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- Real and Complex Forms
- E6 as An Algebraic Group
- Algebra
- Important Subalgebras and Representations
- E6 Polytope
- Chevalley and Steinberg Groups of Type E6 and 2E6
- Importance in Physics
- See Also
- References
There is a unique complex Lie algebra of type E6, corresponding to a complex group of complex dimension 78. The complex adjoint Lie group E6 of complex dimension 78 can be considered as a simple real Lie group of real dimension 156. This has fundamental group Z/3Z, has maximal compact subgroup the compact form (see below) of E6, and has an outer au...
By means of a Chevalley basis for the Lie algebra, one can define E6 as a linear algebraic group over the integers and, consequently, over any commutative ring and in particular over any field: this defines the so-called split (sometimes also known as "untwisted") adjoint form of E6. Over an algebraically closed field, this and its triple cover are...
Dynkin diagram
The Dynkin diagram for E6 is given by , which may also be drawn as .
Roots of E6
Although they span a six-dimensional space, it is much more symmetrical to consider them as vectorsin a six-dimensional subspace of a nine-dimensional space. Then one can take the roots to be 1. (1,−1,0;0,0,0;0,0,0), (−1,1,0;0,0,0;0,0,0), 2. (−1,0,1;0,0,0;0,0,0), (1,0,−1;0,0,0;0,0,0), 3. (0,1,−1;0,0,0;0,0,0), (0,−1,1;0,0,0;0,0,0), 4. (0,0,0;1,−1,0;0,0,0), (0,0,0;−1,1,0;0,0,0), 5. (0,0,0;−1,0,1;0,0,0), (0,0,0;1,0,−1;0,0,0), 6. (0,0,0;0,1,−1;0,0,0), (0,0,0;0,−1,1;0,0,0), 7. (0,0,0;0,0,0;1,−1,0)...
Weyl group
The Weyl group of E6 is of order 51840: it is the automorphism group of the unique simple group of order 25920 (which can be described as any of: PSU4(2), PSΩ6−(2), PSp4(3) or PSΩ5(3)).
The Lie algebra E6 has an F4subalgebra, which is the fixed subalgebra of an outer automorphism, and an SU(3) × SU(3) × SU(3) subalgebra. Other maximal subalgebras which have an importance in physics (see below) and can be read off the Dynkin diagram, are the algebras of SO(10) × U(1) and SU(6) × SU(2). In addition to the 78-dimensional adjoint repr...
The E6 polytope is the convex hull of the roots of E6. It therefore exists in 6 dimensions; its symmetry group contains the Coxeter group for E6 as an index2 subgroup.
The groups of type E6 over arbitrary fields (in particular finite fields) were introduced by Dickson (1901, 1908). The points over a finite field with q elements of the (split) algebraic group E6 (see above), whether of the adjoint (centerless) or simply connected form (its algebraic universal cover), give a finite Chevalley group. This is closely ...
N = 8 supergravity in five dimensions, which is a dimensional reduction from 11 dimensional supergravity, admits an E6 bosonic global symmetry and an Sp(8) bosonic local symmetry. The fermions are in representations of Sp(8), the gauge fields are in a representation of E6, and the scalars are in a representation of both (Gravitons are singlets with...
Adams, J. Frank (1996), Lectures on exceptional Lie groups, Chicago Lectures in Mathematics, University of Chicago Press, ISBN 978-0-226-00526-3, MR 1428422.Baez, John (2002). "The Octonions, Section 4.4: E6". Bull. Amer. Math. Soc. 39 (2): 145–205. arXiv:math/0105155. doi:10.1090/S0273-0979-01-00934-X. ISSN 0273-0979. S2CID 586512. Online HTML version...Cremmer, E.; J. Scherk; J. H. Schwarz (1979). "Spontaneously Broken N=8 Supergravity". Phys. Lett. B. 84 (1): 83–86. Bibcode:1979PhLB...84...83C. doi:10.1016/0370-2693(79)90654-3. Online scanned ve...Dickson, Leonard Eugene (1901), "A class of groups in an arbitrary realm connected with the configuration of the 27 lines on a cubic surface", The Quarterly Journal of Pure and Applied Mathematics,...The Bridge of Sighs ( Italian: Ponte dei Sospiri, Venetian: Ponte de i Sospiri) is a bridge in Venice, Italy. The enclosed bridge is made of white limestone, has windows with stone bars, passes over the Rio di Palazzo, and connects the New Prison ( Prigioni Nuove) to the interrogation rooms in the Doge's Palace.
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