oldandnewonthe exceptionalgroupgoldandnewonthe exceptionalgroupg2 ilka agricola i n a talk delivered...

OldandNewontheExceptionalGroupG2Ilka Agricola

In a talk delivered in Leipzig (Germany) onJune 11, 1900, Friedrich Engel gave thefirst public account of his newly discovereddescription of the smallest exceptional Liegroup G2, and he wrote in the corresponding

note to the Royal Saxonian Academy of Sciences:

Moreover, we hereby obtain a direct defi-nition of our 14-dimensional simple group[G2] which is as elegant as one can wish for.[En00, p. 73]1

Indeed, Engel’s definition of G2 as the isotropygroup of a generic 3-form in 7 dimensions is atthe basis of a rich geometry that exists only on7-dimensional manifolds, whose full beauty hasbeen unveiled in the last thirty years.

This article is devoted to a detailed historicaland mathematical account of G2’s first years, inparticular the contributions and the life of Engel’salmost forgotten Ph. D. student Walter Reichel,who worked out the details of this description in1907. We will also give an introduction to mod-ern G2 geometry and its relevance in theoreticalphysics (in particular, superstring theory).

The Classification of Simple Lie GroupsIn 1887 Wilhelm Killing [Kil89] succeeded in classi-fying those transformation groups that are rightlycalled simple: by definition, these are the Lie groupsthat are not abelian and do not have any nontrivialnormal subgroups.

Every Lie groupGhas a Lie algebra g (the tangentspace to the group manifold at the identity), whichis a vector space endowed with a skew-symmetric

Ilka Agricola is head of a Junior Research Group fundedby the Volkswagen Foundation at Humboldt-Universitätzu Berlin. Her email address is [email protected] Engel’s own words: “Zudem ist hiermit eine direkteDefinition unserer vierzehngliedrigen einfachen Gruppegegeben, die an Eleganz nichts zu wünschen übrig lässt.”

product, the Lie bracket [ , ]; as a purely algebraicobject it is more accessible than the original LiegroupG. IfG happens to be a group of matrices, itsLie algebra g is easily realized by matrices too, andthe Lie bracket coincides with the usual commuta-tor of matrices. In Killing’s and Lie’s time, no cleardistinction was made between the Lie group andits Lie algebra. For his classification, Killing chosea maximal set h of linearly independent, pairwisecommuting elements of g and constructed basevectors Xα of g (indexed over a finite subset R ofelements α ∈ h∗, the roots) on which all elementsof h act diagonally through [ , ]:

[H,Xα] = α(H)Xα for all H ∈ h.

In order to avoid problems when doing so he chosethe complex numbers C as the ground field. Thedimension of the maximal abelian subalgebra h(also called, somehow wrongly, a Cartan subalge-bra) is the rank of the Lie algebra. It is a generalfact that all roots α ≠ 0 appear only once. If wewrite gα := C · Xα (these are the root spaces), weobtain a decomposition of g as

g = h⊕

α∈R

gα

and vectors in gα, gβ for two roots α,β satisfy anextremely easy multiplication rule: [gα, gβ] ⊂ gα+βif α+ β is again a root, while otherwise it is zero.

Two families of complex simple Lie algebraswere well-known at that time:

(1) the Lie algebras so(n,C) consisting ofskew-symmetric complex matrices, whichare the Lie algebras of the orthogonalgroups SO(n,C) (n = 3 or n ≥ 5),

(2) the Lie algebras sl (n,C) consisting oftrace-free matrices, which are the Lie al-gebras of the groups SL(n,C) of matricesof determinant one (n ≥ 2).

922 Notices of the AMS Volume 55, Number 8

It was Killing’s original intent to prove that thesewere the only simple complex Lie algebras.2 Infact, there exists a third family of simple algebras,namely the Lie algebras sp(2n,C) of the symplecticgroups Sp(2n,C) for n ≥ 1, defined as invariancegroups of non-degenerate 2-forms ω on C2n:

Sp(2n,C) = {g ∈ GL(2n,C) : ω = g∗ω}.

Around 1886, Sophus Lie and Friedrich Engel wereaware of their existence, but they had not yetappeared in print anywhere [Ha00, p. 152]. To hisbig surprise, in May 18873 Killing discovered acompletely unknown complex simple Lie algebraof rank 2 and dimension 14, which is just theexceptional Lie algebra g2. By October 1887, hehad basically completed his classification. He dis-covered that besides g2 and the three familiesmentioned above, there exist four additional ex-ceptional simple Lie algebras. In modern notation,they are: f4, e6, e7, e8, and they have dimensions52, 78, 133, and 248 respectively.

There exist many real orthogonal Lie groupswith complexification SO(n,C)—namely all or-thogonal groups SO(p, q) associated with scalarproducts with indefinite signature (p, q) such thatp + q = n; they are called real forms of SO(n,C)(similarly for the Lie algebras), and it is an easyfact that SO(p, q) is compact only for q = 0. Justas well, the complex Lie algebra g2 with complexLie group G2 has two real forms that we are goingto denote by gc2 and g∗2 ; of their (simply connected)Lie groups Gc2 and G∗2 , only the former is compact.

Without doubt, the classification of complexsimple Lie algebras is one of the outstandingresults of nineteenth century mathematics (thiswas not Killing’s point of view, however: his orig-inal aim had been a classification of all real Liealgebras, he was unsatisfied with his own expo-sition and the incompleteness of results, so thathe would not have published his results with-out strong encouragement from Friedrich Engel).Indeed, Killing’s formidable work contains somegaps and mistakes:4 in his thesis (1894), ÉlieCartan gave a completely revised and polishedpresentation of the classification [Ca94], whichhas therefore become the standard reference forthe result.

2Letter from W. Killing to Fr. Engel, April 12, 1886; see[Ha00, p. 153].3Letter from W. Killing to Fr. Engel, May 23, 1887; see[Ha00, p. 161].4For example, he had found two exceptional Lie algebrasof dimension 52 and overlooked that they are isomorphic,and his classification was based on three main theoremswhose statements and proofs were partially wrong.

Élie Cartan(1869–1951) in theyear 1904.

Friedrich Engel(1861–1941) around1922.

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First Results on G2

We can only conjecture how Killing and his contem-poraries felt about the exceptional Lie algebras—asdisturbances to the symmetry or as exotic andunique objects. But since Lie theory as a wholewas developed in these times, they were investi-gated too, but not with high priority. G2 was thefirst—and, for rather a long time, the only—Liegroup for which further results were obtained.This is natural for dimensional considerations,but we will see later that it has also much deeperreasons.

From the weight lattice, which one obtains auto-matically during the classification, one can easilydetermine the lowest dimensional representationof any simple Lie algebra. This Cartan did in thelast section of his thesis, and he rightly observedthat g2 admits a 7-dimensional complex represen-tation, which furthermore possesses a symmetricnondegenerate g2-invariant bilinear form [Ca94,p. 146]:

β := x20 + x1y1 + x2y2 + x3y3.

This scalar product has real coefficients, hence canbe interpreted over the reals as well; in that caseit has signature (4,3), and one can understandCartan’s result as giving a real representation ofthe noncompact form g∗2 inside so(4,3).

At this stage, the question about explicit con-structions of the exceptional Lie algebras becomespressing. Élie Cartan and Friedrich Engel obtainedthe first breakthrough, published in two simulta-neous notes to the Académie des Sciences de Paris[Ca93], [En93]: For every point a ∈ C5, considerthe 2-plane πa in the tangent space TaC5 that isthe zero set of the Pfaffian system

dx3 = x1 dx2 − x2 dx1,

dx4 = x2 dx3 − x3 dx2,

dx5 = x3 dx1 − x1 dx3.

September 2008 Notices of the AMS 923

The 14 vector fields on C5 whose local flows mapthe planes πa to each other satisfy the commu-tator relations of the Lie algebra g2. Both authorsthen gave in the same papers a second geometricrealization of g2: Engel derived it from the firstby a contact transformation, while Cartan identi-fied g2 as the symmetries of the solution spaceof the system of second order partial differentialequations5 (f = f (x, y))

fxx =4

3(fyy)

3, fxy = (fyy)2.

Both viewed their second realization as beingdifferent from the one through the Pfaffian system.Of course, this is correct: stated in modern terms,the complex Lie group G2 has two non-conjugate9-dimensional parabolic subgroups P1 and P2, andG2 acts on the two compact homogeneous spacesM5i := G2/Pi , i = 1,2. It is a detail that Engel and

Cartan did not describe the g2 action on the fullspacesM5

i , but rather on an open subset; this wasthe common way at that time.

Let us have a closer look at these two homoge-neous spaces. For this, we need the lattice insideh∗ spanned by the 12 roots of g2, the root lattice.It is the usual hexagonal planar lattice, in whichthe roots are denoted by arrows:

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−α1

−α2

α1

ω1

ω2

α2

R−

R+

Figure 1. The g2 root system and lattice.

The g2 root system is the only one in whichtwo roots include an angle of π/6, indicating itsexceptional standing among all root systems. Theroots above respectively below the dashed lineare called positive respectively negative roots, andR = R+ ∪ R−. The two positive roots marked α1

and α2 alone already generate the lattice; theyare called simple roots. Keeping in mind the rootsystem with the multiplication rule for root spaces

5In 1910, Élie Cartan returned to his description of G∗2by Pfaffian systems and differential equations [Ca10];a modern treatment and further investigation can befound in the worthwhile article by P. Nurowski [Nu05].It is quite remarkable that the thesis of yet anotherof Engel’s students plays a decisive role here (KarlWünschmann, Greifswald, 1905).

gα stated before, one sees that the direct sum ofh (corresponding, loosely speaking, to the origin),the six positive root spaces, and the root space ofone negative of a simple root, span a subalgebra:

pi := h⊕ g−αi⊕

α∈R+

gα.

Subalgebras of this kind are called parabolic sub-algebras. The 9-dimensional groups P1 and P2

above are now exactly the subgroups of G2 withLie algebras p1 and p2. By general results, thespace G2/Pi is a compact homogeneous variety,and it can be realized in the projectivization of therepresentation space Vi with highest weight ωi

(see figure) as the G2 orbit of some distinguishedvector vi ; butω1 generates the 7-dimensional rep-resentation (spanned by the six short roots andzero with multiplicity one), whileω2 is the highestweight of the adjoint representation (spanned byall roots and zero with multiplicity two). Hence,we obtain

M51 = G2/P1 = G2 · [v1] ⊂ P(C

7) = CP6,

M52 = G2/P2 = G2 · [v2] ⊂ P(g2) = CP

13.

The first spaceM51 is thus a quadric in CP6; we will

come back to the second space later.Let us look again at the real situation. There

are two real 9-dimensional subgroups P∗i insidethe noncompact real form G∗2 corresponding tothe complex parabolic groups Pi ⊂ G2; but theyhave no counterparts in the compact Lie groupGc2 (roughly speaking, gc2 ⊂ so(7) consists of skew-symmetric matrices, while parabolics are alwaysupper triangular): a maximal subgroup ofGc2 is iso-morphic to SU(3) and thus 8-dimensional. Hencea geometric realization of the compact form Gc2 isstill missing.

G2 and 3-forms in Seven VariablesIn his talk on June 11, 1900, in Leipzig, FriedrichEngel presented some results on the complex Liegroup G2 that finally led to the missing realizationof its compact form Gc2. Engel’s geometric insightinto the geometry ofM5

1 ⊂ CP6 was so good that he

realized that it can be written as the zero set of anequation depending solely on the coefficients of ageneric 3-form in seven variables [En00, p. 220].

By a generic p-form we mean an element ω ∈

Λp(Cn)∗ with open GL(n,C) orbit. For dimensional

reasons, n2 ≥(n

p

)is a necessary condition for the

existence of generic p-forms; it holds for all n ifp = 2, but only for n ≤ 8 when p = 3, and, indeed,generic p-forms do exist for these values. Theisotropy group of a differential form (or, for thatmatter, any tensor) consists of all group elementsleaving the form invariant,

Gω := {A ∈ GL(n,C) : ω = A∗ω}.


For a generic 3-form in dimension 7, its dimensionis

dimGω = dim GL(7,C)− dimΛ3(C7)∗ = 14.

Friedrich Engel observed that all generic 3-formsare equivalent under GL(7,C) and proved thefollowing theorem:

Theorem 1 (F. Engel, 1900). There exists exactlyone GL(7,C) orbit of generic complex 3-forms[En00, p. 74]. One such generic form is given by

ω0 := (e1e4 + e2e5 + e3e6)e7 − 2e1e2e3 + 2e4e5e6.

For every generic complex 3-form ω ∈ Λ3(C7)∗,the following holds:

(1) The isotropy group of ω is isomorphic tothe simple Lie group G2 [En00, p. 73];

(2) ω defines a non-degenerate symmetric bi-linearform βω [En00, p. 222] that is cubicin the coefficients of ω, and the quadricM5

1 is its isotropic cone in CP6. In particu-lar, every isotropy group Gω is containedin some SO(7,C).

(3) There exists a G2-invariant polynomialλω ≠ 0 of degree 7 in the coefficients of ω[En00, p. 231].

In fact, Engel had already conjectured in a letterto Killing of April 1886 that the isotropy group ofa 3-form might be a simple 14-dimensional group,but apparently neither he nor Killing had pursuedthis idea at that time.6 In modern notation, we candefine βω through [Br87]

βω(X, Y) := (X ω)∧ (Y ω)∧ω,

which is a symmetric bilinear form with values inthe one-dimensional vector space Λ7(C7)∗. Overthe reals, it can be turned into a true real-valuedscalar product gω after taking an additional squareroot [Br87], [Hi00]. Geometrically, this means thatevery generic 3-form on a real 7-dimensional man-ifold induces a (pseudo)-Riemannian metric. Aneasy dimension count shows that the isotropygroup of a generic 3-form can be a subset ofSO(n,C) only for n = 7,8.

Since βω is cubic inω, its determinant is a poly-nomial of degree 21 in the ω coefficients; Engelunderstood that it is the third power of a degree7 element λω, and its non-vanishing is equivalentto the nondegeneracy of βω.

Engel’s arguments still hold over the reals, aslong as the isotropic cone does not degeneratecompletely. For the 3-form ω0 cited above, gω0

is a real scalar product on R7 with signature(4,3) [En00, p. 64]. In particular, there exists ex-actly one GL(7,R) orbit of real generic 3-formsω ∈ Λ3(R7)∗ with non-degenerate isotropic conefor gω. Its isotropy group is again isomorphic tothe real noncompact real form G∗2 ⊂ SO(4,3).

6Letter from Fr. Engel to W. Killing, April 8, 1886; see[Ha00, p. 152].

In the same article, Engel invested a lot of en-ergy in a description of the second homogeneousspace M5

2 ⊂ P(g2) through the coefficients of ω.For this, he used a symbolic method for invariantsof alternating forms that was communicated tohim by Eduard Study; however, Study’s formalismis not in use anymore, hence his computationsare rather hard to follow. Today, we know thatM5

2 = G2/P2 ⊂ CP13 is a rather complicated pro-

jective algebraic variety: it has degree 18 and itscomplete intersection with three hyperplanes is aK3 surface of genus 10 [Bor83]. For a geometricdescription of G2/P2 in terms of ω, observe thatthe 21-dimensional representationΛ2C7 splits un-der G2 into g2 ⊕ C

7, hence G2/P2 is a subvarietyof P(Λ2C7) as well. By the Plücker embedding, the14-dimensional Grassmann variety G(2,7) of 2-planes in C7 lies in P(Λ2C7). Now, G2/P2 is just theintersection of G(2,7) with P(g2) inside P(Λ2C7).As a subvariety ofG(2, 7),G2/P2 consists preciselyof those 2-planes π ⊂ C7 on which βω andω bothdegenerate (see [LM03] for a modern account), i. e.,such that

π βω = 0, π ω = 0.

It was the realization ofG2 presented in Theorem 1that led Friedrich Engel to the comment cited inthe introduction. Besides its elegance, Theorem 1has far-reaching consequences for modern differ-ential geometry (see last section); furthermore, itwill provide the missing realization of Gc2, as isexplained now.

Walter Reichel and the Invariants of G2

While a professor at Greifswald University (1904-1913), Friedrich Engel turned again to this topicand assigned to his Ph. D. student Walter Reichelthe task of computing a complete system of in-variants for complex 3-forms in six and sevendimensions in Study’s formalism. The thesis de-fended by Reichel in 1907 indeed contains thedetailed description of the invariants and rela-tions among them as well as normal forms of all3-forms under the action of GL(7,C). The vanish-ing of λω for non-generic forms and the drop ofrank of the bilinear form βω play an essential rolehere.

Furthermore, Walter Reichel described theisotropy algebra gω of any generic 3-form ωdirectly through its coefficients [Rei07, p. 48],whereas Friedrich Engel had only computed it forone representative. Over the complex numbers,this makes no difference; but it turns out thatwhen passing to real numbers, the one orbit ofcomplex, GL(7,C) equivalent 3-forms splits intotwo orbits of real generic GL(7,R) equivalent3-forms. If one interprets the scalar product βω asa real one, it turns out to have signature (4,3) onone orbit and signature (7,0) on the other orbit.It does not come as a surprise that the isotropy


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Figure 2. Title page of W. Reichel’s thesis.

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Walter Reichel(1883–1918) in

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W. Reichel (undated);one can recognize a

portrait of Kant onthe desk.

group gω is isomorphic to g∗2 ⊂ so(4,3) in the firstcase and isomorphic to gc2 ⊂ so(7) in the secondcase.

Thus, Reichel obtained a uniform geometricdescription of both real forms of G2. Unfortu-nately, the result was slowly forgotten afterwards;J. A. Schouten described the normal forms of3-forms on C7 in 1931 by simpler methods (with-out invariants, mainly by reduction to smallerdimensions) and observed that Walter Reichelhad missed two out of the nine normal forms[Sch31]. Based on these results, Gurevich solved

the problem in dimension 8 [Gu35]. Up to ourknowledge, the next authors to cite Reichel’s the-sis again are E. B. Vinberg and A. G. Elashvili in1978, who worked out the details of the extremelyinvolved case n = 9 [VE78].

It is well known that Gc2 is the automorphismgroup of the octonians O. Élie Cartan includ-ed this as a comment in his long article oncomplex numbers and their generalizations from1908 [Ca08, p. 467] (see also [Ca14, p. 298]), butapparently never returned to this topic. This ap-proach to exceptional Lie groups became popularthrough the work of Hans Freudenthal, startingwith the article [Fr51], and made the memory of the3-form approach vanish. In fact, these descriptionsare equivalent (a third equivalent description isthrough so-called “vector cross-products”), as isexplained with great care in the article by J. Baez[Ba02, p. 37-39].

The Mathematician Walter ReichelWhereas the life and work of all mathematiciansmentioned up to here are well known, virtuallynothing was known about Walter Reichel, despitethe fact that his thesis has been cited widely inrecent years. The 100th anniversary of his thesislast year was a further motivation to investigatehis story.

Walter Reichel was born on November 3, 1883,in a little Silesian village then called Gnadenfrei(now Piława Górna, Poland). This village had beenfounded by members of the Moravian Unity, ofwhich Reichel’s father was deacon and, later, bish-op. The Moravian Unity, or Unitas Fratrum (Unityof Brethren), emerged in the middle of the fif-teenth century from the Bohemian ReformationMovement around Jan Hus (1369–1415) and wasrenewed in the early eighteenth century in Herrn-hut (Saxony, not far from the Czech and Polishborders), where the management of its Europeanbranch and its archive are still hosted today. Thehistory of the Reichel family is closely linked tothe Brüdergemeine, as the Moravian Unity is calledin German.

In his handwritten CV (which can be foundin his Ph. D. files at Greifswald University), WalterReichel describes how he went to school first in hishome village, followed by four years at the “Päda-gogium” in Niesky (another town founded by theUnitas Fratrum, close to Herrnhut) and three yearsat the Gymnasium in Schweidnitz (now Swidnica,Poland), from where he received his high schooldegree (“Reifezeugnis”) at Easter 1902. He thenstudied mathematics, physics, and philosophy atthe Universities of Greifswald, then Leipzig, Halle,and again, Greifswald.

Among others, he attended lectures by FriedrichEngel and Theodor Vahlen (in Greifswald); by Carl


The “Old Pädagogium” in Niesky, now a publiclibrary, built in 1741 as the first parish houseof the newly founded community in Niesky.Between 1760 and 1945 it was used as anadvanced boarding school.

Neumann (in Leipzig), who formulated the Neu-mann boundary condition in analysis and foundedthe Mathematische Annalen together with AlfredClebsch; by Georg Cantor and Felix Bernstein (inHalle), to whom we owe the foundations of set the-ory and the Cantor-Bernstein-Schröder Theoremin logic; by the theoretical physicist Gustav Mie (inGreifswald), who made important contributionsto electromagnetism and general relativity; by theexperimental physicist Friedrich Ernst Dorn (inHalle), who discovered the gas Radon in 1900. Mo-roever, he took courses in philosophy, chemistry,zoology, and art history.

In July 1907, Walter Reichel passed the ex-amination for high school teachers in “pure andapplied mathematics, physics and philosophicalpropaedeutics” with distinction. He spent one yearas teacher-in-training in Görlitz, and was then ap-pointed in Fall 1908 at the “Realprogymnasium”in Sprottau (now Szprotawa, Poland). In April 1914he moved for an “Oberlehrer” position to Schwei-dnitz (now Swidnica, Poland). With the beginningof the First World War he was drafted, and he diedin France on March 30, 1918. He has no grave,but an inscription on the WWI memorial on the“God’s acre” of the Moravian community in Nieskycommemorates his death.

Walter Reichel married Gertrud, née Müller(1889–1956) in 1909. They had three sons (born1910, 1913, and 1916), who left no children, anda daughter (born March 11, 1918). After the firstWorld War, Reichel’s widow moved with her chil-dren to Niesky, where she was supported by theMoravian Unity. For many years, she accommodat-ed pupils of the “Pädagogium” who did not live inthe boarding school’s dormitories. Walter Reichel’sdaughter Irmtraut Schiller now lives in Bremen and

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Memorial stone on the “God’s acre” in Niesky.

Detail of the inscription on the memorialstone. Walter Reichel’s name and date of death(“30.3.18”) are in the second to last line; thestone has been damaged and repaired abovehis name.

has three children; one of her granddaughters is ateacher of mathematics.

G2 Geometry in Dimension 7The classical symmetry approach to differentialgeometry is based on the notion of the isom-etry group of a Riemannian manifold, i. e., thegroup of all transformations acting on the man-ifold that preserve the metric. In the twentiethcentury, the concept of (Riemannian) holonomygroup became fundamental to Riemannian ge-ometry: it is the group generated by all paralleltransports along closed null-homotopic loops onthe manifold. Here, parallel transport is under-stood with respect to the Levi-Civita connection


∇g, i. e., the straightforward—but not the onlypossible—generalization of the directional deriva-tive of vectors. Marcel Berger’sHolonomy Theoremfrom 1955 states that for an irreducible non-symmetric manifold, the holonomy group is eitherSO(n) or from a finite list—and Gc2 is the only ex-ceptional Lie group on that list. In this case,the manifold is necessarily 7-dimensional, theholonomy group Gc2 acts on the tangent bundleby its 7-dimensional real representation, and themanifold would be called a parallel or integrableGc2-manifold. The argument of the proof is basi-cally reduced to the question of which compactLie groups admit transitive sphere actions, andthis is the case for Gc2 on S6, thought of not as asymmetric space but rather as the homogeneousspace Gc2/SU(3).

Since Berger’s Holonomy Theorem lists onlythe possible holonomy groups without actuallyconstructing manifolds admitting them, his clas-sification result was not an end point, but rathera research program asking for a more detailedgeometric investigation (and a long breath); inparticular, no Riemannian manifolds with holo-nomy group Gc2 were known. In 1966, EdmondBonan observed in a note preceding his thesis(supervised by André Lichnerowicz) that mani-folds with Gc2 holonomy admit a global 3-form ωwith ∇gω = 0, and, in consequence, have to beRicci flat—a very restrictive geometric conditionthat intrigued some contemporaries [Bon66]. This3-form ω is of course precisely the form whosestabilizer is Gc2 as described by Engel and Reichel,but had already been forgotten in this time. For hiswork, Edmond Bonan had started from Cartan’sdescription of Gc2 as the automorphism group ofthe octonians, and saw how to derive an invariant3-form from its multiplication law.

The credit for having made the most creativeuse of Gc2 and its defining 3-form in differen-tial geometry goes without doubt to Alfred Gray.Since the 1960s he had investigated vector crossproducts and their geometric properties in a se-ries of papers. In 1971 he had the radical ideaof weakening the classical holonomy concept tocover interesting manifolds that do not appearin Berger’s list. In particular, he defined nearlyparallel Gc2-manifolds: they have structure groupGc2, but instead of being parallel, the 3-form ωsatisfies the differential equation

dω = λ∗ω

for a real constant λ ≠ 0, and they are Einstein withstrictly positive scalar curvature. Later, he provedtogether with Marisa Fernández that there are infact four basic classes of Gc2-manifolds dependingon the possible nature of the tensor ∇gω [FG82].Maybe even more importantly, they initiated the(still-ongoing) construction of many interestingexamples—ranging from S7 = Spin(7)/Gc2 to the

Allow-Wallach spheres SU(3)/S1, from extensionsof Heisenberg groups to clever non-homogeneousexamples. Since then, these non-integrable geome-tries (not only for Gc2, but also for contact struc-tures, almost Hermitian structures, 8-dimensionalSpin(7)-structures, etc.) have been studied inten-sively [Ag06]. Today, the main philosophy is thatfor many Riemannian geometries (M, g) definedby tensors that are not∇g-parallel, it is possible toreplace the Levi-Civita connection by a more suit-able metric connection ∇ with skew-symmetrictorsion T ∈ Λ3(M) (the characteristic connection)

g(∇XY,Z) := g(∇gXY,Z)+

1

2T(X, Y ,Z)

such that the object becomes parallel, and theholonomy group of this new connection plays therole of the classical Riemannian holonomy group.For example, a Gc2-manifold (M, g,ω) admits acharacteristic connection if and only if there is avector field β such that δ(ω) = −β ω (this ex-cludes one of the four basic types), and its torsionis then given by [FI02]

T = −∗ dω−1

6(dω,∗ω)ω+∗(β∧ω).

For integrable Gc2 geometries, the first break-through was obtained a few years after Gray’swork. In 1987 and 1989, Robert Bryant and SimonSalamon succeeded in constructing local completemetrics with Riemannian holonomy Gc2 ([Br87],[BrSa89]). It was only in 1996—more than fortyyears after Berger’s original paper—that DominicJoyce was able to show the existence of compactRiemannian 7-manifolds with Riemannian holono-my Gc2 [Joy00]: this comes down to proving theexistence of solutions of nonlinear elliptic partialdifferential equations on compact manifolds byvery difficult and involved analytical methods.

One distinguished property of Gc2 is still miss-ing in our discussion—this is the one that makesit attractive for mathematical physics. The groupGc2 can be lifted to the universal covering Spin(7)of SO(7), and Spin(7) has an 8-dimensional irre-ducible real representation ∆7, the spin represen-tation, that decomposes under Gc2 ⊂ Spin(7) intothe trivial and the 7-dimensional representation.Thus, a 7-dimensional spin manifold endowedwith a connection∇ has a∇-parallel spinor field ifand only if the holonomy of ∇ lies inside Gc2, andGc2 is the isotropy group of a generic spinor. In fact,any spinor field defines a global 3-form and viceversa, so this last characterization ofGc2 is, in itself,nothing new. But it explains the intricate relationbetween Gc2 and spin geometry. For example, thenearly parallelGc2-manifolds discovered by Gray in1971 are precisely those 7-manifolds that admita real Killing spinor field [FK90]. More recently,superstring theory has stimulated a deep interestin 7-manifolds with integrable or non-integrableGc2 geometry [Du02]. In this approach, a∇-parallel


spinor field is interpreted as a supersymmetrytransformation: by tensoring with a spinor field,bosonic particles can be transformed into fermion-ic particles (and vice versa). The torsion of ∇ (ifpresent) is related to the B-field, a higher orderversion of the classical field strength of Yang-Millstheory (which would be a 2-form and not a 3-form,of course).

The story of G2 and its relatives is far from con-cluded. The algebraic foundations for the manylines of development sketched in this article werelaid more than a hundred years ago by FriedrichEngel and his student Walter Reichel in a work ofremarkable mathematical insight.

AcknowledgementsI thank the Archive of Greifswald University, theArchive of the Moravian Unity in Herrnhut, theResearch Library Gotha of Erfurt University, theantiquarian Wilfried Melchior in Spreewaldheidenear Berlin, and Walter Reichel’s daughter, Irm-traut Schiller, as well as her family, for theirsupport during the investigations for this article.

I am also grateful to Robert Bryant (MSRI Berke-ley) and Joseph M. Landsberg (Texas) for fruitfulmathematical discussions, and Philip G. Spain(Glasgow) for reading and commenting on earlyversions of this text. Finally, special thanks goto Thomas Friedrich (Humboldt Universität zuBerlin) for his ongoing support, endless hours ofdiscussions bent together over the manuscripts ofEngel, Reichel, and Cartan, and many more thingsthat need not be listed here.

The research for this article was funded bythe Volkswagen Foundation and the DeutscheForschungsgemeinschaft (DFG).

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