the Cartan matrix

As we have seen,#roots = dim(g)− rk(g) rk(g). (1.53)

The roots are therefore in general linearly dependent.Thanks to the Euclidean inner product on h∗, we can have a geometric interpretation of

the roots ("root vector"). We can in particular introduce a hyperplane in root space, whichitself must not contain any roots, and use it to split the root system into positive roots ∆+

which are on one side of the hyperplane, and negative roots ∆− on the other side. Notethat this choice of hyperplane and therefore the splitting into positive and negative roots isnot unique and amounts to the choice of a particular basis. Since whenever α is a root, also−α is a root, we have

∆− = −∆+. (1.54)

We can thus decompose the algebra as

g = h⊕ g+ ⊕ g−, (1.55)

where g± = spanCE±α, α > 0 are the subalgebras spanned by the step operators forpositive and negative roots. This is the so-called Gauss or triangle decomposition. A step

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operator Eα associated to a positive root α ∈ ∆+ is called a raising operator, while E−α

with α ∈ ∆+ is called a lowering operator.A simple root αi is defined to be a root that cannot be written as the sum of two positive

roots. Geometrically speaking, the simple roots are those positive roots that are closest tothe hyperplane used to separate positive and negative roots, where the notion of the normis induced by the Killing form.

There are exactly r = rk(g) simple roots:

∆s = α1, . . . , αr, (1.56)

where the subscript i is a labeling index (not the index that refers to the root component!).The set of simple roots provides the most convenient basis for root space. Immediateconsequences of the above definition are

• αi − αj /∈ ∆,

• any positive root is a sum of positive roots.

The basis of simple roots is however in general not orthonormal. This fact is captured inthe Cartan matrix, which is defined as follows:

Aij =2(αi, αj)

|αj|2. (1.57)

We will see that the Cartan matrix summarizes the structure of a semisimple complex Liealgebra completely. Using Eq. (1.52), we see that the entries of the Cartan matrix arenecessarily integers. Moreover, Aii = 2 and in general, Aij 6= Aji. The Cauchy–Schwarzinequality implies that

Aij Aji < 4, i 6= j. (1.58)

Thus, for i 6= j, the entries of the Cartan matrix are negative integers and can be equalto 0, −1, −2, −3. For Aij 6= 0, at least one of Aij, Aji must be equal to −1. In the set ofroots of a simple Lie algebra at most two different lengths of roots are possible (long andshort). The ratio between the squared lengths of long and short roots can be either twoor three. When all the roots have the same length, the algebra is called simply laced. Forconvenience, we introduce the notation

α∨i =2αi

|αi|2. (1.59)

α∨i is called the coroot associated to the simple root αi. The scalar product between rootsand coroots is always an integer. We can now write the Cartan matrix very compactly as

Aij = (αi, α∨j ). (1.60)

The highest root θ is the unique root for which ∑i mi is maximized in the expansion∑i miαi. All elements of ∆ can be reached by repeated subtraction of simple roots from θ.

θ =r

∑i=1

aiαi =r

∑i=1

a∨i α∨i , ai, a∨i ∈N. (1.61)

The coefficients ai are called the marks or Kac labels, while the coefficients a∨i are calledthe comarks or dual Kac labels. Marks and comarks are related via

ai = a∨i2|αi|2

. (1.62)

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The dual Coxeter number (which we have already used in the definition of the normalizedKilling form Eq. (1.32)) is defined as

g =r

∑i=1

α∨i + 1. (1.63)

1.6 The Chevalley basis

We will see that the full set of roots can be reconstructed from the simple roots, which inturn can be straightforwardly extracted from the Cartan matrix. This fact is made manifestin the so-called Chevalley basis. In this basis, to each simple root correspond the threegenerators

ei = Eαi , f i = E−αi , hi =2αi · H|αi|2

, (1.64)

with commutation relations

[hi, hj] = 0, [hi, ej] = Ajiej, [hi, f j] = −Aji f j, [ei, f j] = δijhj. (1.65)

We see that all the structure constants in the Chevalley basis are integers. The remainingstep operators are obtained by repeated commutations of the basic generators, subject tothe Serre relations,

[ad(ei)]1−Aji ej = 0, (1.66)

[ad( f i)]1−Aji f j = 0. (1.67)

These constraints encode the rules for reconstructing the full root system from the simpleroots. Finally,

K(hi, hj) = (α∨i , α∨j ). (1.68)

The fact that the Serre relations do not mix the generators ei and f i corresponds to the factthat the root system is separated into positive and negative roots. Since the commutationrelations and the Serre relations can be expressed in terms of the Cartan matrix A, wesee that A encodes all the information about the structure of the Lie algebra g. Theabstract formulation of Lie algebras via the Cartan matrix is in fact a good starting pointfor generalizations.

1.7 Dynkin diagrams

All the information in the Cartan matrix can be encoded in a planar diagram, the so-calledDynkin diagram. To each Cartan matrix, a diagram made of vertices and connecting linesis associated:

• Each vertex in the diagram represents a simple root (therefore #vertices = rk(g)).

• Long roots are marked by , short roots by •.

• The vertices i and j are joined by Aij Aji lines, in particular:

– Orthogonal roots correspond to disjoint vertices.

– Vertices connected by one line correspond to simple roots spanning an angle of2π3 = 120.

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Figure 1.1: Root systems of the rank 2 Lie algebras

– Vertices connected by two lines correspond to simple roots spanning an angle of3π4 = 135.

– Vertices connected by three lines correspond to simple roots spanning an angleof 5π

6 = 150.

Dynkin diagrams in which vertices are joined only by a single line correspond to Lie algebrasin which all simple roots have the same length, hence the name simply laced. Cartanmatrices which differ only by a renumbering of roots lead to the same Dynkin diagram.

Example: Cartan matrices and Dynkin diagrams of rank 2 Lie algebras. This is thesimplest non-trivial example, where we can see the power of the Cartan matrix. For ranktwo, the Cartan matrix is a 2× 2 matrix. Based on its properties listed after the definitionin Eq. (1.57), we have the following inequivalent possibilities:

(1) :(

2 00 2

), (2) :

(2 −1−1 2

), (3) :

(2 −2−1 2

), (4) :

(2 −3−1 2

). (1.69)

They correspond to the following Dynkin diagrams:

(1) : , (2) : , (3) : , (4) : (1.70)

(1) is actually not a simple Lie algebra (the Cartan matrix is block-diagonal), but semi-simple: it corresponds to A1 ⊕ A1. (2) corresponds to A2, (3) to B2 and (4) to theexceptional Lie algebra G2. The root diagrams are shown in Figure 1.1. We will meet thesesimple Lie algebras again soon.

1.8 The Cartan classification for finite-dimensional simple Liealgebras

The enumeration of all possible Cartan matrices is purely combinatorial. It is possible toclassify the finite-dimensional simple Lie algebras completely via their Cartan matrices,respectively their Dynkin diagrams. There are four infinite series:

Ar (r ≥ 1), Br (r ≥ 3), Cr (r ≥ 2), Dr (r ≥ 4), (1.71)

plus five isolated cases:

E6, E7, E8, G2, F4, (1.72)

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where the subscript denotes the rank of the group. The algebras in the infinite series arecalled the classical Lie algebras. They are isomorphic to the following matrix algebras:

Ar ∼= sl(r + 1), Br ∼= so(2r + 1), Cr ∼= sp(r), Dr ∼= so(2r). (1.73)

The five isolated cases are called the exceptional Lie algebras. The restrictions on theranks of the classical Lie algebras are imposed to avoid overcounting. Including all valuesof r leads to the following isomorphisms:

A1∼= B1

∼= C1∼= D1, B2 ∼= C2, D2 ∼= A1 ⊕ A1, D3 ∼= A3. (1.74)

The simple Lie algebras of the types Ar, Dr, E6, E7 and E8 are all simply laced. For Br, Crand F4, the long roots are

√2 times longer than the short roots, while for G2, the long root

is√

3 times longer than the short root.The Cartan matrices of the simple finite-dimensional Lie algebras are:

Ar:

A =

2 −1 0 . . . 0 0−1 2 −1 . . . 0 00 −1 2 −1 . . 0 0. . . . . . . .0 0 0 . . −1 2 −10 0 0 . . 0 −1 2

(1.75)

Br:

A =

2 −1 0 . . . 0 0−1 2 −1 . . . 0 00 −1 2 −1 . . 0 0. . . . . . . .0 0 0 . . −1 2 −20 0 0 . . 0 −1 2

(1.76)

Cr:

A =

2 −1 0 . . . 0 0−1 2 −1 . . . 0 00 −1 2 −1 . . 0 0. . . . . . . .0 0 0 . . −1 2 −10 0 0 . . 0 −2 2

(1.77)

Dr:

A =

2 −1 0 . . . 0 0−1 2 −1 . . . 0 00 −1 2 −1 . . 0 0. . . . . . . .0 0 0 . . 2 −1 −10 0 0 . . −1 2 00 0 0 . . −1 0 2

(1.78)

E6:

A =

2 −1 0 0 0 0−1 2 −1 0 0 00 −1 2 −1 0 −10 0 −1 2 −1 00 0 0 −1 2 00 0 −1 0 0 2

(1.79)

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E7:

A =

2 −1 0 0 0 0 0−1 2 −1 0 0 0 00 −1 2 −1 0 0 −10 0 −1 2 −1 0 00 0 0 −1 2 −1 00 0 0 0 −1 2 00 0 −1 0 0 0 2

(1.80)

E8:

A =

2 −1 0 0 0 0 0 0−1 2 −1 0 0 0 0 00 −1 2 −1 0 0 0 00 0 −1 2 −1 0 0 00 0 0 −1 2 0 0 −10 0 0 0 −1 2 −1 00 0 0 0 0 −1 2 00 0 0 0 −1 0 0 2

(1.81)

F4:

A =

2 −1 0 0−1 2 −2 00 −1 2 −10 0 −1 2

(1.82)

G2:

A =

(2 −3−1 2

)(1.83)

Table 1.1 shows the Dynkin diagrams of the finite-dimensional simple Lie algebras.

1.9 Fundamental weights and Dynkin labels

Weights and roots live in the same r-dimensional vector space. The weights can be expandedin the basis of simple roots. However, for irreducible finite-dimensional representations,their coefficients are not integers. A more convenient basis is the dual of the simple corootbasis. It is denoted by ωi and defined by the relation

(ωi, α∨j ) = δij. (1.84)

The ωi are called the fundamental weights and the basis

ωi, i = 1, . . . , r (1.85)

is called the Dynkin basis. The components λi of a weight in the Dynkin basis are calledthe Dynkin labels,

λ =r

∑i=1

λiωi ⇔ λi = (λ, α∨i ). (1.86)

The Dynkin labels of weights in finite-dimensional irreducible representations are alwaysintegers. Such weights are called integral. Whenever we write a weight in component

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Ar 1 2 3 r− 1 r E61 2 3 4 5

6

Br 1 2 r− 1 r E71 2 3 4 5 6

7

Cr 1 2 r− 1 r E81 2 3 4 5 6 7

8

Dr

r

r− 11 2

r− 2 F4 1 2 3 4

G2 1 2

Table 1.1: Dynkin diagrams of the finite-dimensional simple Lie algebras

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form λ = (λ1, . . . , λr), we understand these components to be Dynkin labels. Note that theelements of the Cartan matrix are the Dynkin labels of the simple roots:

αi = ∑j

Aijωj, (1.87)

i.e. the ith row of A is the set of Dynkin labels for the simple root αi. The Dynkin labels arethe eigenvalues of the Chevalley generators of the Cartan subalgebra:

hi|λ〉 = λ(hi)|λ〉 = (λ, α∨i )|λ〉, (1.88)

that is,hi|λ〉 = λi|λ〉. (1.89)

Note the role of the position of the index:

hi : eigenvalue of hi (Dynkin label), λi : eigenvalue of Hi. (1.90)

A weight of special importance is the one for which all Dynkin labels are equal to one:

ρ = ∑i

ωi = (1, . . . , 1). (1.91)

This is the Weyl vector or principal vector and is alternatively defined as

ρ = 12 ∑

α∈∆+

α. (1.92)

The scalar product of weights can be expressed in terms of a symmetric quadratic formmatrix Fij:

(ωi, ωj) = Fij. (1.93)

Fij is the transformation matrix relating the Dynkin basis ωi and the simple coroot basisα∨i :

ωi = ∑j

Fijα∨j . (1.94)

The relation between the quadratic form and the Cartan matrix is given by

Fij = (A−1)ij|αj|2

2. (1.95)

The scalar product between the weights λ = ∑ λiωi and µ = ∑ µiωi takes the form

(λ, µ) = ∑i,j

λiµj(ωi, ωj) = ∑i,j

λiµjFij. (1.96)

1.10 The Weyl group

The root system of a simple Lie algebra has a high degree of symmetry. Also, there aremany equivalent choices for the basis of simple roots. The symmetries of the root system ofa Lie algebra form a group, the automorphism group Aut(∆). A particularly interestingsubgroup of Aut(∆) is the Weyl group W which we will study in detail in the following.

A distinguished element of W can be inferred from the fact that if α is a root, so is −α:the mapping

α 7→ −α, α ∈ ∆ (1.97)

is clearly a symmetry of the root system.

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Example: Automorphisms of A1. For A1, there are only two roots, α and −α, each ofwhich can take the role of the simple root. Therefore, Eq. (1.97) is the only non-trivial map.For any weight λ in a finite-dimensional representation of A1, also −λ is a weight of thatrepresentation. This reflection therefore also leaves the weight system invariant. Togetherwith 1, Eq. (1.97) forms the group

Z2 = ±1, (1.98)

hence Aut(∆A1) is isomorphic to Z2.

F

A map on the set of roots must fulfill the following two conditions in order to qualify asa symmetry of the root system:

• the map must be linear and invertible

• the map must be a permutation of the roots.

Therefore, Aut(∆) ⊂ Sdim g−r, the symmetric group with dim g− r elements. In general,the automorphism group is however much smaller than Sdim g−r, as arbitrary permutationscannot be described by a linear map.

It is however not difficult to give elements of W explicitly. Consider the reflection sα

with respect to the hyperplane (through 0) in root space perpendicular to a fixed root α:

sαβ = β− 2(β, α)

(α, α)α, (1.99)

i.e. we subtract from each root β ∈ ∆ twice the component in the α-direction. This isindeed a permutation of the roots. The set of all such reflections with respect to roots formsthe Weyl group W. The product of two Weyl reflections is given by the composition ofmaps

sαsα′ = sα′sα. (1.100)

Since sα is a reflection, it is its own inverse, and the unit element of the composition isthe identity map. Note that the composition of reflections leads to reflections as well asrotations. A generic w ∈W is therefore not of the form (1.99).

By linearity, the action of the Weyl group extends naturally to the weight space of g:

sαλ = λ− (α∨, λ)α. (1.101)

Since an arbitrary element w ∈ W is a product of reflections, it leaves the inner productinvariant:

(sαλ, sαµ) = (λ, µ). (1.102)

Thus any w ∈W is an isometry.The Weyl group is generated by a small number of specific reflections of type (1.99),

namely the reflections corresponding to the simple roots. These are the simple or funda-mental Weyl reflections:

si ≡ sαi , i = 1, . . . , r. (1.103)

Every w ∈W can be written as a word (Weyl word)

w = sisj . . . sk (1.104)

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in the letters si. This decomposition is however not unique. The length l(w) of w is theminimum number of si among all possible decompositions of w. The signature of w isdefined as

ε(w) = (−1)l(w). (1.105)

The si fulfill the relations

s2i = 1, sisj = sjsi, if Aij = 0. (1.106)

These relations generalize to

(sisj)mij = 1, where mij =

2 if i = j

ππ−θij

if i 6= j,(1.107)

and θij is the angle between the simple roots αi and αj. Eq. (1.107) can be used as thedefining relation of the Weyl group. Any group having such a representation is called aCoxeter group.

On the simple roots, the action of si takes the simple form

siαj = αj − Ajiαi. (1.108)

We have seen, that W maps ∆ into itself. In consequence, it provides a simple way togenerate the full root system ∆ from the simple roots by acting with all the elements of Won ∆s:

∆ = wα1, wα2, . . . , wαr| w ∈W. (1.109)

This makes it clear that any set w′αi for fixed w′ could serve as a basis of simple roots.The Weyl group induced a natural splitting of the r–dimensional weight vector space

into a fan of open cones Cw. These cones, whose number is equal to the order of W, aredefined as

Cw = λ|(wλ, αi) ≥ 0, i = 1, . . . , r, w ∈W (1.110)

and are called Weyl chambers. They intersect only at the reflecting hyperplanes, theirboundaries (wλ, αi) = 0. One of the chambers (depending on the choice of simple roots) isdistinguished: the unique chamber whose points have only positive Dynkin labels λi ∈ Z.This is the fundamental or dominant Weyl chamber C0 (also called fundamental Weyldomain). A weight in this chamber is said to be dominant. The highest root θ is an exampleof a dominant weight.

W acts transitively and freely on the Weyl chambers. When acting with W on C0including its boundary, one obtains the whole root space. Conversely, for any weightλ /∈ C0, there exists a unique w ∈W such that wλ ∈ C0. The W orbit of a weight λ is givenby wλ|w ∈W. The W orbit of every weight has exactly one point in C0.

We can label the Weyl chambers by the elements of W. The fundamental chamber C0corresponds to the identity element of the Weyl group.

1.11 Normalization convention

Until now, all normalizations have been fixed with respect to the square lengths of theroots. To fully fix the notation, we need to give a specific value to these lengths. In thestandard convention, the square length of the long roots is set equal to 2. We can fix ournormalization completely by setting

|θ|2 = 2, (1.111)

since θ is necessarily a long root.

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1.12 Examples: rank 2 root systems and their symmetries

Example 1: A2. As we have seen, A2 has Cartan matrix(2 −1−1 2

)(1.112)

and two simple roots α1, α2 of the same length. Using Eq. (1.87), we find

α1 = (2,−1), α2 = (−1, 2). (1.113)

Making use of Eq. (1.95), we can write down its quadratic form matrix:

F =13

(2 11 2

). (1.114)

The Weyl group of A2 contains for sure the elements 1, s1 and s2. We use Eq. (1.107) withθ12 = 2π

3 to find the relation(s1s2)

3 = 1. (1.115)

From this, we find s1s2s1 = s2s1s2, meaning that no words with more than three elementscan appear in W. The full Weyl group is thus given by

W = 1, s1, s2, s1s2, s2s1, s1s2s1. (1.116)

Using the Weyl group, we can find the full root system of A2 by acting with the elements ofW on the two simple roots. We find

∆ = α1, α2, α1 + α2, −α1 − α2, −α1, −α2. (1.117)

The highest root is θ = α1 + α2. The roots are shown in Figure 1.2. The Weyl chambersand their labels are indicated in red.

Example 2: C2. As we have seen, C2 has Cartan matrix(2 −1−2 2

)(1.118)

and two simple roots α1, α2 of the different length. Using Eq. (1.87), we find

α1 = (2,−1), α2 = (−2, 2). (1.119)

Making use of Eq. (1.95), we can write down its quadratic form matrix:

F =12

(1 11 2

). (1.120)

The Weyl group of A2 contains for sure the elements 1, s1 and s2. We use Eq. (1.107) withθ12 = 3π

4 to find the relation(s1s2)

4 = 1. (1.121)

From this, we find s1s2s1s2 = s2s1s2s1. The full Weyl group is thus given by

W = 1, s1, s1, s1s2, s2s1, s1s2s1, s2s1s2, s1s2s1s2. (1.122)

This time, we start from the highest root θ = 2α1 + α2 and construct the root system byrepeated subtraction of the simple roots. We find

∆ = 2α1 + α2, α1 + α2, α1, α2, −α1, −α2,−α1 − α2, −2α1 − α2, . (1.123)

The root system is shown in Figure 1.3. The Weyl chambers and their labels are indicatedin red.

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↵1

↵2 ↵1 + ↵2 =

↵1 ↵2

↵1

↵2

1

s1

s2s1s2

s1s2s1 s1s2

Figure 1.2: Root system and Weyl chambers of A2

↵1

↵2

↵1 ↵2

↵1

↵2

1s1

s2s1s2

s1s2s1

s2s1s2s1s2s1s2

2↵1 + ↵2

2↵1 ↵2

↵1 + ↵2

s2s1

C2

Figure 1.3: Root system and Weyl chambers of C2

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Figure 1.4: Root systems of the rank 3 Lie algebras

1.13 Visualizing the root system of higher rank simple Lie alge-bras

The root systems of the rank two cases can be easily visualized via root diagrams in theplane. Also the rank three cases are still amenable to visualization, see e.g. Fig. 1.4.Higher dimensional cases however, must make recourse to some form of projection to twodimensions. The projection of choice for root systems is the one onto the Coxeter plane.

A Coxeter element of a Lie group is the product of all simple Weyl reflections. Changingthe order of the reflections leads to a conjugate Coxeter element. The Coxeter numberh is the number of roots divided by the rank of the algebra. For a given Coxeter elementw, there is a unique plane (the Coxeter plane) on which w acts by rotation by 2π/h. TheCoxeter plane is used to draw diagrams of root systems: the vertices and edges roots areorthogonally projected onto the Coxeter plane, yielding a polygon with h-fold rotationalsymmetry. No root maps to zero, so the projections of orbits under w form h-fold circulararrangements and there is an empty center. The root systems of F4 (h = 12), E6 (h = 12),E7 (h = 18) and E8 (h = 30) in the Coxeter plane projection are displayed in Figures 1.5 to1.8 (images sources: Wikimedia Commons).

1.14 Lattices

Let (ε1, . . . , εd) be a basis of d-dimensional Euclidean space Rd. A lattice is the set of allpoints whose expansion coefficients (in terms of the specified basis) are all integers:

Zε1 + Zε2 + · · ·+ Zεd, (1.124)

i.e. the Z–span of εi. There are three lattices that are of importance for Lie algebras:

• the weight lattice:P = Zω1 + Zω2 + · · ·+ Zωr, (1.125)

• the root lattice:Q = Zα1 + Zα2 + · · ·+ Zαr, (1.126)

• the coroot lattice:Q∨ = Zα∨1 + Zα∨2 + · · ·+ Zα∨r . (1.127)

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Figure 1.5: Root system of F4 in the Coxeter plane projection with 12-fold symmetry.

Figure 1.6: Root system of E6 in the Coxeter plane projection with 12-fold symmetry.

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α1

α2

ω1

ω2

s2

s1

s1s2

s1s2s1 s2s1

Figure 1.9: Root and weight lattices of A2.

Weights in finite-dimensional representations have integer Dynkin labels, hence they belongto P. The integers specifying the position of a weight in P are the eigenvalues of the Cheval-ley generators hi. The effect of the remaining generators of g is to shift the eigenvalues byan element of Q, the root lattice. Since roots are weights in a particular finite-dimensionalrepresentation, Q ⊆ P. This means that by acting with Eα, a point in P is translated toanother point in P.

For the algebras G2, F4 and E8, we have Q = P. In all other cases, Q is a proper subsetof P and the quotient P/Q is a finite group. Its order |P/Q| is equal to det A.

The distinct elements of the coset P/Q define the so-called congruence classes orconjugacy classes. A weight λ lies in exactly one congruence class. For any algebra g, thecongruence classes take the form

λ · ν =r

∑i=1

λiνi mod |P/Q| (mod Z2 for g = D2l), (1.128)

where the vector (ν1, . . . , νr) is called the congruence vector.Since the bases ωi and α∨i are dual, P and Q∨ are dual lattices. A lattice is said to

be self-dual if it is equal to its dual. For simple Lie algebras, only E8 has a self-dual weightlattice.

1.15 Highest weight representations

Lie algebras play their role in physics not as abstract algebras, but through their represen-tations, which act on suitable representation spaces. The highest weight representations

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are a particularly interesting subclass. any finite-dimensional representation of a simpleLie algebra belongs to this class. As we will see, the key idea for the analysis of theserepresentations is to reduce the problem to the representation theory of sl(2) which wehave studied in Sec. 1.4. There we have seen that to each (simple) root, an sl(2) subalgebraspanned by Eαi , E−αi , Hi belongs. It follows that any representation space has a basis onwhich the whole Cartan subalgebra h acts diagonally.

Any finite-dimensional irreducible representation has a unique highest weight state|λ〉, which is completely specified by its Dynkin labels λ(hi) = λi. Among all the weightsin the representation, the highest weight is the one for which the sum of its coefficients ismaximal when expanded in simple roots. Thus for any α > 0, λ + α cannot be a weight inthe representation, so that

Eα|λ〉 = 0, ∀ α > 0. (1.129)

From Eq. (1.52) (in this case, p = 0), it is clear that the highest weight of a finite-dimensional representation is necessarily dominant. An irreducible finite-dimensionalrepresentation space of a semi-simple Lie algebra is characterized by the fact that it has ahighest weight of multiplicity one and that this weight is dominant integral. Conversely,each dominant integral weight λ is the highest weight of a unique irreducible finite-dimensional representation Lλ. By abuse of notation, representations are often specified bytheir highest weight. The highest weight of the adjoint representation is θ.

Weights and multiplicities. Starting from the highest-weight state |λ〉, all states in therepresentation space Lλ can be obtained by the action

E−βE−γ . . . E−η |λ〉 for β, γ, . . . , η ∈ ∆+. (1.130)

The set of eigenvalues of all states in Lλ is the weight system Ωλ. Any λ′ ∈ Ωλ is suchthat

λ− λ′ ∈ ∆+. (1.131)

All the weights of a given representation lie therefore in exactly one congruence class.For the weight λ′ = λ−∑ niαi, ni ∈ Z+, we call ∑ ni the level or depth. Whenever twoweights λ, µ satisfy λ− µ = β with β of the form β = ∑ niαi, we say µ ≤ λ. This providesa partial ordering of Ωλ. For any irreducible highest weight representation Lλ, there is aunique weight of depth 0, namely the highest weight λ.

In order to find all the weights λ′ ∈ Ωλ, we use again the representation theory ofsl(2), rewriting Eq. (1.52) as

(λ′, α∨i ) = λ′i = −(pi − qi), pi, qi ∈ Z+. (1.132)

λ′ is necessarily of the form

λ′ = λ−∑ niαi, ni ∈ Z+. (1.133)

We now proceed level by level, using a simple algorithm which stems from the fact thatall weights of a finite-dimensional irreducible sl(2)-representation are obtained from thehighest weight by subtracting the positive root α:

• From the highest weight λ = ∑ λiωi, all weights of level 1 are obtained by subtractingthose simple roots αi for which λi > 0.

• More generally, for an arbitrary weight λ′ of Lλ, when λ′i > 0, we can subtract thesimple root αi λ′i times from λ′, producing weights that are all part of Ωλ.

• Proceeding recursively, we can produce in this way all distinct weights of Lλ untilwe arrive at a depth where all weights have negative Dynkin labels and the processterminates.

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↵1 ↵2

(0, 0)

(2,1)

(1, 1)

(2, 1) (1,2)

(1,1)

↵1

↵1

↵1

↵2

↵2

↵2

(1, 2)

Figure 1.10: Weight diagram for the adjoint representation of A2.

Example: the adjoint representation of A2. The highest weight of the adjoint represen-tation of A2 corresponds to the highest root, λ = (1, 1). As we have learned before, thetwo simple roots are α1 = (2,−1) and α2 = (−1, 2). The weight diagram resulting fromthe above algorithm is given in Figure 1.10.

F

Example: the adjoint representation of C2. The highest weight of the adjoint represen-tation of C2 corresponds again to the highest root, θ = 2α1 + α2. So λ = (2, 0). The twosimple roots are α1 = (2,−1) and α2 = (−2, 2). The weight diagram resulting from theabove algorithm is given in Figure 1.11.

F

Example: a representation of D5. Let us study a representation of D5 with highestweight (1, 0, 0, 0, 0). The roots can be read off from the Cartan matrix given in Eq. (1.78).The weight diagram resulting from the above algorithm is given in Figure 1.12.

F

The above examples show that the multiplicity of a (non-highest) weight might belarger than one. In general, the "quantum numbers" λi do not characterize a weight vectorcompletely. A complete specification requires

12(dim g− r) = |∆+| (1.134)

labels, i.e. 12 (dim g− 3r) quantum numbers in addition to the Dynkin labels. To compute

these multiplicities, we can use the Freudenthal recursion formula, which gives the

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↵1 ↵2

(0, 0)

(2,1)

(2, 1)

↵1

↵1

↵1

↵2

↵2

↵2

↵1

↵1

(2, 0)

(0, 1)

(2, 2)

(2,2)

(0,1)

(2, 0)

Figure 1.11: Weight diagram for the adjoint representation of C2.

(1, 0, 0, 0, 0)

↵1

↵2

↵1

(1, 0, 0, 0, 0)

(1, 1, 0, 0, 0)

(0,1, 1, 0, 0)

↵2

(0, 0,1, 1, 1)

(0, 0, 0,1, 1) (0, 0, 0, 1,1)

(0, 0, 1,1,1)

(0, 1,1, 0, 0)

(1,1, 0, 0, 0)

↵3

↵4 ↵5

↵5 ↵4

↵3

Figure 1.12: Weight diagram for the representation of D5 with highest weight (1, 0, 0, 0, 0).

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multiplicities of the weight λ′ in the representation λ in terms of all the weights above it:

[|λ + ρ|2 − |λ′ + ρ|2

]multλ(λ

′) = 2 ∑α>0

∞

∑k=1

(λ′ + kα, α)multλ(λ′ + kα). (1.135)

Example: multiplicity of the weight (0, 0) in the adjoint representation of A2. Weapply the recursion formula (1.135) to compute the multiplicity of (0, 0) from the lastexample. We know that k = 1 and that the multiplicities of all three weights above are eachone. For the three positive roots, (λ′ + kα, α) = 2. Using λ = θ = ρ = α1 + α2, we find

(8− 2)multθ(0, 0) = 2(2 + 2 + 2), (1.136)

therefore multθ(0, 0) = 2.

F

Generally, the weights of the adjoint representation of any semi-simple Lie algebra gare the g–roots, each occurring with multiplicity one, and in additions the weight λ = 0with multiplicity r.

Note that all weights in a given W–orbit have the same multiplicity:

multλ(wλ′) = multλ(λ′) ∀ w ∈W. (1.137)

Finally, note that a finite-dimensional irreducible representation Lλ is always unitary.

(Hi)† = Hi, (Eα)† = E−† (1.138)

results in the norm of any state |λ〉 ∈ Lλ being positive definite.

1.16 Conjugate representations

In a finite-dimensional irreducible representation, there is obviously also a unique lowestweight state. It lies in the W–orbit of the highest weight λ, in the Weyl chamber exactlyopposite to C0. This chamber corresponds to the longest element w0 of W. The lowestweight state is thus given by w0λ. The conjugate representation λ∗ has for its highestweight the negative of the lowest weight state λ∗ of Lλ:

λ∗ = −(w0λ). (1.139)

The Weyl vector ρ is the highest weight of a self-conjugate representation:

ρ = −(w0ρ). (1.140)

More generally, all the weights in Ωλ∗ are the negatives of those in Ωλ.The conjugation is related to the reflection symmetries of the Dynkin diagram. For Ar,

the conjugation amounts to reversing the order of the finite Dynkin labels. A representationis self-conjugate if and only if its weight system is invariant under the change of sign.As for any root α, also −α is a root, the adjoint representation of any Lie algebra is self-conjugate. The Dynkin diagrams with reflection symmetry are Ar, Dr and E6. The operationof conjugation corresponds to this reflection, which is an automorphism of weight space(the nodes of the Dynkin diagrams corresponding to the fundamental weights). In order for

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complex algebra compact form real formsl(n, C) su(n) sl(n, R)

so(2l + 1, C) so(2l + 1) so(l + 1, l)sp(n, C) sp(n, 0) sp(n, R)so(2l, C) so(2l) so(l, l)

Table 1.2: Compact and normal real forms for the classical Lie algebras

representations of these three cases to be self-conjugate, the Dynkin labels of the highestweights must be invariant under the reflections, i.e. they have to satisfy

Ar : λi = λr−i+1, (1.141)

Dr (r even) : λr = λr−1, (1.142)

E6 : λ1 = λ5, λ2 = λ3. (1.143)

1.17 Remark about real Lie algebras

So far, we have always assumed that the base field of g is C, making use of the fact thatC is algebraically closed. This is not the case for R. This means in particular that theeigenvalue equation (1.6) which determines the roots and therefore encodes the abstractstructure of a Lie algebra need not have a solution in R. The theory of real Lie algebras istherefore much more involved. It is however possible to construct real forms of the complexLie algebras we have studied. A simple Lie algebra has several non-isomorphic real forms.There are two standard real forms one can construct for any complex simple Lie algebra,based on the observation that in a Cartan–Weyl basis, all structure constants are real. As aconsequence, the real vector space spanned by all real linear combinations

∑i

ξi Hi + ∑α∈∆

ξαEα (1.144)

is a real Lie algebra. This Lie algebra is the normal or split real form. It is the least compactreal form.

On the other hand, the compact real form is spanned with real coefficients by iHifor the Cartan subalgebra and

(i√(α, α)/2)(Eα + E−α) ∪ (

√(α, α)/2)(Eα − E−α) (1.145)

for its orthogonal complement. All other real forms can be obtained from the compactform by multiplying suitable generators by a factor of i. Table 1.2 collects the compact andnormal real forms for the classical Lie algebras.

1.18 Characteristic numbers of simple Lie algebras

In this section, we collect the characteristic numbers of the simple Lie algebras we haveencountered, see Tables 1.3 and 1.4.

1.19 Relevance for theoretical physics

In the first part of this course, we have studied the structure of abstract simple Lie algebras.In physics, Lie algebras and in particular their representations are omnipresent. In particle

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Ar Br Cr Dr

dimension dim(g) r2 + 2r 2r2 + r 2r2 + r 2r2 − rdual Coxeter number g r + 1 2r− 1 r + 1 2r− 2order of Weyl group |W| (r + 1)! 2rr! 2rr! 2(r− 1)r!highest root θ (1, 0, . . . , 1) (0, 1, . . . , 0) (2, 0, . . . , 0) (0, 1, . . . , 0)weight/root lattice P/Q Zr+1 Z2 Z2 Z4 r odd,

Z2 ×Z2 r evencongruence vector ν (1, 2, . . . , r) (0, . . . , 0, 1) (1, 2, . . . , r) (2, 4, . . . , r− 1, r) r odd

(0, . . . , 0, 1) r odd

Table 1.3: Characteristic numbers of the classical Lie algebras

E6 E7 E8 F4 G2

dimension dim(g) 78 133 248 52 14dual Coxeter number g 12 18 30 9 4order of Weyl group |W| 51840 2903040 696729600 1152 12highest root θ (0, . . . , 0, 1) (1, 0, . . . , 0) (1, 0, . . . , 0) (1, 0, 0, 0) (1, 0)weight/root lattice P/Q Z3 Z2 1 1 1

congruence vector ν (1, 2, 0, 1, 2, 0) (0, 0, 0, 1, 0, 1, 1) - - -

Table 1.4: Characteristic numbers of the exceptional Lie algebras

physics, we mostly deal with the A-series (e.g. when studying the gauge groups of thestandard model, SU(3)× SU(2)×U(1)). For spin representations, orthogonal and specialorthogonal Lie algebras play a role.

Also in integrable systems such as spin chains, Lie algebras are of great importance.In the simplest model, the XXX spin chain, each lattice point carries a representation ofsu(2) (corresponding to a spin pointing either up or down), but in more general models,representations of any simple Lie algebra are admitted.

In modern theoretical physics beyond the standard model, the whole machinery ofsimple Lie algebras is of paramount importance. Lie algebras show up in many contexts,such as grand unified gauge groups (e.g. SU(5), SO(10), SU(8), E6, O(16)), and globalsymmetries. In string theory, gauge groups of the A and D–series can be created viaD–brane constructions in type II superstring theory, whereas the heterotic string has gaugegroup SO(32) or E8 × E8.

In quantum systems, however, classical symmetries do not carry over directly, and theconcept of the central extension becomes necessary. Other generalizations of Lie algebrasthat lead to infinite-dimensional algebras also appear naturally in physical contexts. Inintegrable models, super algebras and infinite dimensional Yangian algebras appear. Thesecond part of this course will therefore be dedicated to extensions and generalizations ofsimple Lie algebras.

Literature

This part of the course presents material that is contained both in [DMS97] and [FS97].The notation of [DMS97] is used throughout, but some sections follow more closely theexposition of [FS97]. [DMS97] is sometimes a little terse, while [FS97] takes a moremathematical approach.

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References

[DMS97] P. Di Francesco, P. Mathieu, and D. Sénéchal. Conformal field theory. Graduate textsin contemporary physics. New York: Springer, 1997. ISBN: 0-387-94785-X. URL: http://opac.inria.fr/record=b1119694.

[FS97] J. Fuchs and C. Schweigert. Symmetries, Lie Algebras and Representations: A GraduateCourse for Physicists. Cambridge Monographs on Mathematical Physics. Cambridge:Cambridge University Press, 1997. ISBN: 978-0521541190.

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http://opac.inria.fr/record=b1119694


Part 2

Generalizations and extensions:Affine Lie algebras

When considering quantum systems in physics, we often reach the limits of applicabilityof (semi-)simple Lie algebras. While at the level of classical mechanics or field theory,the symmetries of a physical systems are described by a Lie algebra g, in the quantumdescription of the same system, the Lie brackets are not recovered completely. Additionalterms appear in the commutation relations (in physics, this phenomenon is referred toas a quantum anomaly). In order to describe the quantum theory in Lie algebraic terms,we must interpret these new constant terms as the eigenvalues of some new operatorswhich have constant eigenvalues on any irreducible representation space of g. These newoperators extend g to a closely related algebra g, via a so-called central extension. Not everygiven Lie algebra admits however a non-trivial central extension. The (semi-)simple Liealgebras we have studied so far, in particular, do not.

Our aim in this part of the course is to construct (untwisted) affine Lie algebras g.In order to do so, we need to extend our familiar simple Lie algebras g to an infinite-dimensional loop algebra which in turn receives a central extension and is supplementedby a derivation. We will associate to each finite-dimensional g an affine extension g byadding an extra node related to the highest root θ to the Dynkin diagram of g. Theintroduction of this extra simple root will make the root system and the Weyl group of ginfinite dimensional. Also the highest weight representations become infinite dimensional,however, they can be organized in terms of a new parameter, the level. The discussionin Part 2 of this course will follow the one of Part 1 very closely, pointing out importantdifferences to the finite-dimensional case.

2.1 From simple to affine Lie algebras

Let us consider a generalization of g in which the elements of the algebra are also Laurentpolynomials in some variable t. The set of these polynomials is denoted by C[t, t−1]. Thegeneralization

g = g⊗C[t, t−1] (2.1)

is called the loop algebra g. By a loop in a topological spaceM, one means the smoothembedding of a circle intoM, together with a chosen parametrization. The loop algebraassociated to g consists of the space of analytic mappings from the circle S1 to g via themap

t = eiγ, γ ∈ R. (2.2)

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Part 2. Generalizations and extensions: Affine Lie algebras

The generators of g are given by Ja ⊗ tn. The algebra multiplication rule is the naturalextension from g to g:

[Ja ⊗ tn, Jb ⊗ tm] = ∑c

i f abc Jc ⊗ tn+m. (2.3)

A central extension is obtained by adjoining to g a central element:

[Ja ⊗ tn, Jb ⊗ tm] = ∑c

i f abc Jc ⊗ tn+m + k n K(Ja, Jb)δn+m,0, (2.4)

where k commutes with all Jas and K is the Killing form of g. Assuming again the generatorsJa to be orthonormal with respect to the Killing form and using the notation

Jan ≡ Ja ⊗ tn, (2.5)

we can rewrite the commutation relations as

[Jan, Jb

m] = ∑c

i f abc Jc

n+m + k n δa,bδn+m,0, (2.6)

These relations must be supplemented by

[Jan, k] = 0. (2.7)

While seemingly ad hoc, the central extension is actually unique, as we will see in thefollowing. Let us start with the generic cocommutatormutator

[Jan, Jb

m] = ∑c

i f abc Jc

n+m +l

∑i=1

ki(dabi )nm, (2.8)

containing l central terms. Except for when n + m = 0, the central terms can be eliminatedby a redefinition of the generators. So

[Ja0 , Jb

n] = ∑c

i f abc Jc

n, (2.9)

i.e. the generators Jan transform in the adjoint representation of of g (ad(Ja

0)). Sincethe central extensions commute with all the generators Ja

n, they are invariant tensors ofthe adjoint representation. There is, however only one (up to normalization) such tensor,namely the Killing form itself. Therefore, only one central element can be added to theloop extension of a simple Lie algebra. The only central extension of a simple Lie algebracompatible with the antisymmetry of the commutators and the Jacobi identity is indeedthe one of Eq. (2.6).

As in the simple case, we want to rewrite everything in the (affine) Cartan–Weyl basis.With the non-zero Killing norms

K(Hi, H j) = δij, K(Eα, E−α) =2|α|2 , (2.10)

the commutation relations become

[Hin, H j

m] = k n δi,jδn+m,0, (2.11)

[Hin, Eα

m] = αiEαn+m (2.12)

[Eαn, Eβ

m] =

Nα,βEα+β

n+m if α + β ∈ ∆2|α2|

(α · Hn+m + k n δn+m,0

)if α = −β

0 otherwise.

(2.13)

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The set of generators H10 , . . . , Hr

0, k is manifestly Abelian. In the adjoint representation,the eigenvalues of ad(Hi

0) and ad(k) on the generators Eα are, respectively, αi and 0. Beingindependent of n, the eigenvector (α1, . . . , αr, 0) is the same for all Eα

m, m = 0, . . . , ∞, i.e.it is infinitely degenerate. H1

0 , . . . , Hr0, k is therefore not a maximal Abelian subalgebra.

It must be augmented by a new grading operator L0, whose eigenvalues in the adjointrepresentation depend on n:

L0 = −tddt

. (2.14)

In the mathematics literature, usually the operator D = −L0 is used instead, it is called aderivation. The action of L0 on the generators is

ad(L0)Ja ⊗ tn = [L0, Ja ⊗ tn] = −nJa ⊗ tn ⇒ [L0, Jan] = −n Ja

n. (2.15)

The element L0 measures therefore the mode number n of the generators Jan, i.e. the

degrees with respect to the gradation are given by the eigenvalues of L0. The maximalCartan subalgebra is generated by

H10 , . . . , Hr

0, k, L0. (2.16)

The other generators, Eαn for any n and Hi

n for n 6= 0, play the role of ladder operators.With the addition of L0, the resulting algebra is denoted as g:

g = g⊕Ck⊕CL0. (2.17)

It is referred to as an untwisted affine Lie algebra.The addition of L0 has made the Killing form on g non-degenerate and allows thus for a

non-degenerate inner product on g. Having an infinite number of generators Jan, n ∈ Z,

it is clearly an infinite-dimensional algebra. From the perspective of g, g is referred to asthe corresponding finite algebra, generated by the zero modes Ja

0.In the physics literature, affine Lie algebras are often referred to as Kac–Moody algebras.

However the name Kac–Moody refers to a more general construction.

Example: the Heisenberg algebra. An already familiar infinite-dimensional example isthe algebra generated by the modes of a free boson:

[an, am] = n δn+m,0. (2.18)

This is the so-called Heisenberg algebra. It is the affine extension of the u(1) algebragenerated by the element a0. Comparing to the commutation relations Eq. (2.6), the levelappears to be 1, however, the central terms can be changed arbitrarily by rescaling themodes. For the case u(1), the level has no meaning.

F

2.2 The Killing form

In parallel to Part 1 of this course, we want to equip g with an inner product. This meansthat we need to extend the Killing form from g to g. We use again the identity Eq. (1.36).With X, Y ∈ Ja

n, Z = L0, we get

K(Jan, Jb

m) = 0, unless n + m = 0. (2.19)

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For n + m = 0, the t-factors disappear, leaving us with the Killing form of g, implying

K(Jan, Jb

m) = δa,bδn+m,0. (2.20)

Note that the affine Killing form is still orthonormal with respect to the finite algebraindices. The choice X, Z ∈ Ja

n, Y = k yields

K(Jan, k) = 0, K(k, k) = 0, (2.21)

whereas Y = L0 leads toK(Ja

n, L0) = 0, K(L0, k) = −1. (2.22)

The only unspecified norm is K(L0, L0), which by convention is chosen to be

K(L0, L0) = 0. (2.23)

This arbitrariness is related to the possibility of redefining

L0 → L0 + ak, a constant, (2.24)

without affecting the algebra. This redefinition of L0 changes its Killing norm by −2a.As in the finite case, the Killing form leads to an isomorphism between the elements of

the Cartan subalgebra and those of its dual and defines a scalar product for the latter. Takea state that is a simultaneous eigenvector of all the generators of the Cartan subalgebra.The components of the vector λ are given by the eigenvalues of this state:

λ = (λ(H10), λ(H2

0), . . . , λ(Hr0); λ(k); λ(−L0)). (2.25)

λ = (λ; kλ; nλ) is called an affine weight. The scalar product induced by the Killing formis

(λ, µ) = (λ, µ) + kλnµ + kµnλ. (2.26)

As for the simple Lie algebras, affine weights in the adjoint representation are called affineroots.

Since k commutes with all the generators of g, its eigenvalues on the states of theadjoint representation are 0. Affine roots are therefore of the form

β = (β; 0; n). (2.27)

The scalar product between affine roots is therefore the same as of their simple counterparts,(β, α) = (β, α). The affine root associated to the generator Eα

n is

α = (α; 0; n), n ∈ Z, α ∈ ∆. (2.28)

If we defineδ = (0; 0; 1), (2.29)

then nδ is the root associated to Hin. In the following, we use the notation

α ≡ (α; 0; 0), (2.30)

so that we can write the roots in Eq. (2.28) as

α = α + nδ. (2.31)

The full set of roots is

∆ = α + nδ | n ∈ Z, α ∈ ∆ ∪ nδ | n ∈ Z, n 6= 0. (2.32)

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The root δ is unusual as it has zero length,

(δ, δ) = 0. (2.33)

For this reason, it is called an imaginary root. All the roots in nδ are imaginary,

(nδ, mδ) = 0, ∀ n, m. (2.34)

All the imaginary roots have multiplicity r. The other roots have multiplicity one and arecalled real.

2.3 Simple roots, the Cartan matrix and Dynkin diagrams

As the next step, we want to identify a basis of simple roots for the affine Lie algebra g.This corresponds again to splitting the root space into positive and negative roots. In such abasis, the expansion coefficients of any root are either all positive or all negative. The basismust contain r + 1 elements, r of which are necessarily the finite simple roots αi, while theremaining simple root must be a linear combination involving δ. The proper choice for thelatter is

α0 ≡ (−θ; 0; 1) = −θ + δ. (2.35)

The basis of simple roots is thus given by

αi, i = 0, . . . , r. (2.36)

The set of positive roots is given by

∆+ = α + nδ | n > 0, α ∈ ∆ ∪ α | α ∈ ∆+. (2.37)

This makes sense as α + nδ = α + nα0 + nθ = nα0 + (n− 1)θ + (θ + α) and the expansioncoefficients in terms of finite simple roots of the last two factors are necessarily non-negative.Note, that in the affine case, there is no highest root. This implies also that the adjointrepresentation is not a highest weight representation.

Example: the root system of A1. The algebra A1 has exactly two simple roots, namelythe simple root α = θ of the simple algebra A1 and α0 = −θ + δ = −α + δ. Its root systemis given by

∆ = ±α + nδ | n ∈ Z. (2.38)

F

Given a set of simple roots and a scalar product, we can now define the extendedCartan matrix as

Aij = (αi, α∨j ), 0 ≤ i, j ≤ r, (2.39)

where the affine coroots are given by

α∨ =2|α|2 (α; 0; n) =

2|α|2 (α; 0; n) = (α∨; 0;

2|α|2 n). (2.40)

As for simple roots, we omit the hat for simple coroots, such that

α∨0 = α0, α∨i = (α∨i ; 0; 0), i 6= 0. (2.41)

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Compared to the finite Cartan matrix, Aij contains an extra row and column. The extraentries are easily calculated in terms of the marks defined in Eq. (1.61). We have

(α0, α∨0 ) = |θ|2 = 2 (2.42)

and

(α0, α∨j ) = (θ, α∨j ) =r

∑i=0

ai(αi, α∨j ). (2.43)

The zeroth mark is defined to be a0 = 1. Since the finite part of α0 is a long root, the zerothcomark is also 1:

a∨0 =|α0|2

2= 1. (2.44)

By construction, the extended Cartan matrix satisfies

r

∑i=0

ai Aij =r

∑i=0

Aija∨j = 0. (2.45)

The dual Coxeter number is given by

g =r

∑i=0

a∨i . (2.46)

Again, all the information contained in the extended Cartan matrices can be encoded inextended Dynkin diagrams. The Dynkin diagram of g is obtained from that of g by theaddition of an extra node associated to α0. This node is linked to the αi-nodes by A0i Ai0lines.

2.4 Classification of the affine Lie algebras

Just as the simple Lie algebras, affine Lie algebras can be classified completely via theirCartan matrices. The linear dependence between the rows of the extended Cartan matrixmeans that it has one zero eigenvalue, a reflection of the semi-positive nature of the affinescalar product. The rank of an (r + 1)× (r + 1) affine Cartan matrix is thus r. As we willsee, the classification of the extended Cartan matrices relies directly on the classification ofthe simple Lie algebras.

Example: The affine Lie algebras with r = 1. For r = 1, the extended Cartan matricesare 2× 2 matrices and the requirement of det A = 0 is enough to determine the off-diagonalelements:

A11A22 − A12A21 = 4− A12A21 = 0. (2.47)

This leads directly to the two possibilities

A =

(2 −2−2 2

), A =

(2 −4−1 2

). (2.48)

The corresponding Dynkin diagrams are

0 1 , 0 1 , (2.49)

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so differently to the simple case, two nodes can be joined by four lines.

F

For higher rank cases, we start from the observation that by deleting the ith row andcolumn for any i ∈ 0, . . . , r from an extended Cartan matrix, we obtain a Cartan matrixof a (semi)-simple Lie algebra. By deleting further rows and columns, we always remain inthe class of semi-simple Lie algebras. In particular, any 2× 2 matrix obtained by deletingr− 1 rows and their corresponding columns from an extended Cartan matrix A must beone of the rank 2 Cartan matrices we have studied. This restricts the matrix elements fori 6= j to

Aij Aji = 0 or min|Aij|, |Aji| = 1, max|Aij|, |Aji| ≤ 3. (2.50)

We see that the rank 1 cases with A12A21 = 4 are unusual.Implementing the above constraints, the enumeration of all possible affine Cartan

matrices of a given rank can be done by straightforward combinatorics.

Example: r = 2 with the submatrix of G2. For illustration, we study the rank 2 casewith a 2× 2 submatrix corresponding to the Cartan matrix of G2:

A =

2 p qr 2 −3s −1 2

. (2.51)

The determinant is given by

det A = 2− p(2r + 3s)− q(r + 2s). (2.52)

We find that det A = 0 is only fulfilled for two combinations of the integers p, q, r, s,namely

p = r = −1, q = s = 0 and p = r = 0, q = s = −1. (2.53)

The affine Lie algebras corresponding to these two solutions are denoted G2 = G(1)2 and

G(3)2 .

F

Tables 2.1 and 2.2 show all the possible extended Dynkin diagrams resulting from thisclassification. The Dynkin diagrams in Table 2.1 are all untwisted affine Lie algebras, alsodenoted by A(1)

r , B(1)r , etc. In the examples we have however seen, that we also produce

additional Cartan matrices. These additional Dynkin diagram are given in Table 2.2 andbelong to the class of twisted affine Lie algebras which we have not discussed so far. Forthe latter class, a variety of naming conventions exist in the literature.

In total, we have seven infinite series of affine Lie algebras

A(1)r (r ≥ 2), B(1)

r (r ≥ 3), C(1)r (r ≥ 2), D(1)

r (r ≥ 4), (2.54)

B(2)r (r ≥ 3), C(2)

r (r ≥ 2), B(2)r (r ≥ 0), (2.55)

and nine exceptional cases,

A(1)1 , E(1)

6 , E(1)7 , E(1)

8 , F(1)4 , G(1)

2 , (2.56)

A(2)1 , F(2)

4 , G(3)2 . (2.57)

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Ar

1 2 3 r− 1 r

0E6

1 2 3 4 5

0

6

Br

0

12 3 r− 1 r

E71 2 3 4 5 60

7

Cr 0 1 2 r− 1 r E8

0 1 2 3 4 5 6 7

8

Dr

0

1

r

r− 12 3

r− 2 F4 0 1 2 3 4

A1 0 1 G2

0 1 2

Table 2.1: Dynkin diagrams of the untwisted affine Lie algebras

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A(2)1 0 1 B(2)

r , D(2)r+1

0 1 2

B(2)r , A(2)

2r 0 1 2 C(2)r , A(2)

2r−1

0

12 3

F(2)4 , E(2)

6

1 2 3 4 0G(3)

2 , D(3)4

1 2 0

Table 2.2: Dynkin diagrams of the twisted affine Lie algebras

For many purposes, it is natural to regard A(1)1 as the first element of the series A(1)

r and

A(2)1 as the first element of B(2)

r . Note, that in B(2)r three lengths of simple roots occur,

necessitating a new notation for the Dynkin diagram: we have, in decreasing length, thetypes of nodes

. (2.58)

Inspecting Tables 2.1 and 2.2, we find that indeed, by removing any node from an extendedDynkin diagram, we get the Dynkin diagram of a simple Lie algebra.

2.5 A remark on twisted affine Lie algebras

We have only constructed the untwisted affine Lie algebras so far, which is where our maininterest lies. Yet we have seen that the twisted cases arise naturally in the classificationof affine Lie algebras via their Cartan matrices. The twisted cases can be constructed in asimilar way to the untwisted ones, by giving up the requirement of single-valuedness of themap from the circle S1 to g in the construction of the loop algebra, and instead imposingtwisted boundary conditions. We shall however not do this in this course.

2.6 The Chevalley basis

The commutation relations of the generators of the Chevalley basis given in Eq. (1.65) havethe following affine extension:

[hin, hj

m] = (α∨i , α∨j )knδijδn+m,0 =4|αi|2

knδijδn+m,0, (2.59)

[hin, ej

m] = Ajiejn+m, (2.60)

[hin, f j

m] = −Aji f jn+m, (2.61)

[ein, f j

m] = δijhjn+m +

2|αi|2

knδijδn+m,0, i, j = 1, . . . , r. (2.62)

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These relation however do not involve only the generators of the r + 1 simple roots of gand are not expressed in terms of the Cartan matrix of g. In order to construct a genuineChevalley basis, we need to add the generators

e0 = E−θ1 , f 0 = Eθ

−1, h0 = k− θ · H0 (2.63)

to the set of finite generators ei and f i. e0 and f 0 are the raising and lowering operators forα0.

From now on, we will omit the mode index 0 from the finite g Chevalley generators.The commutation relations for the generators associated to the simple roots of g can be

written as

[hi, hj] = 0, (2.64)

[hi, ej] = Ajiejn+m, (2.65)

[hi, f j] = −Aji f jn+m, (2.66)

[ei, f j] = δijhj, i, j = 0, . . . , r. (2.67)

These commutation relations have to be supplemented by the affine Serre relations

[ad(ei)]1−Aji ej = 0, (2.68)

[ad( f i)]1−Aji f j = 0, i 6= j. (2.69)

In this form, it is manifest that A encodes the whole structure of g. Its infinite-dimensionalnature is however not apparent.

Example: the Serre relations and roots of A1. Using the Cartan matrix of A1, we find1− A01 = 1− A10 = 3, so

[ad(e0)]3e1 = [e0, [e0, [e0, e1]]] = 0, (2.70)

[ad(e1)]3e0 = [e1, [e1, [e1, e0]]] = 0. (2.71)

This corresponds to the fact, that 3α0 + α1 and α0 + 3α1 are not roots, while for exampleα0 + 2α1 is a root. Using the fact, that δ = α0 + α1, we can rewrite the roots. In particular,the root system given in Eq. (2.38) can be rewritten as

∆ = nα0 + mα1 | |m− n| ≤ 1, n, m ∈ Z. (2.72)

We see again, that unlike in the simple cases, the root system of A1 contains infinitely manyroots.

F

2.7 Fundamental weights

As in the simple case, the fundamental weights ωi, 0 ≤ i ≤ r are defined to be theelements of the basis dual to the simple coroots. The fundamental weights are assumed tobe eigenstates of L0 with zero eigenvalue. For i 6= 0, the affine fundamental weights aregiven by

ωi = (ωi; a∨i ; 0). (2.73)

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Their finite part makes them dual to the finite simple roots, while the k eigenvalue is fixedby the condition

(ωi, α∨0 ) = 0, i 6= 0. (2.74)

The zeroth fundamental weight must have zero scalar product with all finite αis and satisfy

(ω0, α∨0 ) = 1. (2.75)

Hence it must beω0 ≡ (0; 1; 0). (2.76)

It is called the basic fundamental weight. With

ωi ≡ (ωi; 0; 0), (2.77)

it follows thatωi = a∨i ω0 + ωi. (2.78)

The scalar product between the fundamental weights is given by

(ωi, ωj) = (ωi, ωj) = Fij, i, j 6= 0, (2.79)

(ω0, ωi) = (ω0, ω0) = 0, i 6= 0, (2.80)

where Fij is the quadratic form matrix of g. Affine weights can be expanded in terms of theaffine fundamental weights and δ as

λ =r

∑i=0

λiωi + lδ, l ∈ R. (2.81)

Since each fundamental weight contributes to the k eigenvalue by a factor of a∨i , we define

k ≡ λ(k) =r

∑i=0

a∨i λi. (2.82)

k is called the level. This implies that the zeroth Dynkin label λ0 is related to the finiteDynkin labels λi, i = 1, . . . , r and the level by

λ0 = λ(k)−r

∑i=1

a∨i λi, (2.83)

i.e.λ0 = k− (λ, θ). (2.84)

Modulo a possible δ factor, the relation between λ and its finite counterpart is simply

λ = kω0 + λ. (2.85)

Note, that roots are weights at level zero.Affine weights are generally given in terms of Dynkin labels of the form

λ = [λ0, λ1, . . . , λr], (2.86)

e.g.ω0 = [1, 0, . . . , 0], ω1 = [0, 1, 0, . . . , 0], ωr = [0, . . . , 0, 1]. (2.87)

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Note, however, that this notation does not keep track of the eigenvalue of L0. The Dynkinlabels of simple roots are given by the rows of the affine Cartan matrix:

αi = [Ai0, Ai1, . . . , Air]. (2.88)

The affine Weyl vector is defined as

ρ =r

∑i=0

ω = [1, 1, . . . , 1], (2.89)

andρ(k) = g, (2.90)

the dual Coxeter number. Note, that unlike the simple case, ρ cannot be written as 1/2 thesum of positive affine roots.

As in the simple case, affine weights whose Dynkin labels are all non-negative integersplay a special role, they are called dominant. This property is however level-dependent, asλ0 is fixed by k and λi, i 6= 0, see Eq. (2.84). The set of all dominant weights of level k isdenoted by Pk

+. The finite part of an affine dominant weight is itself a dominant weight,

λ ∈ Pk+ ⇒ λ ∈ P+. (2.91)

2.8 The affine Weyl group

The Weyl reflection with respect to the real affine root α is defined in complete analogy tothe case of the simple Lie algebra:

sαλ = λ− (λ, α∨)α, (2.92)

and the set of all such reflections generates the affine Weyl group W. Just like the Weylgroup of a simple Lie algebra, it acts on the weight space by linear maps. Many of itsproperties are analogous to the simple case, but we will see that important differencesarise. W is generated by the reflections si with respect to the simple roots. Each of theseelementary reflections permutes the set of positive roots.

The new feature in the affine case are related to the existence of the imaginary roots.As (δ, α) = 0, the imaginary roots are unaffected by the affine Weyl reflections:

sαδ = δ− (δ, α∨)α = δ. (2.93)

Thus any Weyl reflection acts on the set of imaginary roots nδ | n 6= 0 as the identitymap.

For i = 1, . . . , r, the si act precisely as the simple Weyl reflections of g. We will seehowever, that the reflection s0 with respect to α0 acts differently, namely as a reflectionsupplemented by a translation. The affine Weyl group acts therefore on the weight space ofg as an affine mapping – hence the name affine for the affine algebras.

With λ = (λ; k; n) and α = (α; 0; m), we find

sαλ = (sα(λ + kmα∨); k; n− [(λ, α) + km]2m|α|2 ). (2.94)

To analyze the structure of W, we want to rewrite this as

sαλ = sα(tα∨)mλ, (2.95)

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↵1

↵2 ↵1 + ↵2 =

↵1 ↵2

↵1

↵2

1

s1

s2s1s2

s1s2s1 s1s2

k = 1

k = 2

k = 3

Figure 2.1: Shifted hyperplanes for s0 in the case A2.

with tα∨ defined astα∨ = s−α+δsα = sαsα+δ, (2.96)

i.e.tα∨ λ = (λ + kα∨; k; n + [|λ|2 − |λ + kα∨|2]/k2). (2.97)

The action of tα∨ on the finite part of λ corresponds thus to a translation by the coroot α∨.Since

(tα∨)(tβ∨) = tα∨+β∨ , (2.98)

the set of all tα∨ generates the coroot lattice Q∨. An affine Weyl reflection is thus a productof a finite Weyl reflection times a translation by an appropriate coroot. As the group of suchtranslations is infinite, the affine Weyl group is infinite-dimensional. Q∨ is an invariantsubgroup of W:

w(tα∨)w−1 = twα∨ , ∀ w ∈ W. (2.99)

As Q∨ and W only have the identity in common, W is isomorphic to the semi-direct productof W and the Abelian group T of translations by k-multiples of elements of Q∨:

W ∼= W n T = W n kQ∨. (2.100)

While the affine Weyl group is independent of the level, its action on those weights whichhave a definite value k of the level depends in a non-trivial manner on k.

Let us now study the action of s0 in Eq. (2.94):

s0λ = (λ + kθ − (λ, θ)θ; k; n− k + (λ, θ)) = sθt−θ(λ). (2.101)

With sθ = −θ, the finite part of s0λ is sθλ + kθ. This mapping can be described as areflection with respect to an appropriately shifted hyperplane, see the example of A2 shownin Figure 2.1. Just as in the simple case, the affine Weyl chambers are defined as thoseopen subsets of the vector space of affine weights which are obtained by removing allhyperplanes that are left invariant by some Weyl reflection,

Cw = λ | (wλ, αi) ≥ 0, i = 0, 1, . . . , r, w ∈ W. (2.102)

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α1

α2

λ1

λ2

Figure 2.2: Affine Weyl chambers of A2 at for the level k = 2.

Due to the presence of the subgoup T ⊂ W, the Weyl chambers are now polytopes of finitevolume rather than infinite cones. In particular, they contain a finite number of weights.They are also referred to as Weyl alcoves. Note that because of the level dependenceof T ∼= kQ∨, the size of the chambers depends on the level. Figure 2.2 shows the Weylchambers of A2 at for the level k = 2.

The fundamental or dominant chamber corresponds to the element w = 1. Weightsin this chamber have all Dynkin labels positive:

λ =r

∑i=0

λiωi + lδ, l ∈ R, λi ≥ 0. (2.103)

To any weight λ which does not lie on the boundary of some chamber, there is a uniqueWeyl transformation wλ associated such that wλ(λ) lies in the fundamental affine Weylchamber.

The affine Weyl group preserves the scalar product,

(sαλ, sαλ) = (sα(λ + kmα∨), sα(λ + kmα∨)) + 2k(n− [(λ, α) + km]2m|α|2 )

= (λ, λ) + 2kn = (λ, λ). (2.104)

Thus, all weights in a given Weyl orbit have the same length. A W orbit contains infinitelymany weights and has a unique weight in he fundamental Weyl chamber.

43 FS2016


k = 0

k = 1

k = 2

k = 3

k = 4

1s1 s0s0s1s1s0s1 s1s0

!1 !1 !2!2!3 !3 !4!4 · · ·· · ·

Figure 2.3: Affine Weyl chambers of A1 at levels k = 0, . . . , 4.

Example: the Weyl group of A1. The affine Weyl group of A1 is generated by thereflections s0, s1. Their actions on a weight λ = [λ0, λ1] are given by

s0λ = λ− λ0α0 = [−λ0, λ1 + 2λ0], (2.105)

s1λ = λ− λ1α1 = [λ0 + 2λ1,−λ1]. (2.106)

Let the level of λ be k. Using λ0 = k− λ1, we can rewrite the simple affine Weyl reflectionsas

s0λ = [−k + λ1, 2k− λ1], (2.107)

s1λ = [k + λ1,−λ1]. (2.108)

We finds0s1λ = [−k− λ1, 2k + λ1]. (2.109)

This means that s0s1 translates the finite part of λ by 2kω1 = kα1 = kα∨1 . This means thats0s1 corresponds to the basic translation operator tα∨1

. The structure of W is

W = (s0s1)n, s1(s0s1)

n | n ∈ Z. (2.110)

The translation s0s1 has no finite order. The above expressions for the actions of the groupelements can be used together with Eq. (2.102) to determine the boundaries of the affineWeyl chambers, which are displayed in Figure 2.3. We see that the size of chambersincreases with the level.

F

Example: the Weyl group of A2. The affine Weyl group of A2 is generated by thereflections s0, s1, s2. Their actions on a weight λ = [λ0, λ1, λ2] are given by

s0λ = [−λ0, λ0 + λ1, λ0 + λ2], (2.111)

s1λ = [−λ0 + λ1,−λ1, λ1 + λ2], (2.112)

s2λ = [−λ0 + λ2, λ1 + λ2,−λ2]. (2.113)

Using again λ0 = k− λ1 − λ2, we find

tα∨1= s2s0s2s1, tα∨2

= s1s0s1s2. (2.114)

The Weyl chambers of A2 change in size with the level. For k = 2, they are shown inFigure 2.2.

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Ar Br

Cr

D2l+1

D2l D2l

E6 E7

Figure 2.4: Outer automorphisms of the Dynkin diagrams of the untwisted affine Liealgebras

2.9 Outer automorphisms

Let D(g) be the symmetry group of the g Dynkin diagram and D(g) the symmetry group ofthe g Dynkin diagram. These are the sets of symmetry transformations of the simple rootsthat preserve the scalar product and hence the Cartan matrix. Since the scalar productof the affine roots only depends on their finite parts, it is enough to consider the finiteprojection of the system of simple roots. A simple root is thus mapped into another simpleroot with the same mark and comark.

We define the group of outer automorphisms of the Dynkin diagram of g as

O(g) = D(g)/D(g). (2.115)

This quotient makes sense as D(g) is a subgroup of D(g). O(g) is thus the set of symmetrytransformations of the Dynkin diagram of g that are not symmetry transformations of theDynkin diagram of g. The outer automorphisms of the Dynkin diagrams of the untwistedaffine Lie algebras are given in Figure 2.4.

In terms of the action of their generating element on an arbitrary weight λ = [λ0, . . . , λr],the outer automorphism groups of the affine Lie algebras are given in Table 2.3 (for thecases, where O(g) 6= 1). Note that the action of O(g) does not change the level sinceevery fundamental weight is mapped into another fundamental weight of the same comark.It is clear that the set of dominant weights Pk

+ is mapped into itself. Thus, O(g) preservesthe fundamental Weyl chamber.

45 FS2016


g O(g) action of generatorsAr Zr+1 aλ = [λr, λ0, . . . , λr−2, λr−1]Br Z2 aλ = [λ1, λ0, . . . , λr−1, λr]Cr Z2 aλ = [λr, λr−1, . . . , λ1, λ0]D2l Z2 ×Z2 aλ = [λ1, λ0, λ2 . . . , λr, λr−1]

aλ = [λr, λr−1, . . . , λ1, λ0]D2l+1 Z4 aλ = [λr−1, λr, λr−2, . . . , λ1, λ0]

E6 Z3 aλ = [λ1, λ5, λ4, λ3, λ6, λ0, λ2]E7 Z2 aλ = [λ6, λ5, λ4, λ3, λ2, λ1, λ0, λ7]

Table 2.3: Action of the generating element of the outer automorphisms

Let A be a generic element of O(g). Its action on an affine weight is given by

Aλ = kAω0 +r

∑i=1

λi A(ωi − a∨i ω0), (2.116)

where k is the level of λ. This follows directly from Eq. (2.83). The second term on ther.h.s. of Eq. (2.116) acts on the finite part of λ like an automorphism on the finite weightlattice that leaves the origin fixed. It is, in fact, an element wA of the finite Weyl group.The sum in Eq. (2.83) is the affine extension of wAλ at level zero, wAλ− kω0. Therefore,

Aλ = k(A− 1)ω0 + wAλ. (2.117)

In general, wA can be characterized as follows. Let wi be the longest element of W(i), thesubgroup of the finite Weyl group generated by all sj, j 6= i. Then,

wA = wiw0 for i such that Aω0 = ωi, (2.118)

where w0 is the longest element of W.Note, that outer automorphisms must preserve the commutation relations of the algebra.

Example: the outer automorphism group of A1. For A1, the only non-trivial outerautomorphism is

a := ω0 ↔ ω1. (2.119)

Since W = 1, s1, wa = s1. Comparing

a[λ0, λ1] = [λ1, λ0] = [λ1, k− λ1] (2.120)

to

a[λ0, λ1] = k(a− 1)ω0 + s1[λ0, λ1] (2.121)

= k(ω1 −ω0) + [λ0 + 2λ1,−λ1] = [λ1, k− λ1],

we find that this is correct.

F

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Part 2. Generalizations and extensions: Affine Lie algebras3.1 Hasse diagrams 55

(1,0) (0,1)

(1,1)

Figure 3.2: Hasse diagram of the positive roots of A2. The numbers (m1, m2) denote theroot vector.

The first step is to distribute the simple roots evenly on a horizontal line aroundthe origin. This is achieved by the following horizontal projection Px of the simpleroots ↵i:

Px(↵i) =i 1

n 1 1

2 xi, (3.3)

where n is the rank of the Lie algebra. The horizontal position of a generic root↵ = mi↵i can now be defined as

Px (↵) = mi Px (↵i) = mixi. (3.4)

Note that the explicit summation of the index i has been dropped. From now on,any contracted index will be summed over. We can formalize the above a bit byintroducing a projection vector ' that satisfies

(↵i|') = xi. (3.5)

Expanded in the basis of simple co-roots, the projection vector ' explicitly reads

' = (A1)ijxj ↵_i . (3.6)

When we take its inner product with a generic root ↵, we see that it indeed gives usthe desired projection (3.4): (↵|') = mixi. The complete projection P = (Px, Py)can then be written as

Px(↵) = (↵|') , (3.7a)

Py(↵) = (↵|_) , (3.7b)

where the projection in the vertical coordinate y is just the height of the root.Note that the horizontal coordinate (3.3) of a simple root ↵i strongly depends on

its number i. If the order of the simple roots is changed, the Hasse diagram changesshape too. The best looking diagrams are produced when the ordering of nodes inthe Dynkin diagram (and thus the ordering of simple roots) matches the connectionsbetween the nodes. See also Figure 3.3.

Figure 2.5: Hasse diagram for the roots of A2.

Example: the outer automorphism group of A2. For A2, the basic element a maps

a := ω0 → ω1 → ω2 → ω0. (2.122)

We find that i = 1 and the longest element of W(1) is s2. Using w0 = s2s1s2 = s1s2s1, wefind

wa = s2s2s1s2 = s1s2. (2.123)

We find indeed by direct calculation that

a[λ0, λ1, λ2] = k(a− 1)ω0 + s1s2[λ0, λ1, λ2] (2.124)

= k(ω1 −ω0) + [λ0 + 2λ2 + λ1,−λ1 − λ2, λ1]

= [λ2, k− λ1 − λ2, λ1] = [λ2, λ0, λ1].

F

2.10 Visualizing the root systems of affine Lie algebras

We have seen how higher rank root systems of simple Lie algebras can be visualized inSec. 1.13. Affine root systems, being infinite, necessitate other means of visualizations.

A good means of visualization are Hasse diagrams. A Hasse diagram is a graph thatdisplays the ordering between the different elements of a set, in our case, the roots. Wehave seen that we can define an ordering on the roots by saying one root is larger thanthe other, α ≥ β, if their difference is a positive root. Additionally, we need to introduce acover relation: a root α is said to cover β, α β, if there is no root γ smaller than α andbigger than β. We can now draw a Hasse diagram using the following rules:

• If α ≥ β, the vertical coordinate for β is less than that for α.

• If α β there is a straight line connecting α and β.

Of course, we can also use Hasse diagrams for finite root systems, resulting in a finitegraph. As a first simple example, the roots of A2 are displayed in Figure 2.5. The linesjoining the roots correspond to the Weyl reflections with respect to simple roots which turnthe simple root below into the one above. In the figures, each color and angle of a lineencodes a Weyl reflection, see e.g. in Fig. 2.6 for the roots of A4. We can also use a Hassediagram to visualize the Serre construction, where all elements of the algebra are producedvia commutators of the Chevalley generators, see Figure 2.7 for the Serre construction ofA4. While the root systems of simple Lie algebras can be visualized by other means, thetrue power of the Hasse diagrams is their ability to depict the infinite affine root systems.Figure 2.8 shows the roots systems of A1, C2, D4, A8, D7 and E7. The graphs display the

47 FS2016

Part 2. Generalizations and extensions: Affine Lie algebras56 Chapter 3 Visualizations

(1,0,0,0)

(0,1,0,0) (0,0,1,0)

(0,0,0,1)

(1,1,0,0)

(0,1,1,0)

(0,0,1,1)

(1,1,1,0) (0,1,1,1)

(1,1,1,1)

1 2 3 4

(1,0,0,0)

(0,1,0,0) (0,0,1,0)

(0,0,0,1)

(1,1,0,0) (1,0,1,0) (0,1,0,1)

(1,1,1,0) (1,1,0,1)

(1,1,1,1)

3 1 2 4

Figure 3.3: Two Dynkin diagrams (below) and Hasse diagrams (above) of the same Liealgebra, A4. The ordering of nodes in the left Dynkin diagram, indicated withnumbers below the nodes, is canonical. The ordering of nodes in the rightDynkin diagram does not match the connections between them, resulting in aHasse diagram with crossing lines.

The lines drawn in a Hasse diagram represent the Weyl reflections in the simpleroots. Say there is a root ↵ projected to the point (x, y). Then the root ↵ + ↵i

connected to it by the line of the fundamental Weyl reflection wi gets projected tothe point (x + xi, y + 1). The line of a fundamental reflection is therefore drawn atan angle given by

wi = tan1 1

xi. (3.8)

Because xi is unique for all i, the n distinct fundamental reflections wi all are drawnat di↵erent angles, and reflections in the same simple root are drawn parallel. Todistinguish between them even further they will get drawn in di↵erent colors, rangingfrom blue (the first fundamental reflection) to red (the nth).

The Hasse diagram of the full root system is symmetric around the origin, becauseof the Chevalley involution (2.21). It is therefore customary to draw only the positiveroots in a Hasse diagram.

Following the above procedure it is straightforward, though sometimes tedious,to draw a Hasse diagrams of any root system. Figure 3.4 displays for example theHasse diagrams of various root systems.

Figure 2.6: Hasse diagram for the roots of A4.58 Chapter 3 Visualizations

ade1ade2

ade3ade4

(a) Legend

e1 e2 e3 e4[e1, [e2, e3]] [e2, [e3, e4]]

(b) Positive Chevalley generators

[e1, e2] [e2, e3] [e3, e4]

(c) Single commutators

[e1, [e2, e3]] [e2, [e3, e4]]

(d) Double commutators

[e1, e2] [e3, e4]

[e1, [e2, [e3, e4]]]

(e) Triple commutators

Figure 3.5: The Serre construction for A4.

3.1.1 Visualizing the Serre construction

Hasse diagrams can serve as a neat tool to visualize the results of the Serre con-struction, the step-by-step construction of the full algebra from the Cartan matrix(see Example 2.2). One then has to interpret the points in the diagram not as roots,but as the generator they belong to. Furthermore, the lines can then be interpretedas the adjoint action of the respective positive Chevalley generators. Starting atthe bottom, the vertical steps in the diagram then represent the steps of the Serreconstruction.

Figure 3.5 displays the Serre construction for the Lie algebra A4. One startsout with just the positive Chevalley generators (Figure 3.5b). The first step is totake all (single) commutators [ei, ej ] of the positive Chevalley generators that areconsistent with the Chevalley relations (2.16), the Serre relations (2.17), and theJacobi identity (2.3), which results in Figure 3.5c. This procedure is then iterated(Figure 3.5d and 3.5e) until it no longer yields new generators.

The analogy presented above is only valid up to a certain point. The Serre

Figure 2.7: Hasse diagram for the Serre construction of A4.

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Figure 2.8: Hasse diagram for the roots systems of A1, C2, D4, A8, D7 and E7.

49 FS2016


symmetry structure of the various affine Lie algebras and give a more intuitive idea of theirstructure. All illustrations in this section are reproduced with permission from [Nut10],where the visualization via Hasse diagrams is explained in more detail.

2.11 Highest weight representations

In contrast to the case of simple Lie algebras, all non-trivial representations of an affineLie algebra are infinite-dimensional. Again, the most interesting representations are thehighest weight representations, as they allow for an easy investigation of unitarity andhave many other similarities to the finite-dimensional case.

Highest weight representations are characterized by a unique highest state |λ〉 which isannihilated by the action of all ladder operators for positive roots,

Eα0 |λ〉 = E±α

n |λ〉 = Hin|λ〉 = 0, n > 0, α > 0. (2.125)

The eigenvalue of this state, λ, is the highest weight of the representation

Hi0|λ〉 = λi|λ〉, k|λ〉 = k|λ〉, L0|λ〉 = 0, i 6= 0. (2.126)

Setting the L0 eigenvalue to zero is a matter of convention; a redefinition of L0 would yieldany desired value. In the Chevalley basis, the eigenvalues are the Dynkin labels:

hi0|λ〉 = λi|λ〉, i = 0, . . . , r. (2.127)

All states in this representation are generated by the action of the lowering operators on|λ〉. Since k commutes with all the generators, these states all have the same k eigenvalue.From now on, k will be identified with its eigenvalue k, the level. In most applications, k isfixed from the onset.

The analogues of the irreducible highest weight representations of g are those representa-tions whose projections onto the sl(2) subalgebra associated to any real root are finite. It issufficient to concentrate on simple roots.

The weight system Ωλ is the set of all weights in the representation of the highest-weight state |λ〉. An analysis parallel to the simple case shows that any weight λ′ ∈ Ωλsatisfies

(λ′, α∨i ) = −(pi − qi), i = 0, . . . , r (2.128)

for some positive integers pi, qi. This implies that

λ′i ∈ Z, i = 0, . . . , r. (2.129)

For the highest weight λ, pi = 0 for all i, therefore

λ′i ∈ Z+, i = 0, . . . , r. (2.130)

This requires in particular that

λ0 = k− (λ, θ) ∈ Z+. (2.131)

Since (λ, θ) ∈ Z+, this means that k must be a positive integer, bounded from below by(λ, θ):

k ∈ Z+, k ≥ (λ, θ). (2.132)

A far-reaching consequence of this constraint is that for a fixed value of k, there can onlybe a finite number of dominant highest-weight representations. For example, for k = 1, the

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only such representations are those with highest weight ωi such that the correspondingsimple root αi has unit comark. Since a∨0 = 1 for all g, ω0 is always dominant. The level-1representation with highest weight ω0 is called the basic representation.

For Ar, all comarks are one. There are thus r + 1 dominant highest-weight representa-tions at level 1 whose highest weights are the ωi, i = 1, . . . , r.

In the following, we will use the notation gk for the algebra g at level k.

Example: Dominant highest weight representations of A2 at level 2. For A2, all thecomarks are one, so the set of all dominant highest-weight representations at level two isgiven by

[2, 0, 0], [0, 2, 0], [0, 0, 2], [1, 1, 0], [1, 0, 1], [0, 1, 1]. (2.133)

F

Example: Dominant highest weight representations of G2 at level 2. For G2, thecomarks are a∨0 = a∨2 = 1, a∨1 = 2, which leads to the set of all dominant highest-weightrepresentations at level two being

[2, 0, 0], [0, 0, 2], [1, 0, 1], [0, 1, 0]. (2.134)

F

Representations that decompose into finite irreducible representations of sl(2) and canfurther be written as a direct sum of finite-dimensional highest-weight spaces are calledintegrable. The adjoint representation, although not a highest weight representation isintegrable. The first requirement is obviously met, and the direct sum decomposition is theroot space decomposition into finite roots and imaginary roots. Dominant highest-weightrepresentations are also integrable.

Moreover, if(Ja

n)† = Ja

−n, or (Hin)

† = Hin, (Eα

n)† = E−α

−n, (2.135)

dominant highest-weight representations are easily checked to be unitary. For instance,

|Eα−n|λ〉|2 = 〈λ|E−α

n Eα−n|λ〉 =

2|α|2 [nk− (α, λ)]〈λ|λ〉 ≥ 0 (2.136)

since for n > 0, any α and λ dominant

nk− (α, λ) = [k− (λ, θ)] + (n− 1)k + (θ − α, λ) ≥ 0. (2.137)

The condition for the simple case given in Eq. (1.52) is for dominant highest weightsequivalent to the existence of the singular vectors

Eαi0 |λ〉 = E−θ

1 |λ〉 = 0 (2.138)

and(E−αi

0 )λi+1|λ〉 = (Eθ−1)

k−(λ,θ)+1|λ〉 = 0, i 6= 0. (2.139)

In the Chevalley basis, these vectors read

ei|λ〉 = ( f i)λi+1|λ〉 = 0, i = 0, . . . , r. (2.140)

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In sharp contrast to the simple Lie algebras, the representation space Lλ resulting fromquotienting out these singular vectors is not finite dimensional. The imaginary root δ canbe subtracted from any weight without leaving the representation:

If λ′ ∈ Ωλ, then λ′ − nδ ∈ Ωλ ∀ n > 0. (2.141)

The source of the infinity lies in the absence of a singular vector similar to Eq. (2.139), butinvolving δ.

We will now study how the various weights in Ωλ can be obtained. The algorithm weused for g also works for g, just involving one additional Dynkin label. In the affine case,however, the algorithm never terminates.

We define the grade to be the L0 eigenvalue shifted such that L0|λ〉 = 0 for the higheststate |λ〉. At grade zero, the states are obtained from |λ〉 by applying the finite Lie algebragenerators, as they do not change the L0 eigenvalue. The finite projection of all the weightsat grade zero are all the weights in the g irreducible finite-dimensional representation withhighest weight λ. The weights at grade one are obtained from those at grade zero thathave λ0 > 0 by subtraction of α0, followed again by the subtraction of all possible finitesimple roots. The analysis of the higher grades follows the same pattern.

An important point is that the finite projections of the affine weights at a fixed value ofthe grade are organized into a direct sum of finite-dimensional weight spaces. This showsthat dominant highest-weight representations are integrable.

Finally, we must give the multiplicity of each weight. When the L0 eigenvalue is takeninto account, the multiplicities of the weights are clearly finite. It can be calculated from amodified version of the Freudenthal recursion formula Eq. (1.135) which keeps track ofthe root multiplicities:[|λ + ρ|2 − |λ′ + ρ|2

]multλ(λ

′) = 2 ∑α>0

mult(α)∞

∑p=1

(λ′ + pα, α)multλ(λ′ + pα). (2.142)

Recall that real roots have multiplicity one, while imaginary roots have multiplicity r.Using our convention for the L0 eigenvalue of the highest weight states, the scalar

product of two affine highest weights does not differ from its finite form:

(λ, µ) = (λ, µ) for λ(L0) = µ(L0) = 0. (2.143)

Thus, with λ = (λ; k; 0) and ρ = (ρ; g; 0),

|λ + ρ|2 = |λ + ρ|2. (2.144)

However, at grade m, λ′ = (λ; k;−m) and

|λ′ + ρ|2 = |λ′ + ρ|2 − 2m(k + g). (2.145)

Multiplicity calculations using the Freudenthal recursion formula are rather involved.The constancy of the weight multiplicities along W-orbits greatly simplifies the analysis.The generating function for such multiplicities is called a string function. Let µ be a weightin Ωλ such that µ + δ /∈ Ωλ and denote the set of such weights as Ωmax

λ. The multiplicity

of the various weights in the string µ, µ− δ, µ− 2δ, . . . is given by the string function

σλµ (q) =

∞

∑n=0

multλ(µ− nδ)qn. (2.146)

For more complicated representations, several string functions are required. The completeinformation about the multiplicities of all the weights in the representation is contained in

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[1, 0]1

[1, 0]1

[1, 0]2

[1, 0]3

[1, 0]5

[1, 2]1

[1, 2]1

[1, 2]2

[1, 2]3

[3,2]1

[3,2]1

[3,2]2

[3,2]3[3, 4]1 [5,4]1

↵0

↵1

Figure 2.9: Weights at the first few grades of the basic representation of (A2)1. Themultiplicity is given in the subscript. The colors encode the orbits under the Weyl group.

the set of string functions σλµ (q) for all µ ∈ Ωmax

λ. However, since weight multiplicities are

constant along Weyl orbits, that is,

σwλµ (q) = σλ

µ (q), (2.147)

it is sufficient to know the string functions for those weights in Ωmaxλ

that are also dominant(recall that a Weyl orbit contains exactly one element in the fundamental chamber). Wenote further, that all the weights in Ωλ must also be in the same congruence class as λ.The number of independent string functions required to fully specify the representationof highest weight λ is thus equal to the number of integrable weights at level k that arein the same congruence class as λ. For example, in (A1)2, there are three integrableweights, [2, 0], [0, 2], [1, 1]. The first two are in the same conjugacy class, therefore twostring functions are needed in this case.

Example: the basic representation of (A2)1. Let us consider the basic representationof (A2)1 with highest weight [1, 0]. Using the algorithm described above, it is easy to writedown the weights in the first few grades, see Figure 2.9. The first weight with non-trivialmultiplicity is [1, 0] at grade 2. With λ = (0; 1; 0) and λ′ = (0; 1;−2), we find |λ + ρ|2 = 1

2and |λ′ + ρ|2 = 1

2 − 12. To calculate the r.h.s. of Eq. (2.142), we must consider all theweights λ′+ pα for p, α > 0, up to grade zero. The positive roots of A2 are α1, ±α1 + nδ, nδfor n > 0, they all have multiplicity one. Finally, we find that the r.h.s. of Eq. (2.142) isequal to 24, resulting in multiplicity 2 for [1, 0] at grade 2.

We have seen that we can simplify the calculation of the multiplicities by taking intoaccount the Weyl orbits. We have already studied the action of the Weyl group A1 on theDynkin labels in an earlier exercise. We start with the dominant weight [1, 0] at grade 0.We find the orbit

[1, 0]s0−→ [−1, 2]

s1−→ [3,−2]s0−→ [−3, 4]

s1−→ [5,−4] · · · (2.148)

It is important to take into account that s0 increases the L0 eigenvalue of the weight it actson and thus the grade by λ0. Thus, the second weight in the above sequence is at grade 1

53 FS2016


and the fourth and fifth are at grade 4. s1 does not change the grade. In Figure 2.9, theW-orbits are color coded. We see that each of the weights [1, 0] = (0; 1;−m) represents oneorbit. The orbits are consistent with the multiplicities of the weights given in the figure. Wesee that the first few coefficients of the string function σ

[1,0][1,0] are given by 1, 1, 2, 3, 5, . . . .

This is the number p(n) of inequivalent decompositions of n into positive integers (numberof integer partitions) for which a closed formula exists:

σ[1,0][1,0] (q) =

∞

∑n=0

multλ(µ− nδ)qn =∞

∑n=0

p(n)qn =∞

∏n=1

11− qn , (2.149)

it is the inverse of the Euler function.

F

Literature

The discussion on affine Lie algebras follows again [DMS97] and [FS97], using the notationof [DMS97] and supplementing some material from [FS97]. [Fuc92] treats affine Liealgebras in more mathematical detail, in particular the twisted case. The discussion of thevisualization of affine root systems via Hasse diagrams as well as the illustrations can befound in [Nut10].

References

[Nut10] T. Nutma. “Kac-Moody symmetries and gauged supergravity”. PhD thesis. Rijksuniver-siteit Groningen, Dec. 2010. URL: http://www.rug.nl/research/portal/files/14628607/15_thesis.pdf.



[Fuc92] J. Fuchs. Affine Lie Algebras and Quantum Groups: An Introduction, with Applicationsin Conformal Field Theory. 1st. Cambridge Monographs on Mathematical Physics. Cam-bridge University Press, 1992. ISBN: 0521415934.

FS2016 54

http://www.rug.nl/research/portal/files/14628607/15_thesis.pdf

http://www.rug.nl/research/portal/files/14628607/15_thesis.pdf



Part 3. Advanced topics: Beyond affine Lie algebras

Part 3

Advanced topics: Beyond affine Liealgebras

In this last part of the course, we are entering advanced territory of further extensions ofsimple Lie algebras beyond the affine case or in different directions. The motivation forthese extensions comes again from physics, they are the mathematical answers to somenecessities that arise naturally in the physics context. Due to the advanced nature of thesetopics, this part of the course will have more the character of an overview. We will firstvisit the Virasoro algebra which arises naturally in two-dimensional conformal field theoryand is a further extension of the untwisted affine Lie algebras. Next, we will consider Liesuper-algebras, which bring the physical concept of fermions to Lie algebras by the inclusionof fermionic Dynkin nodes. Applications of this structure can be found for example in thestudy of integrable spin chains (the tJ model). Lastly, we will have a look at so-calledquantum groups that solve the problem of how to deform a Lie algebra by a new parameterq. The simplest example of a supergroup is motivated by the XXZ spin chain, where thespins on the chain are subject to an anisotropy in the form of an external magnetic fieldin the Z-direction. Another important example of a quantum group is the Yangian, whichappears in the algebraic Bethe ansatz resp. quantum inverse scattering method to solveintegrable spin chains.

3.1 The Virasoro algebra

In the second part of this course, we have constructed the untwisted affine Lie algebrasby extending the simple Lie algebras by a central element k and a derivation −L0. Animportant extension of the affine Lie algebras arises when we introduce another centralelement c and infinitely many generators Ln, n ∈ Z. Their commutation relations are givenby

[Ln, Lm] = (n−m)Ln+m +c

12n(n2 − 1)δn+m,0, (3.1)

[Ln, Lm] = 0, (3.2)

[Ln, Lm] = (n−m)Ln+m +c

12n(n2 − 1)δn+m,0, (3.3)

[Ln, c] = 0. (3.4)

55 FS2016


These bracket relations define a Lie algebra, the so-called Virasoro algebra. The relationsinvolving both the Ln and the generators of g read

[Lm, Jan] = −n Ja

n+m, (3.5)

[Ln, k] = [c, Jan] = [c, k] = 0. (3.6)

Comparing with the commutation relations of L0 in Eq. (2.15) in the affine case, thenaming convention for the derivation suddenly makes sense, as it is now identified withthe Virasoro generator −L0.

Let us state a few important facts about the Virasoro algebra Vir. Vir possesses atriangle decomposition,

Vir+ ⊕Vir0 ⊕Vir− (3.7)

into the subalgebras generated by the positive, zero , and negative modes

Vir+ = spanLn | n > 0, Vir0 = spanL0, c, Vir− = spanLn | n < 0. (3.8)

Vir0 is maximal Abelian subalgebra. Owing to the existence of a triangular decomposition,the representation theory of the Virasoro algebra parallels to a large extent the one of affineLie algebras. There are, in particular, highest-weight representations.

We will briefly study the representations with respect to the Ln. The representations oftheir anti-holomorphic counterparts are constructed along the same lines. As we have seen,the holomorphic and anti-holomorphic components of the overall algebra decouple.

Since no pairs of generators in (3.1) commute, we choose a single operator, L0, whichwill be diagonal in the representation space, which in this context is also called a Vermamodule. We denote the highest-weight state by |h〉, with eigenvalue h of L0:

L0|h〉 = h|h〉. (3.9)

Since [L0, Lm] = −mLm, Lm for m > 0 is a lowering operator for h, and L−m, m > 0 araising operator. We adopt the convention,

Ln|h〉 = 0, n > 0. (3.10)

A basis for the other states of the representation, the descendant states, is obtained byapplying the raising operators in all possible ways:

L−k1 L−k2 . . . L−kn |h〉, 1 ≤ k1 ≤ · · · ≤ kn, (3.11)

where, by convention, the L−ki appear in increasing order of ki. This state is an eigenstateof L0 with eigenvalue

h′ = h + k1 + k2 + · · ·+ kn = h + N, (3.12)

where N is the level of the state (which differs from what we called the level in the affinecase, as there, it referred to the eigenvalue of the operator k).

The lowest levels of a Verma module. The first few levels of the representation withhighest state |h〉 are spanned by the states given in Table 3.1. Looking at the numberof distinct, linearly independent states at level N, we find again the coefficients of theseries of number of partitions p(N) of the integer N that we have already encountered inEq. (2.149).

F

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level number of states states0 1 |h〉1 1 L−1|h〉2 2 L2

−1|h〉, L−2|h〉3 3 L3

−1|h〉, L−1L−2|h〉, L−3|h〉4 5 L4

−1|h〉, L2−1L−2|h〉, L−1L−3|h〉, L2

−2|h〉, L−4|h〉

Table 3.1: The first few levels of the Verma module generated by |h〉

Using L†m = L−m, we define an inner product on the Verma module. The inner product

of two states L−k1 L−k2 . . . L−kn |h〉 and L−l1 L−l2 . . . L−ln |h〉 is

〈h|Lkm . . . Lk1 L−l1 . . . L−ln |h〉, (3.13)

where the dual state 〈h| satisfies

〈h|Ll = 0, j < 0. (3.14)

Note that the inner product of two states vanishes unless they belong to the same level. Ingeneral, two eigenspaces of a Hermitian operator (here, L0) having different eigenvaluesare orthogonal.

Since c is a central generator, all vectors in the Verma module generated by |h〉 havethe same eigenvalue of c, called the central charge.

The Virasoro algebra is intimately linked with conformal transformations in 2d andtherefore of paramount importance in the study of 2d conformal field theory (CFT), i.e. aquantum field theory which is covariant under conformal transformations.

A conformal transformation of the coordinates in d spacetime dimensions is an in-vertible mapping x → x′ which leaves the metric tensor gµν invariant up to a scale:

g′µν(x′) = Λ(x)gµν(x). (3.15)

A conformal transformation is locally equivalent to a (pseudo-)rotation and a dilation. Thename conformal comes from the fact that the conformal group preserves angles.

In d = 2, conformal invariance takes a new meaning. The reason is that here, there is aninfinite variety of locally conformal transformations, namely the holomorphic mappings ofthe complex plane onto itself. Consider the coordinates (z0, z1) on the plane. The covariantmetric tensor gµν transforms under a change of the coordinate system given by zµ → wµ(x)as

gµν →(

∂wµ

∂zα

)(∂wν

∂zβ

)gαβ. (3.16)

The condition that the above transformation is conformal is g′(w) ∝ g(z), in other words,(∂w0

∂z0

)2

+

(∂w0

∂z1

)2

=

(∂w1

∂z0

)2

+

(∂w1

∂z1

)2

, (3.17)

∂w0

∂z0∂w1

∂z0 +∂w0

∂z1∂w1

∂z1 = 0. (3.18)

This corresponds in turn to

∂w1

∂z0 =∂w0

∂z1 and∂w0

∂z0 = −∂w1

∂z1 , (3.19)

57 FS2016


which are the Cauchy–Riemann equation for holomorphic maps. The result of the above isthat CFTs in 2d can be solved exactly, which is why they hold such an important place intheoretical physics.

Any holomorphic infinitesimal transformation can be expressed as

z′ = z + ε(z), ε(z) =∞

∑n=−∞

cnzn+1 (3.20)

for the complex variable z = z0 + iz1, where by hypothesis, the infinitesimal mappingadmits a Laurent expansion around z = 0. The effect of such a mapping on a spinless anddimensionless field φ(z, z) living on the complex plane is

δφ = −ε(z)∂φ− ε(z)∂φ = ∑n[cnlnφ(z, z) + cn lnφ(z, z)], (3.21)

where we have introduced the generators

ln = −zn+1∂z, ln = −zn+1∂z. (3.22)

These generators obey the following commutation relations:

[ln, lm] = (n−m)ln+m, (3.23)

[ln, lm] = 0, (3.24)

[ln, lm] = (n−m)ln+m. (3.25)

Thus the conformal algebra is the direct sum of two isomorphic algebras (generatedby the ln and ln, respectively). It is also called the Witt algebra. Each of these twoinfinite-dimensional algebras contains a subalgebra generated by l−1, l0 and l1. This is thesubalgebra associated with the global conformal group. Indeed, from Eq. (3.20) we seethat l−1 = −∂z generates translations on the complex plane, l0 = −z∂z generates scaletransformations and rotations, and l1 = −z2∂z generates special conformal transformations.

In CFT, however, one needs unitarizable representations of the symmetry algebra.However, just like in the case of the loop algebra, the Witt algebra does not have anynon-trivial unitary representations. Therefore, we must again introduce a central extension.It can be shown on quite general grounds that it must just consist of a complex number onthe right hand side of the bracket relation (the central charge). In physics language, we saythat the conformal symmetry develops an anomaly.

3.2 Lie superalgebras

Superalgebras are related to the concept of particles of different statistics, i.e. fermions andbosons. They contain even and odd generators, the odd ones being of fermionic nature.Mathematically speaking, these algebras have a Z2–grading.

Lie superalgebras present a different kind of generalization of complex semi-simple Liealgebras from the ones we have studied so far. While they share many of the key conceptsof simple Lie algebras such as the Cartan subalgebra, roots, weights, the Cartan matrix andDynkin diagrams, new features arise due to the possibility of vanishing Killing forms. Thelatter give rise to a host of pathological possibilities, which makes many definitions involvedin the study of superalgebras more bulky as they must either encompass or exclude them.

We only have time for a brief look at simple Lie superalgebras and, apart from thebasic definitions, will focus on their classification. We will concentrate on one class of

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complex simple Lie superalgebras, namely the basic classical ones, which are nearest to thesimple case we are familiar with. They also hold the most interest from the point of viewof mathematical physics.

The Lie superalgebras are however not "the superalgebra" that appears in particlephysics beyond the standard model, which is an extension of the Poincaré algebra. Ofcourse, all superalgebras share the basic Z2–grading. In the context of theoretical physics,Lie superalgebras appear as symmetry algebras of integrable spin chains, the simplestexample being the tJ model.

First of all, we need to introduce the grading. Inspired by the properties of integernumbers, we introduce the following structure with product rules

even · even = even, (3.26)

even · odd = odd, (3.27)

odd · odd = even. (3.28)

The above grading is also called a Z2–grading. Let us first define a graded vector spaceV. Let V be a complex vector space of dimension m + n, m, n ∈ Z+, and let A1, . . . , Am+n

be a basis of V. Then any A ∈ V can be written as

A =m+n

∑j=1

aj Aj, aj ∈ C. (3.29)

V can be graded by saying that

A =m

∑j=1

aj Aj, is even and (3.30)

A =m+n

∑j=m+1

aj Aj, is odd. (3.31)

Thus the even elements only involve the first m basis elements, while the odd elementsinvolve only the remaining n basis elements. Any A ∈ V that is either even or odd is saidto be homogeneous, and the degree (or parity) of such elements is defined as

deg A =

0 if A is even1 if A is odd.

(3.32)

The set of even elements of V forms the even subspace V0, the odd elements form the oddsubspace V1:

V = V0 ⊕V1. (3.33)

If we supplement the graded vector space V by an associative product, the resultingstructure is an associative superalgebra.

Let gs be a graded vector space with g0 and g1 being its even, respectively odd subspacesand

dim g0 = m, dim g1 = n, m ≥ 0, n ≥ 0, m + n ≥ 1. (3.34)

Assume that ∀A, B ∈ gs, there exists a generalized Lie product or supercommutator[A, B] with the following properties:

• [A, B] ∈ gs ∀ A, B ∈ gs,

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• ∀ A, B ∈ gs, ∀ a, b ∈ C,

[aA + bB, C] = a[A, C] + b[B, C], (3.35)

• for any two homogeneous A, B ∈ gs, also [A, B] is homogeneous with degree

deg([A, B]) = (deg A + deg B) mod 2. (3.36)

• for any two homogeneous A, B ∈ gs,

[B, A] = −(−1)(deg A)(deg B)[A, B], (3.37)

• for any three homogeneous A, B, C ∈ gs,

[A, [B, C]](−1)(deg A)(deg C) + [B, [C, A]](−1)(deg B)(deg A)

+ [C, [A, B]](−1)(deg C)(deg B) = 0. (3.38)

(generalized Jacobi identity)

Then gs is called a complex Lie superalgebra with even dimension m and odd dimensionn. We choose a homogeneous basis of gs in which the basis elements A1, . . . , Am+n aresuch that A1, . . . , Am ∈ g0 and Am+1, . . . , Am+n ∈ g1. Then, the structure constants cpq

r canbe defined by

[Ap, Aq] =m+n

∑r=1

cpqr Ar. (3.39)

Any generalized Lie product can be evaluated from the knowledge of the structure constants.The grading implies that

cpqr = −(−1)(deg Ap)(deg Aq)cqp

r . (3.40)

For m ≥ 1, the even subspace g0 is an ordinary Lie algebra. For m ≥ 1, n ≥ 1, theodd subspace g1 of gs is a carrier space for a representation of the Lie algebra g0. Thisrepresentation is called the representation of g0 on g1. Every Lie algebra can be regardedas a special case of a Lie superalgebra for m ≥ 1, n = 0.

When represented as matrices, the elements of a superalgebra are block matrices

M =

(A BC D

). (3.41)

If M ∈ g0, it has the form

M =

(A 00 D

), (3.42)

while if M ∈ g1, it has the form

M =

(0 BC 0

). (3.43)

As in the simple case, we define ad(X)Y = [X, Y], but now using the supercommutator.The Killing form is defined as

K(X, Y) = str(adX adY), X, Y ∈ gs, (3.44)

where str is the supertrace, defined as

strM = Tr A− Tr D (3.45)

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for the block matrix M.

A Lie superalgebra is said to be Abelian or commutative if

[A, B] = 0 ∀A, B ∈ gs. (3.46)

A subalgebra g ⊂ gs is a subset of elements of gs that form a vector subspace of gs andthat is closed under the generalized Lie product,

[A, B] ∈ g for A, B ∈ g. (3.47)

A graded subalgebra g′s ⊂ gs is itself a Lie superalgebra and its even subspace g′0 is asubspace of g0 and its odd subspace g′1 is a subspace of g1. g′s is said to be a proper gradedsubalgebra of gs if at least one element of gs is not contained in g′s. A graded subalgebra g′sis invariant if

[A, B] ∈ g′s ∀A ∈ g′s, B ∈ gs. (3.48)

A Lie superalgebra is said to be simple if it is not Abelian and does not possess a properinvariant graded subalgebra. If gs is a simple Lie superalgebra, its Killing form K( , ) iseither non-degenerate or identically zero. We see that the situation is very different fromthe case of Lie algebras, where a Lie algebra with a zero Killing form cannot be semi-simple.

Unlike in the case of Lie algebras, for Lie superalgebras, there do exist semi-simple Liesuperalgebras that are not expressible as the direct sum of simple Lie superalgebras. Inorder to define them correctly, we first need to introduce the concept of solvability. Letg(0)s = gs and g(k)s = [g(k−1)

s , g(k−1)s ] for each k = 1, 2, 3, . . . . Then the Lie superalgebra gs is

solvable if there exists a value of k for which g(k)s = 0.A Lie superalgebra gs is said to be semi-simple if it does not possess a solvable invariant

graded subalgebra.

Just as for the complex finite-dimensional simple Lie algebras, we want to classify thesimple Lie superalgebras. We will however only look at the so-called classical ones, whichare the only ones of interest for mathematical physics. As before, we will only considercomplex Lie superalgebras.

A simple Lie superalgebra is said to be classical if the representation of its even part g0on its odd part g1 is either irreducible or if it is reducible, it is completely reducible, i.e. canbe written as a direct sum of irreducible representations. If gs is a complex classical simpleLie superalgebra, its even part has the form

g0 = gA0 ⊕ gss

0 , (3.49)

where gA0 is an Abelian complex Lie algebra and gss

0 is a semi-simple complex Lie algebra.Let hss

0 be a Cartan subalgebra of gss0 . Then,

hs = gA0 ⊕ hss

0 (3.50)

is a Cartan subalgebra of gs. The rank r of a classical simple complex Lie algebra is thedimension of its Cartan subalgebra. We can define the roots analogously to the case ofsimple Lie algebras. If α(H) is a linear functional on hs, we can find at least one elementEα ∈ gs such that

[H, Eα] = α(H)Eα ∀H ∈ hs, (3.51)

and α is called a root of gs, and the set of all elements Eα that satisfy Eq. (3.51) forms theroot subspace gs,α. If Eα ∈ g0, α is said to be an even root of gs, while if Eα ∈ g1, α is saidto be an odd root of gs. The set of all distinct non-zero even roots is denoted by ∆0, and

61 FS2016


the set of all distinct odd roots is denoted by ∆1. The set of distinct roots of gs containedeither in ∆0 or ∆1 or both is denoted by ∆s.

hs can be regarded as the subspace of gs corresponding to zero even roots.

The set of classical simple Lie superalgebras can be divided into

1. Basic classical simple Lie superalgebras which posses a non-degenerate bilinearsupersymmetric consistent invariant form K′. This set can in turn be divided into

(a) those for which the Killing form K is non-degenerate, so K = K′.

(b) those for which the Killing form is identically zero, K 6= K′.

2. Strange classical simple Lie superalgebras, which do not posses any non-degeneratesupersymmetric consistent invariant form.

The following is a complete list of the complex classical simple Lie superalgebras:

1. Basic classical simple Lie superalgebras

(a) with non-degenerate Killing form:

i. simple complex Lie algebrasii. • A(r|s), r > s ≥ 0

• B(r|s), r > 0, s ≥ 1• C(s), s ≥ 2• D(r|s), r ≥ 2, s ≥ 1, r 6= s + 1• F(4)• G(3)

(b) with zero Killing form:

• A(r|r), r ≥ 1• D(s + 1|s), s ≥ 1• D(2|1; α), α ∈ C \ 0,−1, ∞

2. Strange classical simple complex Lie algebras

• P(r), r ≥ 2

• Q(r), r ≥ 2.

In the following, we will collect the necessary material in order to define the Cartanmatrices and Dynkin diagrams of basic complex classical simple Lie superalgebras.

For semi-simple complex Lie algebras, we have learned that (α, α) > 0 for all non-zeroroots α. This is not the case for basic complex classical simple Lie superalgebras:

1. If α ∈ ∆0 is an even, non-zero root of gs, (α, α) 6= 0, but it need not be real andpositive. If gs has non-degenerate Killing form, (α, α) ∈ R, but can be positive ornegative.

2. If α ∈ ∆1 is an odd root of gs, it is possible to have (α, α) = 0, even when α is notidentically zero.

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Example: generators and roots of gs = A(1|0). A(1|0) is the complex Lie superalgebrasl(2|1; C). Its even part is given by

g0 = gA0 ⊕ gss

0 , (3.52)

where gA0 is a one-dimensional Abelian Lie algebra and gss

0 = A1. The Cartan subalgebrais two-dimensional with basis elements H1

1 and C, where C is the generator of gA0 and H1

1is the generator of hss

0 = hA1 . Let Epq be the 3× 3 matrix with entry (p, q) = 1 and allothers zero. The generators of the Cartan subalgebra are given by H1

1 = E11 − E22 andC = −E11 − E22 − 2E33. E12, E21, E13, E31, E23 and E32 generate the root subspaces. It iseasy to work out the commutation relations, we have e.g.

[H11 , E12] = 2 E12, [H1

1 , E13] = E13, [H11 , E23] = −E23, (3.53)

[C, E12] = 0, [C, E13] = E13, [C, E23] = E23, (3.54)

[E12, E13] = 0, [E13, E23] = 0, [E12, E23] = E13. (3.55)

A(1|0) has the even roots ±α1 associated with E12, E21 and the odd roots ±α2, ±α3 associ-ated to E23, E32 and E13, E31. We see that α3 = α1 + α2.

Taking the basis elements of A(1|0) in the order

C, H11 , E12, E21, E13, E31, E23, E32, (3.56)

we can calculate adC, which is a diagonal matrix with elements 0, 0, 0, 0, 1,−1, 1,−1 andadH1

1 which is again diagonal with elements 0, 0, 2,−2, 1,−1,−1, 1. From here, it is easyto find the Killing forms K(C, C) = −4, K(H1

1 , H11) = 4 and K(C, H1

1) = 0.The roots have the scalar products

(α1, α1) = 1, (α2, α2) = (α3, α3) = 0, (3.57)

(α1, α2) = (α2, α3) = − 12 , (α1, α3) =

12 . (3.58)

F

In the following, we need to define the concepts of positive, negative and simple rootssuch that as many results from simple Lie algebras carry over as possible. One maindifference however remains. For simple Lie algebras, all choices for the set of positive rootsare equivalent, which is not the case for simple Lie superalgebras.

Let hss0 be the Cartan subalgebra of the semi-simple Lie algebra gss

0 , and suppose achoice of positive roots of gss

0 has been made. Then the subalgebra bss0 of gss

0 that consists ofall elements of hss

0 together with the root subspaces corresponding to α ∈ ∆+ of gss0 forms a

maximal solvable subalgebra of gss0 . If g0 contains a one-dimensional Abelian Lie algebra

gA0 , then we define the subalgebra b0 as

b0 = bss0 ∪ gA

0 , (3.59)

but if gA0 = 0, then

b0 = bss0 . (3.60)

In both cases, b0 is a maximal solvable subalgebra of g0 and is known as a Borel subalgebraof g0. Let b be a maximal solvable subalgebra of the Lie superalgebra gs and b0 ⊂ b. Asthe Cartan subalgebra is contained in b0, also hs ⊂ b, and so a subspace N+ ⊂ gs can bedefined such that

b = hs ⊕N+. (3.61)

63 FS2016


We can introduce a further subspace N− via

gs = N− ⊕ hs ⊕N+. (3.62)

A root α ∈ gs is said to be positive if the intersection of the root subspace gs,α with N+,gs,α ∩N+, is non-trivial and negative if gs,α ∩N− is non-trivial. This definition impliesthat if α is positive, −α is negative and vice versa. Moreover, every positive even root isan extension of a positive root of gss

0 , and every positive root of gss0 extends to a positive

even root of gs. In the exceptional case gs = A(1|1), dimgs,α = 2 for every odd root α, andgs,α ∩N+ and gs,α ∩N− are both non-trivial, implying that for A(1|1), every odd root isboth positive and negative.

A non-zero root α of gs is said to be simple if α is positive, but cannot be expressed asα = β + γ, with β, γ positive roots of gs.

Example: Positive, negative and simple roots of gs = A(1|0). We have seen that∆s = ±α1, ±α2, ±α3. The corresponding root subspaces are all one-dimensional andgenerated by the E±αi .

1. One choice of b has basis C, H11 , Eα1 , Eα2 , Eα3, so thatN+ has basis Eα1 , Eα2 , Eα3.

With this choice, α1, α2, α3 are all positive roots and−α1, −α2, −α3 are negative roots.Since α3 = α1 + α2, the corresponding simple roots are α1 and α2.

2. Another inequivalent choice of b has basis C, H11 , Eα1 , E−α2 , Eα3, so that N+ has

basis Eα1 , E−α2 , Eα3 and hence N− has basis E−α1 , Eα2 , E−α3. However, nowα3 = α1 + α2 is no longer a sum of positive roots and is hence simple. We rename thepositive roots to α′1 = α3, α′2 = −α2, α′3 = α1. The simple roots are now α′1 and α′2,both being odd. Note that (α′1, α′1) = (α′2, α′2) = 0, (α′1, α′2) =

12 .

F

The number of simple roots R is related to the rank r of gs by

R =

r + 1 if gs = A(p, p), p > 0,r for any other basic gs.

(3.63)

Moreover, the set of simple roots α1, . . . , αR is always linearly independent except forA(p, p), p > 0.

We can define the Cartan matrix A of a basic classical simple complex Lie superalgebrags to be a R× R matrix with matrix elements Ajk defined in terms of the simple roots as

Ajk =2(αj, αk)

(αj, αj)if (αj, αj) 6= 0, (3.64)

Ajk =(αj, αk)

(αj, αj′)if (αj, αj) = 0, (3.65)

where αj′ is another simple root such that (αj, αj′) 6= 0. In the first case, Ajj = 2 and theonly possible values of Ajk, j 6= k are 0, -1, -2 and -3. In the second case, Ajj = 0 andAjj′ = 1, which is clearly very different from the allowed values of a Cartan matrix of anordinary simple Lie algebra.

A is an r× r matrix except for A(p, p), p > 0. For D(2|1; α) with α /∈ R, at least oneoff-diagonal element of A is not real.

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Example: Cartan matrices of of gs = A(1|0). With choice (1) for the simple roots, wehad (α2, α2) = 0, but (α2, α1) 6= 0, so we can take α2′ = α1. Then,

A =

(2 −11 0

). (3.66)

With choice (2) for the simple roots, taking α2′ = α1 and α1′ = α2, we find

A =

(0 11 0

). (3.67)

F

We want to again associate to each Cartan matrix a generalized Dynkin diagram.They are constructed according to the following rules:

1. assign to each simple root αj a vertex j which is drawn as

• if αj is even. This is called a white node.

• if αj is odd and (αj, αj) = 0. This is called a grey node.

• if αj is odd and (αj, αj) 6= 0. This is called a black node.

2. draw ljk lines from the vertex j to the vertex k, where

ljk = max|Ajk|, |Akj|. (3.68)

3. add an arrow pointing from the j vertex to the k vertex if |Akj| > 1 (except forD(2|1; α).

Example: Generalized Dynkin diagrams of gs = A(1|0). The generalized Dynkin dia-gram corresponding to the choice (1) is

1 2 (3.69)

The generalized Dynkin diagram corresponding to the choice (2) is

1 2 (3.70)

F

While each simple complex Lie algebra has a unique Dynkin diagram, we see that thisuniqueness is lost in the superalgebra case. Due to the anomalous cases D(2|1; α) andD(2|s), s > 0, the above prescription cannot be reversed to determine the Cartan matrixfrom the Dynkin diagram. Therefore, the Dynkin diagrams play a less important role forsuperalgebras.

For each basic classical simple complex Lie superalgebra gs with non-trivial odd part,there is a distinguished choice of simple roots, which consists of one odd root, all otherR− 1 simple roots being even. The corresponding Dynkin diagram therefore has R− 1white vertices and one grey or black vertex. In Table 3.2, all the Dynkin diagrams of thebasic classical simple complex Lie superalgebras corresponding to the distinguished choiceof the Borel subalgebra are depicted.

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A(r|s) r ≥ s > 01 2 r r + 1 r + s + 1

A(r|r) r ≥ 11 2 r r + 1 2r + 1

B(r|s), r > 0, s > 0 1 2 r r + 1 r− 1 r

B(s), s > 0 1 2 s− 1 s

C(s), s > 0 1 2 s− 1 s

D(r|s), r > 2, s > 0

r + s

r + s− 11 2 s− 1 s

r + s− 2

D(2|s), s > 0

s + 2

s + 11 2

s

D(2|1; α), α 6= −1, 0, ∞

3

2

1

F4 1 2 3 4

G2 1 2 3

Table 3.2: Basic classical simple complex Lie superalgebras corresponding to the distin-guished choice of the Borel subalgebra

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3.3 Quantum groups

In this last section of the course, we touch on a subject which requires the highest levelof abstraction encountered so far, namely quantum groups. They act as a generalizedsymmetry and arise naturally in several independent contexts in theoretical physics, one ofwhich is the study of integrable models.

The concept of quantum group is mathematically speaking equivalent to the one ofthe Hopf algebra, which is an associative algebra with a host of extra structure. We willconcentrate on a particular class of Hopf algebras, namely Uq(g), a deformation of theenveloping algebra U(g) of a complex simple Lie algebra g. Uq(g) depends in a naturalway on the algebra g alone, so the theory of Uq(g) can be viewed as a self-containedoutgrowth of Lie algebra theory. It should be pointed out, however, that the theory ofgeneral quantum groups is much richer and encompasses new phenomena that have noanalogy in Lie algebra theory.

The fundamental new concept to be introduced is the one of a Hopf algebra. Let a bea vector space and F the base field of the vector space (in the following, we take againF = C). A Hopf algebra a is a vector space endowed with five operations

M : a× a→ a (multiplication) (3.71)

η : F → a (unit map) (3.72)

∆ : a→ a× a (co-multiplication) (3.73)

ε : a→ F (co-unit map) (3.74)

γ : a→ a (antipode), (3.75)

which have the following properties:

M (id×M) =M (M× id) (associativity) (3.76)

M (id× η) = id =M (η × id) (existence of unit) (3.77)

(id× ∆) ∆ = (∆× id) ∆ (co-associativity) (3.78)

(ε× id) ∆ = id = (id× ε) ∆ (existence of co-unit) (3.79)

M (id× γ) ∆ = η ε =M (γ× id) ∆ (3.80)

∆ M = (M×M) (∆ ∆) (connecting axiom). (3.81)

At first sight, this structure might seem quite complicated, but we will see that it arisesquite naturally.

Vector spaces with a multiplication satisfying Eq. (3.76) are called associative algebrasand are very common. Many of them have a unit (unital algebras).

By formalizing the notion of unmultiplying objects, one gets the operations of co-multiplication and co-unit. The co-multiplication of an element X ∈ a is the sum of allthose things in a⊗ a which could give X when combines according to an underlying groupstructure.

Vector spaces endowed with a co-multiplication are called co-algebras. the theory ofco-algebras is dual to the one of algebras, hence it is not necessary to consider them in theirown right. The requirement of co-associativity is also natural. Having co-multiplication asa dual operation to multiplication, it makes sense to require also the existence of a co-unitas the dual operation to the unit. An algebra a that possesses the four operationsM, η, ∆and ε is called a bi-algebra.

Finally, we need a way of connecting the operations of multiplication and co-multiplicationnon-trivially. We need to construct a map as the composition of multiplication and co-multiplication, which leads to the concept of the antipode, and hence to Hopf algebras. The

67 FS2016


antipode is a weaker structure than the inverse, it provides a nonlocal linearized inverse. Itmeans that now not individual elements, but certain linear combinations are invertible.

We can express the defining properties Eq. (3.76)–Eq. (3.81) more explicitly in terms ofthe elements X ∈ a. If we use for the multiplicationM,

M : X⊗Y 7→ M(X⊗Y) ≡ X Y, (3.82)

Eq. (3.76) reads now

X (Y Z) = (X Y) Z ∀ X, Y, Z ∈ a. (3.83)

Similarly, the existence of a unit means that there is an element E ∈ a (the unit element)such that

E X = X = X E ∀ X ∈ a. (3.84)

The map η is then given byη : ξ 7→ ξE ∀ ξ ∈ F. (3.85)

Another way of expressing the properties Eq. (3.76)–Eq. (3.81) in a more explicit way isvia commutative diagrams. A diagram is said to be commutative iff the composite mapswhich are obtained by following the arrows are independent of the path used to link anytwo given spaces in the diagram.

The following diagram shows the associativity property Eq. (3.76).

a× a

a× a× a a

a× a

Mid×M

M×id M

(3.86)

The existence of the unit (Eq. (3.77)) is shown below, with the map s : a× F → a the scalarmultiplication.

a× F a F× a

a× a a a× a

s

id×M

s

M×id

MM

(3.87)

Co-associativity (Eq. (3.78)) is shown below:

a× a

a× a× a a

a× a

id×∆ ∆

∆∆×id

(3.88)

Finally, the existence of the co-unit (Eq. (3.79)) is expressed as follows, with the mapi : a→ a× F the inclusion X 7→ X⊗ 1, where 1 is the multiplicative unit of F.

a× F a F× a

a× a a a× a

ii

id×ε

∆∆

ε×id (3.89)

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We see that the third and fourth diagrams are obtained from the first and second byreversing the arrows, in other words, the rules for ∆ are just the same as the rules formultiplication, with the arrows reversed.

When a basis Ja of a is fixed, then multiplication, co-multiplication and antipode canbe expressed in terms of structure constants, just like the bracket relations of a Lie algebra:

Ja Jb = ∑c

µabc Jc, (3.90)

∆(Jc) = ∑a,b

νcab Ja ⊗ Jb, (3.91)

γ(Jc) = ∑a

τca Ja. (3.92)

The associativity property Eq. (3.76) is then expressed as

µbcd µad

f = µabd µdc

f . (3.93)

Similarly, the structural properties Eq. (3.78) and Eq. (3.80) read

νacdνb

e f = νaecνb

f d, (3.94)

νabcτc

d µbde = νa

dcτdb µbc

e . (3.95)

Example: Universal enveloping algebra U(g) of a complex Lie algebra g. The uni-versal enveloping algebra U(g) of a Lie algebra g consists of all finite formal power seriesin the elements of g. U(g) is an associative algebra with the product given by termwiseformal multiplication. As a vector space, it is generated by all monomials in the generatorsof g, identifying however all monomials which become equal to each other upon use of thebracket relations of g.

U(g) becomes a Hopf algebra by takingM as the usual formal multiplication on U(g),and defining the unit element by

η(ξ) = ξ1 ∀ξ ∈ C. (3.96)

Co-multiplication, co-unit and antipode are defined by

∆(X) = X⊗ 1 + 1⊗ X, (3.97)

ε(X) = 0, (3.98)

γ(X) = −X (3.99)

and

∆(1) = 1⊗ 1, (3.100)

ε(1) = 1, (3.101)

γ(1) = −1. (3.102)

Note, that in this case, γ γ = id. In particular, for any X ∈ g, ξ ∈ C, the formal elementX = eξX ∈ U(g) obeys

∆(eξX) = eξX ⊗ eξX, (3.103)

γ(ξX) = e−ξX. (3.104)

As a consequence, this element satisfies

X γ(X) = 1 = γ(X) X, (3.105)

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which justifies the description of the antipode as the analogue of an inverse.

F

Some general properties of Hopf algebras are the following:

• For a given multiplication and co-multiplication, the co-unit is unique.

• If (a, M, η, ∆, ε, γ) is a Hopf algebra, then the dual vector space a∗ inherits a Hopfalgebra structure by interchangingM, η with ∆, ε.

• The co-multiplication and co-unit are homomorphisms of a, i.e. preserve the multipli-cation. For the co-unit, this means

ε(X Y) = ε(X)ε(Y), ∀ X, Y ∈ a, (3.106)

and for the co-multiplication,

∆(X Y) = ∆(X) ∆(Y), ∀ X⊗Y ∈ a× a. (3.107)

• The antipode is an anti-homomorphism, i.e.

γ(X Y) = γ(Y)γ(X), ∀ X, Y ∈ a. (3.108)

• The antipode is an anti-cohomomorphism, i.e.

(γ× γ) ∆ = π ∆ γ, (3.109)

where π is the permutation map

π : a× a→ a× a (3.110)

X⊗Y → Y⊗ X. (3.111)

• The map ∆′ := π ∆ is also a co-associative multiplication.

An algebra a is said to be commutative iff the multiplication does not depend on the orderof the factors,

M π =M. (3.112)

Analogously, a Hopf algebra is called co-commutative iff the co-multiplication satisfies

π ∆ = ∆, (3.113)

or in other words, iff ∆′ coincides with ∆. An example of a co-commutative Hopf algebrais given by the universal enveloping algebras U(g) we have discussed above. For anycommutative or co-commutative Hopf algebra, the antipode obeys γ γ = id.

Before we can study the class of quantum universal enveloping algebras Uq(g), we mustdiscuss one last property of Hopf algebras, namely quasitriangularity. A quasitriangularHopf algebra is a Hopf algebra for which the co-multiplications ∆ and ∆′ are related byconjugation, i.e.

∆′(X) = R ∆(X) R−1 ∀ X ∈ a (3.114)

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for some element R ∈ a× a which is invertible and satisfies

(id× ∆)(R) = R13 R12, (3.115)

(∆× id)(R) = R13 R23, (3.116)

(γ× id)(R) = R−1. (3.117)

Here, the inverse R−1 of R ∈ a× a is by definition that element of a× a which satisfies

R−1 R = E⊗ E = R R−1. (3.118)

Generally, R has the structureR = ∑

lR(l)

1 ⊗ R(l)2 . (3.119)

In the above definition, R13 is meant as the identity in the second factor of a× a× a and asR on the first and third factors, analogously for R12, R23. A quasitriangular Hopf algebra iscalled triangular iff

R12 R21 = E⊗ E. (3.120)

The universal enveloping algebra U(g) of any Lie algebra g is quasitriangular. Generally, aquasitriangular Hopf algebra is neither commutative nor co-commutative; however, thenon-commutativity is under control, see Eq. (3.114). As a consequence, many properties ofco-commutative Hopf algebras generalize to generic quasitriangular Hopf algebras.

An immediate consequence of Eq. (3.114) is that

R12 (∆× id)(X⊗Y) = (∆′ × id)(X⊗Y) R12 ∀ X⊗Y ∈ a× a. (3.121)

Taking X⊗Y = R and using Eq. (3.115) and (3.116), this yields

R12 R13 R23 = R23 R13 R12. (3.122)

This is the so-called Yang–Baxter equation which plays a fundamental role in the theoryof completely integrable systems. In this context, R is called the universal R-matrix.

The quantum universal enveloping algebra Uq(g) is the algebra of power series inthe 3r + 1 generators

ei, f i, hi | i = 1, . . . , r ∪ 1 (3.123)

modulo the relations

[hi, hj] = 0, (3.124)

[hi, ej] = Ajiej, (3.125)

[hi, f j] = −Aji f j, (3.126)

[ei, f j] = δijbhic, (3.127)

1−Aji

∑p=0

(−1)p⌊

1− Aji

p

⌋i(ei)p(ej)(ei)1−Aji−p = 0 i 6= j, (3.128)

1−Aji

∑p=0

(−1)p⌊

1− Aji

p

⌋i( f i)p( f j)( f i)1−Aji−p = 0 i 6= j, (3.129)

1 X = X = X 1 ∀X ∈ Uq(g), (3.130)

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where Aji are the elements of the Cartan matrix of g. Here, [X, Y] ≡ X Y− Y X withM(X⊗Y) = X Y the formal product in U(g). The q-number symbol b c is defined by

bXc ≡ bXcq :=qX/2 − q−X/2

q1/2 − q−1/2 , (3.131)

together with

bXci ≡ bXcqi with qi ≡ q(αi ,αi)/(θ,θ), (3.132)

bnc! :=n

∏m=1bmc,

⌊ nm

⌋:=

bnc!bmc!bn−mc! . (3.133)

The exponential functions of generators appearing in bhic are defined via the correspondingpower series, i.e.

eξh =∞

∑n=0

ξn

n!hn (3.134)

with hn ≡ hn defined inductively, i.e. hn = h h(n−1), so that in particular

eξh e−ξh = 1. (3.135)

Note that due to the appearance of q±hi/2, we are forced to consider infinite power seriesin the hi. In contrast, it is consistent to restrict to only finite power series in the ei, f i. Inthe limit q→ 1, the relations Eq. (3.124) to (3.127) reduce to the Lie brackets of g and therelations Eq. (3.128), (3.129) to the Serre relations of g.

Example: Serre relations for Aji = −1. In this case, the relation Eq. (3.128) is given by

eieiej − (q1/2 + q−1/2)eiejei + ejeiei = 0, (3.136)

similarly for f i. In the limit q→ 1, we have (q1/2 + q−1/2) ≡ b2c = 2, so that Eq. (3.136)reduces to

0 = eieiej − 2 eiejei + ejeiei = [ei, [ei, ej]] = (ad(ei))2ej. (3.137)

F

We can define the quantum version of the operator ad(X) as

Ad(Eα)Eβ ≡ [Eα, Eβ]q := q−(α,β)/4Eα Eβ − q(α,β)/4Eβ Eα. (3.138)

Example: Uq(A1). The relations Eq. (3.128)-(3.129) become more transparent in thesimplest example Uq(A1), where there are no Serre relations. The defining relations ofUq(A1) are

[h, e] = 2 e, (3.139)

[h, f ] = −2 f , (3.140)

[e, f ] = bhc. (3.141)

The enveloping algebra U(A1) is by definition associative and its unit 1 is given by thetrivial power series 1. These properties are inherited by Uq(A1). Uq(A1) is moreover

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endowed with the structure of a quasitriangular Hopf algebra. The co-multiplication actson the generators h, e, f and on 1 as

∆(h) = h⊗ 1 + 1⊗ h, (3.142)

∆(e) = e⊗ qh/4 + q−h/4 ⊗ e, (3.143)

∆( f ) = f ⊗ qh/4 + q−h/4 ⊗ f , (3.144)

∆(1) = 1⊗ 1, (3.145)

and the antipode is

γ(h) = −h, (3.146)

γ(e) = −q1/2e, (3.147)

γ( f ) = −q1/2 f , (3.148)

γ(1) = 1. (3.149)

Finally, the co-unit is given by

ε(h) = ε(e) = ε( f ) = 0, (3.150)

ε(1) = 1. (3.151)

We need the identity

[qξh, e] = (q2ξ − 1)e qξh, [qξh, f ] = (q2ξ − 1) f qξh (3.152)

to verify the defining property of γ. The antipode γ′ associated to ∆′ = π ∆ is obtainedfrom γ by τ → τ−1 (see Eq. (3.92)). The quantum group Uq(A1)

′ defined by ∆′ and γ′ isrelated to Uq(A1) by

Uq(A1)′ = Uq−1(A1). (3.153)

F

The Hopf algebra structure of Uq(A1) generalizes in a natural way to Uq(g) witharbitrary simple or even affine g. The defining formulae for the co-multiplication are

∆(hi) = hi ⊗ 1 + 1⊗ hi, (3.154)

∆(ei) = ei ⊗ qhi/4 + q−hi/4 ⊗ ei, (3.155)

∆( f i) = f i ⊗ qhi/4 + q−hi/4 ⊗ f i, (3.156)

∆(1) = 1⊗ 1. (3.157)

For the co-unit,

ε(hi) = ε(ei) = ε( f i) = 0, (3.158)

ε(1) = 1, (3.159)

and for the antipode,

γ(hi) = −hi, (3.160)

γ(ei) = −qhρ eiq−hρ , (3.161)

γ( f i) = −qhρ f iq−hρ , (3.162)

γ(1) = 1, (3.163)

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where

hρ =1

(θ, θ) ∑α>0

hα∨ ≡1

(θ, θ) ∑α>0

r

∑i=1

αihi =r

∑i=1

ρihi = (ρ, h), (3.164)

with ρ := 2(θ,θ)ρ, ρ the Weyl vector of g.

It is straightforward to check that the maps defined above satisfy the defining propertiesof a Hopf algebra. To check the co-associativity of ∆, we use

∆(qξh) = qξh ⊗ qξh. (3.165)

Above, we have only presented the Hopf algebra structure for the Chevalley generators. Toidentify it also for the rest of the generators, we have to use the Serre relations togetherwith the homomorphism property of the co-multiplication, etc.

Example: Hopf algebra action on E±(α1+α2) in Uq(A2). Using Eq. (3.138), we find

E±(α1+α2) = q1/4E±α1 E±α2 − q−1/4E±α2 E±α1 . (3.166)

Using the homomorphism property of ∆, its action on the Chevalley generators and thecommutation relations of Uq(A2), we find

∆(E(α1+α2)) =E(α1+α2) ⊗ q(h1+h2)/4 + q−(h

1+h2)/4 ⊗ E(α1+α2) (3.167)

+ (q1/2 + q−1/2)q−h1/4Eα2 ⊗ qh2/4Eα1 ,

∆(E−(α1+α2)) =E−(α1+α2) ⊗ q(h1+h2)/4 + q−(h

1+h2)/4 ⊗ E−(α1+α2) (3.168)

+ (q1/2 + q−1/2)q−h2/4E−α1 ⊗ qh1/4E−α2 .

Similarly, we find for the antipode

γ(E(α1+α2)) = q±1(

q−1/4E±α1 E±α2 − q1/4E±α2 E±α1)

. (3.169)

F

Literature. The Virasoro algebra is briefly introduced in [FS97], and at much more lengthbut from a completely different starting point (i.e. conformal field theory) in [DMS97].The section on Lie superalgebras is summarizing parts of [Cor89], which is very extensive.The part on quantum groups is taken mostly from [Fuc92], supplemented by [Maj95].

References



[Cor89] J. Cornwell. Group Theory in Physics, Volume III. Supersymmetries and Infinite-DimensionalAlgebras. Techniques of Physics. Academic Press, 1989. ISBN: 0-12-189805-9.

[Fuc92] J. Fuchs. Affine Lie Algebras and Quantum Groups: An Introduction, with Applicationsin Conformal Field Theory. 1st. Cambridge Monographs on Mathematical Physics. Cam-bridge University Press, 1992. ISBN: 0521415934.

[Maj95] S. Majid. Foundations of Quantum Group Theory. Cambridge: Cambridge UniversityPress, 1995. ISBN: 0-521-46032-8.

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Epilogue: Lie algebras in integrablespin chains

Despite having treated a mathematical subject, this course is intended for theoreticalphysicists. As this course did not contain any physics examples, let me close by giving threeexamples of integrable spin chains which have a simple Lie algebra, a quantum group anda Lie superalgebra as symmetry algebras. Of course this is just one context in which thesealgebras arise, but it is a particularly simple one. We will see that small modifications of asimple physical problem, such as turning on an external magnetic field or allowing latticesites to be empty, changes the structure of the symmetry algebra of the system from theone of a simple Lie algebra to being a quantum group or a Lie superalgebra.

All these examples are completely solvable and a beautiful subject in themselves,however we won’t touch upon how to actually solve them here.

The XXX1/2 or Heisenberg spin chain. Let us consider a closed linear chain of L identicalatoms with only next-neighbor interactions. Each atom has one electron in an outer shell(all other shells being complete). These electrons can either be in the state of spin up (↑)or down (↓). At first order, the Coulomb- and magnetic interactions result in the exchangeinteraction in which the states of neighboring spins are interchanged:

↑↓ ↔ ↓↑ . (3.170)

In a given spin configuration of a spin chain, interactions can happen at all the anti-parallelpairs. Take for example the configuration

↑↑↓↑↓↓↓↑↑↓↓ . (3.171)

It contains five anti-parallel pairs on which the exchange interaction can act, giving rise tofive new configurations. The Hamiltonian of the XXX1/2 spin chain is given by

H = −JL

∑n=1

Πn,n+1, (3.172)

where J is the exchange integral1 and Πn,n+1 is the permutation operator of states atpositions n, n + 1. The spin operator at position n on the spin chain is given by

~Sn = (Sxn, Sy

n, Szn) =

12~σn, (3.173)

where σin are the Pauli matrices for spin 1/2:

σ1 =

(0 11 0

), σ2 =

(0 −ii 0

), σ3 =

(1 00 −1

). (3.174)

1−J > 0: ferromagnet, spins tend to align, −J < 0: anti-ferromagnet, spins tend to be anti-parallel.

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For the closed chain, the sites n and n + L are identified:

~SL+1 = ~S1. (3.175)

In terms of the spin operators, the permutation operator is given by

Πn,n+1 = − 12 (1 +~σn~σn+1). (3.176)

The Hamiltonian (3.172) now becomes

H = −JL

∑n=1

~Sn~Sn+1

= −JL

∑n=1

12

(S+

n S−n+1 + S−n S+n+1

)+ Sz

nSzn+1,

(3.177)

where S± = Sxn ± iSy

n are the spin flip operators. The term in parentheses corresponds tothe exchange interaction which exchanges neighboring spin states.

The spin flip operators act as follows on the spins:

S+k | . . . ↑ . . . 〉 = 0, S+

k | . . . ↓ . . . 〉 = | . . . ↑ . . . 〉,S−k | . . . ↑ . . . 〉 = | . . . ↓ . . . 〉, S−k | . . . ↓ . . . 〉 = 0,

Szk| . . . ↑ . . . 〉 = 1

2 | . . . ↑ . . . 〉, Szk| . . . ↓ . . . 〉 = − 1

2 | . . . ↓ . . . 〉.(3.178)

The spin operators have the commutation relations

[Szn, S±m ] = ±S±n δnm, [S+

n , S−m ] = 2Sznδnm, (3.179)

which we recognize as those of A1. The Hamiltonian moreover commutes with thegenerators of su(2),

[H, S±] = [H, Sz] = 0, (3.180)

thus we say that su(2) is the symmetry algebra of the XXX1/2 spin chain and each latticesite carries a spin 1/2 representation of su(2).

The XXZ1/2 or anisotropic spin chain. In the anisotropic case, a magnetic field is turnedon in the z-direction, resulting in the Hamiltonian

H∆ = −JL

∑n=1

SxnSx

n+1 + SynSy

n+1 + ∆(SznSz

n+1 − 14 ). (3.181)

The anisotropy is captured by the parameter

∆ =q + q−1

2. (3.182)

∆ = 1 is the isotropic case we have treated so far.The XXZ spin chain admits the quantum group Uq(su(2)) as symmetry algebra. Uq(su(2))

is generated by S+, S− and q±Szunder the relations

qSzS±qSz

= q±1S±, [S+, S−] =q2Sz − q−2Sz

q− q−1 . (3.183)

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These relations reduce to the ones of A1 for q→ 1. For the case of spin 1/2, we find thefollowing representations for the operators:

qSz= qσ3/2 ⊗ · · · ⊗ qσ3/2︸︷︷︸

L

, (3.184)

S± =L

∑n=1

S±n =L

∑n=1

qσ3/2 ⊗ · · · ⊗ qσ3/2︸︷︷︸n−1

⊗ σ±n2⊗ q−σ3/2 ⊗ · · · ⊗ q−σ3/2, (3.185)

where we have the Pauli matrices

σ+ =

(0 10 0

), σ− =

(0 01 0

), σ3 =

(1 00 −1

). (3.186)

and

qσ3=

(q 00 1/q

). (3.187)

The tJ model. The tJ model describes a system of electrons on a lattice with a Hamiltonianthat describes nearest–neighbor hopping (with coupling t) and spin interactions (withcoupling J). Consider a one-dimensional lattice of length L with periodic boundaryconditions. Each site can be either free () or occupied by a spin up (↑) or down (↓)electron. Excluding double occupancy, the Hilbert space at each point k is:

Hk = C(1|2). (3.188)

It is convenient to introduce anticommuting creation–annihilation pairs c†k,s, ck,s, s = ↑, ↓

at each site, acting as

|s〉k = c†k,s|〉k , for s = ↑, ↓, (3.189)

where |〉k is the vacuum, annihilated by ck,s. Let nk,s = c†k,sck,s be the number of s electrons

at position k and nk = nk,↑ + nk,↓. We can further introduce su(2) spin operators at eachsite:

S−k = c†k,↑ck,↓ , S+

k = c†k,↓ck,↑ , Sz

k =12

(nk,↑ − nk,↓

). (3.190)

With these ingredients, we can write down the Hamiltonian

H =L−1

∑k=1

[−tP ∑

s=↑,↓

(c†

k,sck+1,s + h.c.)P + J

(~Sk · ~Sk+1 − 1

4 nknk+1 + 2 nk − 12

)],

(3.191)where P projects out double occupancy.

For J = 2t = 2, the Hamiltonian is invariant under the action of the Lie superalgebrasl(1|2).

The even part g0 = gl(1)⊕ sl(2) of sl(1|2) is generated by the operators S±, Sz, Z withcommutation relations

[Sz, S±] = ±S± , [S+, S−] = 2Sz , [Z, S±] = 0 , [Z, Sz] = 0 . (3.192)

There are two additional fermionic multiplets Q±s , s = ↑, ↓ which transform with respectto g0 as

[Sz, Q±s ] = ± 12 Qs , [S±, Q±s ] = 0 , [Z, Q±↓ ] =

12 Q±↓ , [Z, Q±↑ ] = − 1

2 Q±↑ . (3.193)

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The fermionic generators satisfy the following anticommutation relations:

Q±s , Q∓s = 0 , Q±↑ , Q±↓ = S± , Q±↑ , Q∓↓ = Z± Sz . (3.194)

At each point k of the lattice, also the generators Q±, Z can be represented in terms ofcreation–annihilation operators as

Q−k,↑ =(1− nk,↓

)ck,↑ , Q+

k,↑ =(1− nk,↓

)c†

k,↑ , Q−k,↓ =(1− nk,↑

)ck,↓ , (3.195)

Q+k,↓ =

(1− nk,↑

)c†

k,↓ , Zk = 1− 12 nk . (3.196)

The supersymmetric Hamiltonian can be expressed in terms of these generators as

H = −L

∑k=1

∑s=↑,↓

[Q+

k,sQ−k+1,s + Q+k+1,sQ−k,s

]+

+L

∑k=1

[S+

k S−k+1 + S−k S+k+1 + 2Sz

kSzk+1 − 2ZkZk+1 − 1k1k+1

]. (3.197)

Literature. The XXX1/2 spin chain was introduced by Hans Bethe [Bet31]. The quantumgroup structure of the XXZ chain is discussed in [PS90]. The tJ model and its supergroupstructure are discussed in [EK92].

References

[Bet31] H. Bethe. Zur Theorie der Metalle. Zeitschr. f. Phys. 71 (1931), p. 205.[PS90] V. Pasquier and H. Saleur. Common Structures Between Finite Systems and Conformal

Field Theories Through Quantum Groups. Nucl.Phys. B330 (1990), p. 523.[EK92] F. H. L. Essler and V. E. Korepin. A New solution of the supersymmetric T-J model by

means of the quantum inverse scattering method (1992).

Acknowledgments

I would like to thank all those students who have made these lecture notes better bypointing out typos. Special thanks go to Manuel Meyer who has brought me a list of typosevery week.

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