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OldandNewonthe ExceptionalGroupG2

n a talk delivered in Leipzig () on product, the Lie bracket [ , ]; as a purely algebraic June 11, 1900, Friedrich Engel gave the object it is more accessible than the original Lie first public account of his newly discovered G. If G happens to be a group of matrices, its description of the smallest exceptional Lie g is easily realized by matrices too, and group , and he wrote in the corresponding the Lie bracket coincides with the usual commuta- noteI to the Royal Saxonian Academy of Sciences: tor of matrices. In Killing’s and Lie’s time, no clear Moreover, we hereby obtain a direct defi- distinction was made between the and nition of our 14-dimensional its Lie algebra. For his classification, Killing chose [G2] which is as elegant as one can wish for. a maximal set h of linearly independent, pairwise 1 [En00, p. 73] commuting elements of g and constructed base

Indeed, Engel’s definition of G2 as the isotropy vectors Xα of g (indexed over a finite subset R of group of a generic 3-form in 7 dimensions is at elements α ∈ h∗, the roots) on which all elements the basis of a rich geometry that exists only on of h act diagonally through [ , ]: 7-dimensional , whose full beauty has been unveiled in the last thirty years. [H,Xα] = α(H)Xα for all H ∈ h. This article is devoted to a detailed historical In order to avoid problems when doing so he chose and mathematical account of G ’s first years, in 2 the complex numbers C as the ground field. The particular the contributions and the life of Engel’s dimension of the maximal abelian subalgebra h almost forgotten Ph. D. student Walter Reichel, (also called, somehow wrongly, a Cartan subalge- who worked out the details of this description in bra) is the rank of the Lie algebra. It is a general 1907. We will also give an introduction to mod- fact that all roots α ≠ 0 appear only once. If we ern G geometry and its relevance in theoretical 2 = C · (in particular, ). write gα : Xα (these are the root spaces), we obtain a decomposition of g as The Classification of Simple Lie Groups g = h M gα In 1887 [Kil89] succeeded in classi- α∈R fying those transformation groups that are rightly and vectors in g , g for two roots α,β satisfy an called simple:bydefinition,thesearetheLiegroups α β ⊂ that are not abelian and do not have any nontrivial extremely easy multiplication rule: [gα, gβ] gα+β normal . if α + β is again a root, while otherwise it is zero. Every Lie group G has a Lie algebra g (the tangent Two families of complex simple Lie algebras space to the group at the identity), which were well-known at that time: is a endowed with a skew-symmetric (1) the Lie algebras so(n, C) consisting of skew-symmetric complex matrices, which Ilka Agricola is head of a Junior Research Group funded are the Lie algebras of the orthogonal by the Volkswagen Foundation at Humboldt-Universität groups SO(n, C) (n = 3 or n ≥ 5), zu Berlin. Her email address is agricola@mathematik. C hu-berlin.de. (2) the Lie algebras sl (n, ) consisting of 1In Engel’s own words: “Zudem ist hiermit eine direkte trace-free matrices, which are the Lie al- Definition unserer vierzehngliedrigen einfachen Gruppe gebras of the groups SL(n, C) of matrices gegeben, die an Eleganz nichts zu wünschen übrig lässt.” of determinant one (n ≥ 2).

922 Notices of the AMS Volume 55, Number 8 It was Killing’s original intent to prove that these were the only simple complex Lie algebras.2 In fact, there exists a third family of simple algebras, namely the Lie algebras sp(2n, C) of the symplectic groups Sp(2n, C) for n ≥ 1, defined as invariance groups of non-degenerate 2-forms ω on C2n:

Sp(2n, C) ={g ∈ GL(2n, C) : ω = g∗ω}. Oeuvres Complètes, vol. II;

Around 1886, and Friedrich Engel were aware of their existence, but they had not yet appeared in print anywhere [Ha00, p. 152]. To his big surprise, in May 18873 Killing discovered a completely unknown complex (Photo sources: Cartan, Engel, University Archive Greifswald.) of rank 2 and dimension 14, which is just the Élie Cartan Friedrich Engel (1869–1951) in the (1861–1941) around exceptional Lie algebra g2. By October 1887, he had basically completed his classification. He dis- year 1904. 1922. covered that besides g2 and the three families mentioned above, there exist four additional ex- First Results on G2 ceptional simple Lie algebras. In modern notation, We can only conjecture how Killing and his contem- they are: f , e , e , e , and they have dimensions 4 6 7 8 poraries felt aboutthe exceptionalLie algebras—as 52, 78, 133, and 248 respectively. disturbances to the or as exotic and There exist many real orthogonal Lie groups unique objects. But since as a whole C with complexification SO(n, )—namely all or- was developed in these times, they were investi- thogonal groups SO(p, q) associated with scalar gated too, but not with high priority. G2 was the products with indefinite signature (p, q) such that first—and, for rather a long time, the only—Lie p + q = n; they are called real forms of SO(n, C) group for which further results were obtained. (similarly for the Lie algebras), and it is an easy This is natural for dimensional considerations, fact that SO(p, q) is compact only for q = 0. Just but we will see later that it has also much deeper as well, the complex Lie algebra g2 with complex reasons. Lie group G2 has two real forms that we are going From the weight , which one obtains auto- c ∗ to denote by g2 and g2 ; of their (simply connected) matically during the classification, one can easily c ∗ Lie groups G2 and G2 , only the former is compact. determine the lowest dimensional representation Without doubt, the classification of complex of any simple Lie algebra. This Cartan did in the simple Lie algebras is one of the outstanding last section of his thesis, and he rightly observed results of nineteenth century mathematics (this that g2 admits a 7-dimensional complex represen- was not Killing’s point of view, however: his orig- tation, which furthermore possesses a symmetric inal aim had been a classification of all real Lie nondegenerate g2-invariant bilinear form [Ca94, algebras, he was unsatisfied with his own expo- p. 146]: sition and the incompleteness of results, so that 2 β := x0 + x1y1 + x2y2 + x3y3. he would not have published his results with- This scalar product has real coefficients, hence can out strong encouragement from Friedrich Engel). be interpreted over the reals as well; in that case Indeed, Killing’s formidable work contains some it has signature (4, 3), and one can understand gaps and mistakes:4 in his thesis (1894), Élie Cartan’s result as giving a of Cartan gave a completely revised and polished ∗ the noncompact form g2 inside so(4, 3). presentation of the classification [Ca94], which At this stage, the question about explicit con- has therefore become the standard reference for structions of the exceptional Lie algebras becomes the result. pressing. Élie Cartan and Friedrich Engel obtained the first breakthrough, published in two simulta- neous notes to the Académie des Sciences de Paris 2Letter from W. Killing to Fr. Engel, April 12, 1886; see [Ca93], [En93]: For every point a ∈ C5, consider [Ha00, p. 153]. the 2-plane π in the tangent space T C5 that is 3Letter from W. Killing to Fr. Engel, May 23, 1887; see a a [Ha00, p. 161]. the zero set of the Pfaffian system 4 For example, he had found two exceptional Lie algebras dx3 = x1 dx2 − x2 dx1, of dimension 52 and overlooked that they are isomorphic, = − and his classification was based on three main theorems dx4 x2 dx3 x3 dx2, whose statements and proofs were partially wrong. dx5 = x3 dx1 − x1 dx3.

September 2008 Notices of the AMS 923 5 The 14 vector fields on C whose local flows map gα stated before, one sees that the direct sum of the planes πa to each other satisfy the commu- h (corresponding, loosely speaking, to the origin), tator relations of the Lie algebra g2. Both authors the six positive root spaces, and the root space of then gave in the same papers a second geometric one negative of a simple root, span a subalgebra: realization of g2: Engel derived it from the first p := h ⊕ g g . by a contact transformation, while Cartan identi- i −αi M α α∈R+ fied g2 as the symmetries of the solution space of the system of second order partial differential Subalgebras of this kind are called parabolic sub- 5 equations (f = f (x, y)) algebras. The 9-dimensional groups P1 and P2 above are now exactly the subgroups of G with 4 3 2 2 fxx = (fyy ) , fxy = (fyy ) . 3 Lie algebras p1 and p2. By general results, the Both viewed their second realization as being space G2/Pi is a compact homogeneous variety, different from the one through the Pfaffian system. and it can be realized in the projectivization of the Of course, this is correct: stated in modern terms, representation space Vi with highest weight ωi (see figure) as the G2 orbit of some distinguished the G2 has two non-conjugate vector v ; but ω generates the 7-dimensional rep- 9-dimensional parabolic subgroups P1 and P2, and i 1 resentation (spanned by the six short roots and G2 acts on the two compact homogeneous spaces 5 zero with multiplicity one), while ω2 is the highest Mi := G2/Pi , i = 1, 2. It is a detail that Engel and Cartan did not describe the g2 action on the full weight of the (spanned by 5 all roots and zero with multiplicity two). Hence, spaces Mi , but rather on an open subset; this was the common way at that time. we obtain Let us have a closer look at these two homoge- M5 = G /P = G · [v ] ⊂ P(C7) = CP6, neous spaces. For this, we need the lattice inside 1 2 1 2 1 ∗ M5 = G /P = G · [v ] ⊂ P(g ) = CP13. h spanned by the 12 roots of g2, the root lattice. 2 2 2 2 2 2 It is the usual hexagonal planar lattice, in which 5 6 The first space M1 is thus a quadric in CP ; we will the roots are denoted by arrows: come back to the second space later. Let us look again at the real situation. There ω2 ∗ are two real 9-dimensional subgroups Pi inside ∗ the noncompact G2 corresponding to the complex parabolic groups Pi ⊂ G2; but they α ω 2 1 have no counterparts in the compact Lie group c c G2 (roughly speaking, g2 ⊂ so(7) consists of skew- −α α symmetric matrices, while parabolics are always 1 1 upper triangular): a maximal of Gc is iso- + 2 R morphic to SU(3) and thus 8-dimensional. Hence c −α a geometric realization of the compact form G2 is 2 still missing. R− (Courtesy of the author.) G2 and 3-forms in Seven Variables Figure 1. The g2 and lattice. In his talk on June 11, 1900, in Leipzig, Friedrich Engel presented some results on the complex Lie

group G2 that finally led to the missing realization The g2 root system is the only one in which c of its compact form G2. Engel’s geometric insight two roots include an angle of π/6, indicating its 5 CP6 into the geometry of M1 ⊂ was sogood that he exceptional standing among all root systems. The realized that it can be written as the zero set of an roots above respectively below the dashed line equation depending solely on the coefficients of a are called positive respectively negative roots, and generic 3-form in seven variables [En00, p. 220]. + − R = R ∪ R . The two positive roots marked α1 By a generic p-form we mean an element ω ∈ and α2 alone already generate the lattice; they p(Cn)∗ with open GL(n, C) orbit. For dimensional are called simple roots. Keeping in mind the root reasons,Λ n2 ≥ n is a necessary condition for the system with the multiplication rule for root spaces p existence of generic p-forms; it holds for all n if 5 ∗ p = 2, but only for n ≤ 8 when p = 3, and, indeed, In 1910, Élie Cartan returned to his description of G2 by Pfaffian systems and differential equations [Ca10]; generic p-forms do exist for these values. The a modern treatment and further investigation can be isotropy group of a differential form (or, for that found in the worthwhile article by P. Nurowski [Nu05]. matter, any tensor) consists of all group elements It is quite remarkable that the thesis of yet another leaving the form invariant, of Engel’s students plays a decisive role here (Karl ∗ Wünschmann, Greifswald, 1905). Gω := {A ∈ GL(n, C) : ω = A ω}.

924 Notices of the AMS Volume 55, Number 8 For a generic 3-form in dimension 7, its dimension In the same article, Engel invested a lot of en- is ergy in a description of the second homogeneous 5 3 7 ∗ space M2 ⊂ P(g2) through the coefficients of ω. dim Gω = dimGL(7, C) − dim (C ) = 14. For this, he used a symbolic method for invariants Λ Friedrich Engel observed that all generic 3-forms of alternating forms that was communicated to C are equivalent under GL(7, ) and proved the him by Eduard Study; however, Study’s formalism following theorem: is not in use anymore, hence his computations Theorem 1 (F. Engel, 1900). There exists exactly are rather hard to follow. Today, we know that 5 CP13 one GL(7, C) orbit of generic complex 3-forms M2 = G2/P2 ⊂ is a rather complicated pro- [En00, p. 74]. One such generic form is given by jective algebraic variety: it has degree 18 and its complete intersection with three hyperplanes is a ω := (e e + e e + e e )e − 2e e e + 2e e e . 0 1 4 2 5 3 6 7 1 2 3 4 5 6 of genus 10 [Bor83]. For a geometric 3 C7 ∗ For every generic complex 3-form ω ∈ ( ) , description of G2/P2 in terms of ω, observe that the following holds: Λ the 21-dimensional representation 2C7 splits un- 7 (1) The isotropy group of ω is isomorphic to der G2 into g2 ⊕ C , hence G2/P2 isΛ a subvariety 2 7 the G2 [En00, p. 73]; of P( C ) as well. By the Plücker embedding, the (2) ω defines a non-degenerate symmetric bi- 14-dimensionalΛ Grassmann variety G(2, 7) of 2- 7 2 7 linearform βω [En00, p. 222] that is cubic planes in C lies in P( C ). Now, G2/P2 is just the 2 7 in the coefficients of ω, and the quadric intersection of G(2, 7Λ) with P(g2) inside P( C ). 5 6 M1 is its isotropic cone in CP . In particu- As a subvariety of G(2, 7), G2/P2 consists preciselyΛ 7 lar, every isotropy group Gω is contained of those 2-planes π ⊂ C on which βω and ω both in some SO(7, C). degenerate (see [LM03] for a modern account), i.e., (3) There exists a G2-invariant polynomial such that λω ≠ 0 of degree 7 in the coefficients of ω π βω = 0, π ω = 0. [En00, p. 231]. It was the realization of G2 presented in Theorem 1 In fact, Engel had already conjectured in a letter that led Friedrich Engel to the comment cited in to Killing of April 1886 that the isotropy group of the introduction. Besides its elegance, Theorem 1 a 3-form might be a simple 14-dimensional group, has far-reaching consequences for modern differ- but apparently neither he nor Killing had pursued ential geometry (see last section); furthermore, it this idea at that time.6 In modern notation, we can c will provide the missing realization of G2, as is define βω through [Br87] explained now.

βω(X, Y ) := (X ω) ∧ (Y ω) ∧ ω, which is a symmetric bilinear form with values in Walter Reichel and the Invariants of G2 the one-dimensional vector space 7(C7)∗. Over While a professor at Greifswald University (1904- 1913), Friedrich Engel turned again to this topic the reals, it can be turned into a trueΛ real-valued and assigned to his Ph. D. student Walter Reichel scalar product gω after taking an additional square root [Br87], [Hi00]. Geometrically, this means that the task of computing a complete system of in- every generic 3-form on a real 7-dimensional man- variants for complex 3-forms in six and seven ifold induces a (pseudo)-Riemannian metric. An dimensions in Study’s formalism. The thesis de- easy dimension count shows that the isotropy fended by Reichel in 1907 indeed contains the group of a generic 3-form can be a subset of detailed description of the invariants and rela- SO(n, C) only for n = 7, 8. tions among them as well as normal forms of all 3-forms under the action of GL(7, C). The vanish- Since βω is cubic in ω, its determinant is a poly- nomial of degree 21 in the ω coefficients; Engel ing of λω for non-generic forms and the drop of understood that it is the third power of a degree rank of the bilinear form βω play an essential role here. 7 element λω, and its non-vanishing is equivalent Furthermore, Walter Reichel described the to the nondegeneracy of βω. Engel’s arguments still hold over the reals, as isotropy algebra gω of any generic 3-form ω long as the isotropic cone does not degenerate directly through its coefficients [Rei07, p. 48], whereas Friedrich Engel had only computed it for completely. For the 3-form ω0 cited above, gω0 is a real scalar product on R7 with signature one representative. Over the complex numbers, (4, 3) [En00, p. 64]. In particular, there exists ex- this makes no difference; but it turns out that actly one GL(7, R) orbit of real generic 3-forms when passing to real numbers, the one orbit of ω ∈ 3(R7)∗ with non-degenerate isotropic cone complex, GL(7, C) equivalent 3-forms splits into two orbits of real generic GL(7, R) equivalent for gωΛ. Its isotropy group is again isomorphic to ∗ 3-forms. If one interprets the scalar product βω as the real noncompact real form G2 ⊂ SO(4, 3). a real one, it turns out to have signature (4, 3) on 6Letter from Fr. Engel to W. Killing, April 8, 1886; see one orbit and signature (7, 0) on the other orbit. [Ha00, p. 152]. It does not come as a surprise that the isotropy

September 2008 Notices of the AMS 925 926

(Photos courtesy of Irmtraut (née Reichel) (Courtesy University Archive Greifswald.) group a isdtooto h ienra forms normal solved Gurevich nine results, the these of Reichel on Based out Walter [Sch31]. smaller two that to missed observed had reduction and by dimensions) mainly invariants, out .A cotndsrbdtenra om of forms normal the described on Schouten 3-forms A. afterwards; forgotten J. slowly was result the nately, aeadioopi to isomorphic and case ecito fbt elfrsof forms real both of description case. Schiller, Bremen.) hs ece bandauiomgeometric uniform a obtained Reichel Thus, iue2 il aeo .Rihlsthesis. Reichel’s W. of page Title 2. Figure oebr1914. November g ω 18–98 in (1883–1918) atrReichel Walter sioopi to isomorphic is C 7 n13 ysmlrmtos(with- methods simpler by 1931 in g 2 c g .Rihl(undated); Reichel W. 2 ∗ ⊂ n a eonz a recognize can one ⊂ otato aton Kant of portrait so so ( 7 oie fteASVolume AMS the of Notices ( ) 4 , ntesecond the in 3 G ) 2 ntefirst the in Unfortu- . h desk. the ru fteoctonians the of group i gi r .B ibr n .G lsvl in Elashvili G. extremely A. the case of and involved details the Vinberg out worked our B. who E. 1978, to the- are Reichel’s Up again cite [Gu35]. to sis authors 8 next the dimension knowledge, in problem the ne n hoo aln(nGefwl) yCarl by Greifswald); (in Vahlen Theodor and Engel Greifswald. Halle, again, Leipzig, and then Greifswald, at of then Universities philosophy the He and physics, 1902. school mathematics, Easter studied high at his (“Reifezeugnis”) received degree he where from Poland), itr fteRihlfml scoeylne to linked closely is family The the Reichel today. the hosted still of are history Polish archive European and its its and Czech of branch management the the from where far borders), Herrn- not in (Saxony, century hut eighteenth was early and the fif- (1369–1415) in renewed the Reformation Hus Jan Bohemian of around the middle Movement from the century in teenth (Unity emerged Fratrum Brethren), Unitas or of Unity, of Moravian bish- Unity, The later, op. and, Moravian deacon was the father Reichel’s of which members been had Gnadenfrei by village called This founded then Poland). Górna, village Piława Silesian (now little a in investigate to motivation further story. in thesis his a widely his was cited of year anniversary been last 100th has The thesis years. his recent virtually that despite fact known, Reichel, Walter the well about are known was here nothing all to of up work mentioned and life the Whereas Reichel Walter The Baez J. by article 37-39]. the p. in [Ba02, care is great as is with cross-products”), explained description “vector equivalent so-called third through (a equivalent are descriptions these fact, In the vanish. starting of approach memory 3-form the Freudenthal, made and Hans [Fr51], article the of with work popular the became ap- groups through This on Lie but topic. exceptional 298]), article to this p. proach to [Ca14, long returned also never his (see apparently 467] in p. from [Ca08, comment generalizations 1908 their a and numbers as complex this ed tteGmaimi cwint (now Schweidnitz in Gymnasium years the three at the and Herrnhut) by to founded close Fratrum, town Unitas “Päda- (another the Niesky at in years gogium” four by his followed in village, first school home Walter to went University), he Greifswald how describes at Reichel files D. Ph. his in German. in ti elkonthat known well is It mn tes eatne etrsb Friedrich by lectures attended he others, Among atrRihlwsbr nNvme ,1883, 3, November on born was Reichel Walter nhshnwitnC wihcnb found be can (which CV handwritten his In Brüdergemeine n = [VE78]. 9 steMrva nt scalled is Unity Moravian the as , G O 2 c leCra includ- Cartan Élie . steautomorphism the is 55 Number , ´ Swidnica, 8 The “Old Pädagogium” in Niesky, now a public library, built in 1741 as the first parish house of the newly founded community in Niesky. Between 1760 and 1945 it was used as an advanced boarding school.

Neumann (in Leipzig), who formulated the Neu- mann boundary condition in analysis and founded (Photos this page courtesy of I. Agricola and T. Friedrich.) the Mathematische Annalen together with Alfred Clebsch; by Georg Cantor and Felix Bernstein (in Memorial stone on the “God’s acre” in Niesky. Halle), to whom we owe the foundations of set the- ory and the Cantor-Bernstein-Schröder Theorem in logic; by the theoretical physicist Gustav Mie (in Greifswald), who made important contributions to electromagnetism and general relativity; by the experimental physicist Friedrich Ernst Dorn (in Halle), 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 and applied mathematics, physics and philosophical Detail of the inscription on the memorial propaedeutics” with distinction. He spent one year stone. Walter Reichel’s name and date of death as teacher-in-training in Görlitz, and was then ap- (“30.3.18”) are in the second to last line; the pointed in Fall 1908 at the “Realprogymnasium” stone has been damaged and repaired above in Sprottau (now Szprotawa, Poland). In April 1914 his name. he moved for an “Oberlehrer” position to Schwei- dnitz (now ´Swidnica, Poland). With the beginning of the First World War he was drafted, and he died has three children; one of her granddaughters is a in France on March 30, 1918. He has no grave, teacher of mathematics. but an inscription on the WWI memorial on the “God’s acre” of the Moravian community in Niesky G2 Geometry in Dimension 7 commemorates his death. The classical symmetry approach to differential Walter Reichel married Gertrud, née Müller geometry is based on the notion of the isom- (1889–1956) in 1909. They had three sons (born etry group of a Riemannian manifold, i.e., the 1910, 1913, and 1916), who left no children, and group of all transformations acting on the man- a daughter (born March 11, 1918). After the first ifold that preserve the metric. In the twentieth World War, Reichel’s widow moved with her chil- century, the concept of (Riemannian) dren to Niesky, where she was supported by the group became fundamental to Riemannian ge- Moravian Unity. For many years, she accommodat- ometry: it is the group generated by all parallel ed pupils of the “Pädagogium” who did not live in transports along closed null-homotopic loops on the boarding school’s dormitories. Walter Reichel’s the manifold. Here, parallel transport is under- daughter Irmtraut Schiller now lives in Bremen and stood with respect to the Levi-Civita connection

September 2008 Notices of the AMS 927 ∇g, i. e., the straightforward—but not the only Allow-Wallach spheres SU(3)/S1, from extensions possible—generalization of the directional deriva- of Heisenberg groups to clever non-homogeneous tive of vectors. Marcel Berger’sHolonomy Theorem examples. Since then, these non-integrable geome- c from 1955 states that for an irreducible non- tries (not only for G2, but also for contact struc- symmetric manifold, the holonomy group is either tures, almost Hermitian structures, 8-dimensional c SO(n) or from a finite list—and G2 is the only ex- Spin(7)-structures, etc.) have been studied inten- ceptional Lie group on that list. In this case, sively [Ag06]. Today, the main philosophy is that the manifold is necessarily 7-dimensional, the for many Riemannian geometries (M, g) defined c g holonomy group G2 acts on the tangent bundle by tensors that are not ∇ -parallel, it is possible to by its 7-dimensional real representation, and the replace the Levi-Civita connection by a more suit- manifold would be called a parallel or integrable able metric connection ∇ with skew-symmetric c 3 G2-manifold. The argument of the proof is basi- torsion T ∈ (M) (the characteristic connection) cally reduced to the question of which compact Λ 1 g(∇ Y,Z) := g(∇g Y,Z) + T(X,Y,Z) Lie groups admit transitive sphere actions, and X X 2 c 6 this is the case for G2 on S , thought of not as a such that the object becomes parallel, and the but rather as the homogeneous c holonomy group of this new connection plays the space G2/SU(3). role of the classical Riemannian holonomy group. Since Berger’s Holonomy Theorem lists only c For example, a G2-manifold (M, g, ω) admits a the possible holonomy groups without actually characteristic connection if and only if there is a constructing manifolds admitting them, his clas- vector field β such that δ(ω) = −β ω (this ex- sification result was not an end point, but rather cludes one of the four basic types), and its torsion a research program asking for a more detailed is then given by [FI02] geometric investigation (and a long breath); in 1 particular, no Riemannian manifolds with holo- T = − ∗ dω − (dω, ∗ω) ω + ∗(β ∧ ω). c 6 nomy group G2 were known. In 1966, Edmond c Bonan observed in a note preceding his thesis For integrable G2 geometries, the first break- (supervised by André Lichnerowicz) that mani- through was obtained a few years after Gray’s c work. In 1987 and 1989, Robert Bryant and Simon folds with G2 holonomy admit a global 3-form ω with ∇gω = 0, and, in consequence, have to be Salamon succeeded in constructing local complete c Ricci flat—a very restrictive geometric condition metrics with Riemannian holonomy G2 ([Br87], that intrigued some contemporaries [Bon66]. This [BrSa89]). It was only in 1996—more than forty 3-form ω is of course precisely the form whose years after Berger’s original paper—that Dominic c Joyce was able to show the existence of compact stabilizer is G2 as described by Engel and Reichel, but had already been forgotten in this time. For his Riemannian 7-manifolds with Riemannian holono- c work, Edmond Bonan had started from Cartan’s my G2 [Joy00]: this comes down to proving the c existence of solutions of nonlinear elliptic partial description of G2 as the of the octonians, and saw how to derive an invariant differential equations on compact manifolds by 3-form from its multiplication law. very difficult and involved analytical methods. c The credit for having made the most creative One distinguished property of G2 is still miss- c ing in our discussion—this is the one that makes use of G2 and its defining 3-form in differen- tial geometry goes without doubt to Alfred Gray. it attractive for . The group c Since the 1960s he had investigated vector cross G2 can be lifted to the universal covering Spin(7) products and their geometric properties in a se- of SO(7), and Spin(7) has an 8-dimensional irre- ries of papers. In 1971 he had the radical idea ducible real representation 7, the spin represen- c of weakening the classical holonomy concept to tation, that decomposes under∆ G2 ⊂ Spin(7) into cover interesting manifolds that do not appear the trivial and the 7-dimensional representation. in Berger’s list. In particular, he defined nearly Thus, a 7-dimensional spin manifold endowed c with a connection ∇ has a ∇-parallel field if parallel G2-manifolds: they have structure group c c G , but instead of being parallel, the 3-form ω and only if the holonomy of ∇ lies inside G2, and 2 c satisfies the differential equation G2 is the isotropy group of a generic spinor. In fact, any spinor field defines a global 3-form and vice dω = λ ∗ ω c versa, so this last characterization of G2 is, in itself, for a real constant λ ≠ 0, and they are Einstein with nothing new. But it explains the intricate relation c strictly positive scalar curvature. Later, he proved between G2 and spin geometry. For example, the c together with Marisa Fernández that there are in nearly parallel G2-manifolds discovered by Gray in c fact four basic classes of G2-manifolds depending 1971 are precisely those 7-manifolds that admit on the possible nature of the tensor ∇gω [FG82]. a real Killing spinor field [FK90]. More recently, Maybe even more importantly, they initiated the superstring theory has stimulated a deep interest (still-ongoing) construction of many interesting in 7-manifolds with integrable or non-integrable 7 c c examples—ranging from S = Spin(7)/G2 to the G2 geometry [Du02]. In this approach, a ∇-parallel

928 Notices of the AMS Volume 55, Number 8 spinor field is interpreted as a [Ca08] , Nombres Complexes, Encyclop. Sc. Math., transformation: by tensoring with a spinor field, édition française, 15, 1908, 329–468, d’après bosonic particles can be transformed into fermion- l’article allemand de E. Study. ic particles (and vice versa). The torsion of ∇ (if [Ca10] , Les systèmes de Pfaff à cinq variables et les équations aux dérivées partielles du second present) is related to the B-field, a higher order ordre, Ann. Éc. Norm. 27 (1910), 109–192. version of the classical field strength of Yang-Mills [Ca14] , Les groupes réels simples finis et theory (which would be a 2-form and not a 3-form, continus, Ann. Éc. Norm. 31 (1914), 263–355. of course). [Du02] M. J. Duff, M-theory on manifolds of G2-holono- The story of G2 and its relatives is far from con- my: The first twenty years, hep-th/0201062. cluded. The algebraic foundations for the many [En93] F. Engel, Sur un groupe simple à quatorze lines of development sketched in this article were paramètres, C. R. Acad. Sc. 116 (1893), 786–788. laid more than a hundred years ago by Friedrich [En00] , Ein neues, dem linearen Komplexe 52 Engel and his student Walter Reichel in a work of analoges Gebilde, Leipz. Ber. (1900), 63–76, 220–239. remarkable mathematical insight. [FG82] M. Fernández and A. Gray, Riemannian man- ifolds with structure group G2, Ann. Mat. Pura Acknowledgements Appl. 132 (1982), 19–45. I thank the Archive of Greifswald University, the [Fr51] H. Freudenthal, Oktaven, Ausnahmegruppen Archive of the Moravian Unity in Herrnhut, the und Oktavengeometrie, Utrecht: Mathematisch Research Library Gotha of Erfurt University, the Institut der Rijksuniversiteit (1951), 52 S. [FI02] T. Friedrich and S. Ivanov, Parallel antiquarian Wilfried Melchior in Spreewaldheide and connections with skew-symmetric torsion near Berlin, and Walter Reichel’s daughter, Irm- in theory, Asian Jour. Math. 6 (2002), traut Schiller, as well as her family, for their 303–336. support during the investigations for this article. [FK90] T. Friedrich and I. Kath, 7-dimensional com- I am also grateful to Robert Bryant (MSRI Berke- pact Riemannian manifolds with Killing spinors, ley) and Joseph M. Landsberg (Texas) for fruitful Comm. Math. Phys. 133 (1990), 543–561. mathematical discussions, and Philip G. Spain [Gra71] A. Gray, Weak holonomy groups, Math. Z. 123 (Glasgow) for reading and commenting on early (1971), 290–300. [Gu35] G. B. Gurevich, Classification des trivecteurs versions of this text. Finally, special thanks go ayant le rang huit, C. R. Acad. Sci. URSS 2 (1935), to Thomas Friedrich (Humboldt Universität zu 353–356. Berlin) for his ongoing support, endless hours of [Ha00] T. Hawkins, Emergence of the theory of Lie discussions bent together over the manuscripts of groups, Springer Verlag, 2000. Engel, Reichel, and Cartan, and many more things [Hi00] N. 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