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| G_2 | |
|---|---|
| Name | G₂ |
| Type | Exceptional Lie group |
| Dimension | 14 |
G_2
G_2 is an exceptional compact and complex Lie group and corresponding Lie algebra notable in the classification of simple Lie algebras. It appears alongside E₈, E₇, E₆, F₄, B_n, and C_n in the Cartan classification and connects to structures such as the octonions, Fano plane, Holonomy group, Élie Cartan, and Hermann Weyl. G_2 plays roles in differential geometry, representation theory, and theoretical physics through links with Riemannian manifold, Calabi–Yau manifold, M-theory, and Supersymmetry.
G_2 is the unique 14-dimensional simple Lie algebra of rank 2 over the complex numbers classified by Cartan matrix and Dynkin diagram data introduced by Élie Cartan and systematized by Hermann Weyl and Nicolai Bourbaki. As a compact real form, it is a compact simple Lie group of dimension 14; as a complex Lie algebra it admits a Chevalley basis studied by Claude Chevalley and connections to Kac–Moody algebra methods of Victor Kac. Key invariants include the Killing form used in the work of Élie Cartan and the highest root appearing in the Weyl group action described by H. S. M. Coxeter.
The exceptional list including G_2 emerged from classification programs by Élie Cartan and later refinements by Weyl, Killing, and Bourbaki. The label "G_2" originates from Cartan's notation for rank-2 exceptional cases; contemporaries like Émile Picard and later historians such as Jean-Pierre Serre and Robert Steinberg trace the development. The role of the octonions in understanding G_2 was emphasized by John Baez, building on earlier work of Arthur Cayley and John T. Graves on composition algebras. The connection to holonomy groups in Riemannian geometry was popularized in papers by Marcel Berger and later expositions by Dominic Joyce.
The Lie algebra of G_2 is a 14-dimensional simple algebra with a 2-dimensional Cartan subalgebra; its structure constants can be realized using derivations of the octonion algebra as in constructions by Élie Cartan and Nathan Jacobson. The group admits compact and split real forms, studied by Élie Cartan and classified in tables by Armand Borel and Harish-Chandra. Representation-theoretic tools from Weyl character formula, Peter–Weyl theorem, and the theory of Verma modules elucidate unitary representations relevant to works by Harish-Chandra and Bernstein. The adjoint representation is 14-dimensional; fundamental representations of dimensions 7 and 14 play roles in geometric realizations associated with the Fano plane and with exceptional holonomy.
The root system of G_2 consists of 12 long and short roots forming a nonreduced two-dimensional configuration encoded by the G_2 Dynkin diagram in Cartan's notation; related combinatorial analyses appear in the work of H. S. M. Coxeter and John Conway. Highest-weight theory classifies finite-dimensional irreducible representations via dominant integral weights as in constructions by Élie Cartan, Hermann Weyl, and Harish-Chandra. Notable representations include the 7-dimensional fundamental representation linked to the octonionic action and the 14-dimensional adjoint; tensor product decompositions were computed by Roger Howe and cataloged in tables by Fulton and Harris. The Weyl group of G_2 is a dihedral group of order 12 connected to symmetry considerations by Weyl and Coxeter.
G_2 is the automorphism group of the octonion algebra discovered by Arthur Cayley and John T. Graves and studied by Adolf Hurwitz and Nathanson?; explicit modern expositions are by John Baez and Octonionic researchers. As a stabilizer group of a generic 3-form on a 7-dimensional vector space, G_2 appears in Berger's classification of Riemannian holonomy groups by Marcel Berger and in constructs of G_2-manifolds developed by Dominic Joyce and Robert Bryant. These manifolds connect to calibrated geometry introduced by Harvey and Lawson and to exceptional calibrated cycles relevant to Mirror Symmetry discussions by Maxim Kontsevich.
In mathematics, G_2 structures inform special holonomy metrics studied by Dominic Joyce and Robert Bryant and influence questions in topology addressed by Simon Donaldson and Slawomir Kolodziej?; in representation theory they appear in Langlands duality contexts by Robert Langlands and in automorphic forms by James Arthur. In physics, G_2 compactifications are central to M-theory model-building and to flux compactification scenarios by Cumrun Vafa, Edward Witten, and K. Becker; G_2 gauge theories arise in work by Nathan Seiberg and Edward Witten. Exceptional symmetry patterns with G_2 show up in model-building by Georgi–Glashow-style grand unified theories and in string dualities studied by Ashoke Sen and Juan Maldacena.
G_2 occupies the smallest rank among the five exceptional Lie groups F₄, E₆, E₇, and E₈ cataloged in the Cartan classification. Connections to other groups arise via folding and subalgebra embeddings studied by Dynkin and Onishchik; notable inclusions are embeddings of G_2 into SO(7), SO(8), and relationships with Spin(7) via triality observed by Elie Cartan and later authors. The role of G_2 in the broader exceptional series is surveyed in expository texts by Fulton and Harris, John Baez, and historical treatments by Jean-Pierre Serre.
Category:Exceptional Lie groups