Albert algebra

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In mathematics, an Albert algebra is a 27-dimensional exceptional Jordan algebra. They are named after Abraham Adrian Albert, who pioneered the study of non-associative algebras, usually working over the real numbers. Over the real numbers, there are three such Jordan algebras up to isomorphism.[1] One of them, which was first mentioned by Jordan, Neumann & Wigner (1934) and studied by Albert (1934), is the set of 3×3 self-adjoint matrices over the octonions, equipped with the binary operation

where denotes matrix multiplication. Another is defined the same way, but using split octonions instead of octonions. The final is constructed from the non-split octonions using a different standard involution.

Over any algebraically closed field, there is just one Albert algebra, and its automorphism group G is the simple split group of type F4.[2][3] (For example, the complexifications of the three Albert algebras over the real numbers are isomorphic Albert algebras over the complex numbers.) Because of this, for a general field F, the Albert algebras are classified by the Galois cohomology group H1(F,G).[4]

The Kantor–Koecher–Tits construction applied to an Albert algebra gives a form of the E7 Lie algebra. The split Albert algebra is used in a construction of a 56-dimensional structurable algebra whose automorphism group has identity component the simply-connected algebraic group of type E6.[5]

The space of cohomological invariants of Albert algebras a field F (of characteristic not 2) with coefficients in Z/2Z is a free module over the cohomology ring of F with a basis 1, f3, f5, of degrees 0, 3, 5.[6] The cohomological invariants with 3-torsion coefficients have a basis 1, g3 of degrees 0, 3.[7] The invariants f3 and g3 are the primary components of the Rost invariant.

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Notes[edit]

  1. ^ Springer & Veldkamp (2000) 5.8, p.153
  2. ^ Springer & Veldkamp (2000) 7.2
  3. ^ Chevalley C, Schafer RD (February 1950). "The Exceptional Simple Lie Algebras F(4) and E(6)". Proc. Natl. Acad. Sci. U.S.A. 36 (2): 137–41. doi:10.1073/pnas.36.2.137. PMC 1063148Freely accessible. PMID 16588959. 
  4. ^ Knus et al (1998) p.517
  5. ^ Skip Garibaldi (2001). "Structurable Algebras and Groups of Type E_6 and E_7". Journal of Algebra. 236. pp. 651–691. 
  6. ^ Garibaldi, Merkurjev, Serre (2003), p.50
  7. ^ Garibaldi (2009), p.20

References[edit]

Further reading[edit]

  • Petersson, Holger P.; Racine, Michel L. (1994), "Albert algebras", in Kaup, Wilhelm, Jordan algebras. Proceedings of the conference held in Oberwolfach, Germany, August 9-15, 1992, Berlin: de Gruyter, pp. 197–207, Zbl 0810.17021 
  • Petersson, Holger P. (2004). "Structure theorems for Jordan algebras of degree three over fields of arbitrary characteristic". Communications in Algebra. 32 (3): 1019–1049. doi:10.1081/AGB-120027965. 
  • Albert algebra at Encyclopedia of Mathematics.