G2 (mathematics)

G2 (mathematics)
Name	G2
Type	Exceptional Lie group
Dimension	14
Compact form	Compact real form
Split form	Split real form
Related	F4, E6, E7, E8, SO(7), Spin(7)

G2 (mathematics) is an exceptional simple Lie group and Lie algebra of rank 2 and dimension 14 that occupies a central place among the five exceptional Lie types. It appears in the classification of simple Lie algebras by Wilhelm Weyl and Élie Cartan, plays a role in the study of the octonions associated with Arthur Cayley and John Baez, and manifests in geometric and physical contexts connected to Marcel Grossmann-type structures, Michel Scharlemann-style gauge theories, and string- and M-theory compactifications studied by Edward Witten and Nathan Seiberg.

Definition and basic properties

The group is defined as the automorphism group of a normed algebra structure on an 8-dimensional real vector space arising from the Cayley or octonion algebra studied by John Cayley and Friedrich Zorn, or equivalently as the subgroup of GL(7,ℝ) preserving a generic 3-form initially analyzed by Élie Cartan and later by Shiing-Shen Chern. Its Lie algebra is a 14-dimensional simple Lie algebra over ℝ or ℂ classified in Cartan's list along with types A_n, B_n, C_n, D_n and exceptional types F4, E6, E7, E8 examined by Wilhelm Killing and Élie Cartan. Over the complex numbers it has two real forms: the compact real form often denoted by the compact simple Lie group studied by Hermann Weyl in representation contexts and the split real form related to the split octonions considered by Jacques Tits.

Lie algebra and root system

The Lie algebra has Cartan type G2 in Cartan–Killing classification, with a rank-2 root system consisting of 12 roots forming the unique non-simply-laced configuration with long and short roots described by Wilhelm Weyl and Harold Bourbaki-style root data. The Dynkin diagram consists of two nodes connected by a triple bond with orientation, as catalogued by Élie Cartan and Claude Chevalley; the corresponding Cartan matrix underlies the Serre relations used by Jean Dieudonné and Igor Kac in Kac–Moody generalizations. The small Weyl group is dihedral of order 12, which is significant in the study of Coxeter groups elaborated by H.S.M. Coxeter. The Killing form and structure constants can be described explicitly in bases related to octonionic multiplication rules explored by Arthur Cayley and octonion expositors such as John Baez.

Representations and representation theory

Irreducible representations are indexed by two nonnegative integers corresponding to highest weights relative to the two fundamental weights recorded in the work of Hermann Weyl and I. M. Gelfand. The minimal nontrivial complex representation is 7-dimensional, realized as the action on the imaginary octonions, while the adjoint representation is 14-dimensional; higher-dimensional modules were tabulated by Robert Steinberg and Roger Howe in studies of exceptional group modules. Tensor product decompositions, character formulas, and branching rules are computed using Weyl character formula techniques introduced by Hermann Weyl and implemented in the context of crystal bases by Masaki Kashiwara and Michio Jimbo. Modular and reduction-theoretic aspects appear in the work of George Lusztig and J.E. Humphreys on representation theory over finite fields and in Deligne–Lusztig constructions tied to the finite group of Lie type G2(q) first studied by Claude Chevalley and Robert Steinberg.

Geometry and octonions

G2 preserves a stable 3-form on a 7-dimensional real vector space, a fact used by Nigel Hitchin and Dominic Joyce to construct Riemannian manifolds with G2 holonomy, leading to compact manifolds in the work of Dominic Joyce and to metrics studied by Simon Donaldson and Karen Uhlenbeck. These G2-manifolds feature prominently in calibrated geometry developed by Reese Harvey and H. B. Lawson and in special holonomy theory initiated by Marcel Berger. The octonion algebra links to composition algebra classification by Adolf Hurwitz and Hurwitz's theorem, with automorphism group precisely the compact form of G2 as shown in classical treatments by John Baez and Jacques Tits.

Classification and relation to other groups

G2 sits in the Bott–Samelson and Dynkin diagram family of exceptional groups alongside F4, E6, E7, E8 cataloged by Élie Cartan and Rudolf Steinberg. Embeddings include G2 inside SO(7) and Spin(7) as the stabilizer of a spinor or 3-form; these inclusions are used by Marcel Berger and Michael Atiyah in holonomy classifications. Over finite fields the Chevalley groups of type G2 give rise to G2(q) and twisted Ree groups 2G2 studied by Robert Ree; these finite groups were analyzed by John Thompson and Daniel Gorenstein in the classification of finite simple groups. Connections to Lie types B3 and D4 appear in folding and triality constructions investigated by Claude Chevalley and J. Tits.

Applications and occurrences in mathematics and physics

G2 appears in gauge theory and compactification scenarios in string theory and M-theory examined by Edward Witten, Juan Maldacena, and Michael Green, where G2-holonomy manifolds yield minimally supersymmetric models; it also appears in condensed matter contexts modeled by Nigel Hitchin-type geometric flows and in special solutions of Yang–Mills theories explored by Simon Donaldson and Karen Uhlenbeck. In number theory and automorphic forms G2 features in Langlands program considerations studied by Robert Langlands and Hervé Jacquet, and in the construction of exceptional theta correspondences developed by Roger Howe and David Savin. Finite groups of type G2 enter the classification of finite simple groups worked on by Daniel Gorenstein, and computational group theory implementations are found in systems like Richard Cannon's and Jürgen Bosch-era software used by experimental algebraists.

Category:Lie groups