Albert algebra (original) (raw)

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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 Pascual Jordan, John von Neumann, and Eugene Wigner (1934) and studied by Albert (1934), is the set of 3×3 self-adjoint matrices over the octonions, equipped with the binary operation

x ∘ y = 1 2 ( x ⋅ y + y ⋅ x ) , {\displaystyle x\circ y={\frac {1}{2}}(x\cdot y+y\cdot x),} {\displaystyle x\circ y={\frac {1}{2}}(x\cdot y+y\cdot x),}

where ⋅ {\displaystyle \cdot } {\displaystyle \cdot } 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][4] (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).[5][6]

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.[7]

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, _f_3, _f_5, of degrees 0, 3, 5.[8] The cohomological invariants with 3-torsion coefficients have a basis 1, _g_3 of degrees 0, 3.[9] The invariants _f_3 and _g_3 are the primary components of the Rost invariant.

  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. Bibcode:1950PNAS...36..137C. doi:10.1073/pnas.36.2.137. PMC 1063148. PMID 16588959.
  4. ^ Garibaldi, Petersson, Racine (2024), p. 577
  5. ^ Knus et al (1998) p.517
  6. ^ Garibaldi, Petersson, Racine (2024), pp. 599, 600
  7. ^ Skip Garibaldi (2001). "Structurable Algebras and Groups of Type E_6 and E_7". Journal of Algebra. 236 (2): 651–691. arXiv:math/9811035. doi:10.1006/jabr.2000.8514.
  8. ^ Garibaldi, Merkurjev, Serre (2003), p.50
  9. ^ Garibaldi (2009), p.20