OldandNewonthe ExceptionalGroupG2 Ilka Agricola n a talk delivered in Leipzig (Germany) 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 group G. If G happens to be a group of matrices, its description of the smallest exceptional Lie Lie algebra g is easily realized by matrices too, and group G2, and he wrote in the corresponding the Lie bracket coincides with the usual commuta- Inote 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 Lie group and nition of our 14-dimensional simple group 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 manifolds, 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 · physics (in particular, superstring theory). 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 Wilhelm Killing [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 subgroups. 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 manifold at the identity), which were well-known at that time: is a vector space 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, Sophus Lie 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 simple Lie algebra (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 symmetry or as exotic and There exist many real orthogonal Lie groups unique objects. But since Lie theory 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 lattice, 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 real representation 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 complex Lie group 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.