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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.〔Springer & Veldkamp (2000) 5.8, p.153〕 One of them, which was first mentioned by and studied by , 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.〔Springer & Veldkamp (2000) 7.2〕 (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'').〔Knus et al (1998) p.517〕 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. 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.〔Garibaldi, Merkurjev, Serre (2003), p.50〕 The cohomological invariants with 3-torsion coefficients have a basis 1, ''g''3 of degrees 0, 3.〔Garibaldi (2009), p.20〕 The invariants ''f''3 and ''g''3 are the primary components of the Rost invariant. ==See also== *Euclidean Jordan algebra for the Jordan algebras considered by Jordan, von Neumann and Wigner *Euclidean Hurwitz algebra for details of the construction of the Albert algebra for the octonions 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Albert algebra」の詳細全文を読む スポンサード リンク
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