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| E6 (group) | |
|---|---|
| Name | E6 |
| Type | Lie group |
| Dimension | 78 |
| Root system | E6 |
| Dynkin diagram | E6 Dynkin diagram |
E6 (group) is a complex, simply connected, exceptional simple Lie group of rank 6 and dimension 78 arising in the classification of simple Lie groups by Wilhelm Élie Cartan and Cartan–Killing theory. Its compact real form and split real form play central roles in the work of Cartan classification and the classification of simple Lie algebras by Dynkin diagram methods due to Eugene Dynkin. E6 appears across mathematics in the theory of Jordan algebra, Freudenthal magic square, and the study of exceptional structures connected to Cayley algebra and the Octonions.
E6 is defined abstractly as the unique (up to isomorphism) connected, simply connected, complex simple Lie group whose complex Lie algebra has Dynkin diagram of type E6 in Cartan matrix classification. The complex Lie algebra has dimension 78, rank 6, and a nondegenerate Killing form giving a compact real form commonly denoted E6^cpt and a split real form denoted E6(6) in real form notation due to Elie Cartan real forms. Structure constants and Chevalley bases allow construction over arbitrary fields, yielding finite groups of Lie type such as groups of type E6 over finite fields studied by Claude Chevalley and Robert Steinberg.
The E6 root system is an exceptional irreducible root system in a 6-dimensional Euclidean space associated to the E6 Dynkin diagram introduced by Eugene Dynkin. The diagram features six nodes with a single branching node; the corresponding Cartan matrix appears in the classification of simple roots and coroots used in the Weyl group construction. The Weyl group of E6 is a finite reflection group of order 51,840, related to symmetry groups studied by Coxeter and appearing in the context of reflection groups and root lattices. The E6 root lattice embeds into the E7 and E8 lattices; these embeddings are important in the lattice constructions of Niemeier lattice theory and the study of Leech lattice connections.
The complex Lie algebra e6 admits a Cartan decomposition and highest-weight theory; fundamental representations correspond to the six fundamental weights labeled by the nodes of the Dynkin diagram introduced by Eugene Dynkin. Notable representations include the 27-dimensional minimal (or fundamental) representation and its dual 27*, as well as the adjoint representation of dimension 78. Representation theory of e6 connects to Weyl character formula, Branching rules under embeddings into classical algebras such as sl(3) and so(10). Exceptional invariant forms and cubic invariants on the 27 arise from the connection with Albert algebra (the 27-dimensional exceptional Jordan algebra) and the Freudenthal triple system, studied by Hans Freudenthal.
Real forms of the complex E6 algebra are classified by Cartan involution and Satake diagrams; the real forms include the compact form E6^cpt, the split form E6(6), the Hermitian real form E6(2) with a noncompact Hermitian symmetric space, and the intermediate form E6(-14) connected to a noncompact symmetric domain. These real forms appear in the classification of simple Lie groups by Helgason and in the list of real simple Lie algebras compiled by Élie Cartan. The Hermitian symmetric space associated to one real form yields a bounded symmetric domain relevant to work by Élie Cartan on symmetric domains and by Harish-Chandra on discrete series.
E6 contains notable maximal subgroups and regular embeddings linking exceptional and classical types. Prominent embeddings include E6 ⊃ F4 via the fixed points of diagram automorphisms, E6 ⊃ so(10) × u(1) via the 27 branching into spinor and vector parts, and embeddings into E7 and E8 as nodes are added in the Dynkin diagram chain studied by Bourbaki. Finite subgroups and centralizers in E6 are analyzed via Dynkin index and the classification of semisimple subalgebras by Dynkin. These subgroup structures are crucial in the construction of homogeneous spaces such as E6/F4 and in studying exceptional holonomy and special geometries explored by Marcel Berger and Berger classification.
E6 plays a role in grand unified theories in high-energy physics, where models by Howard Georgi and Sheldon Glashow inspired unified gauge proposals employing E6 gauge symmetry and 27-dimensional matter multiplets. In string theory and compactification, E6 arises in heterotic string model building and in the study of gauge enhancements associated with Calabi–Yau manifold singularities and F-theory constructions. Geometrically, E6 symmetry appears in the study of exceptional holonomy, projective planes over the Cayley numbers (the octonionic projective plane), and in the classification of special algebraic varieties linked to the 27 lines on a cubic surface studied since the work of Arthur Cayley and George Salmon. E6-structures also inform studies of vertex operator algebras and sporadic groups such as connections between E6-related lattices and the Monster group via lattice and moonshine phenomena investigated by John Conway and Richard Borcherds.
Category:Exceptional Lie groups