Английская Википедия:Albert algebra
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 Шаблон:Harvs and studied by Шаблон:Harvtxt, is the set of 3×3 self-adjoint matrices over the octonions, equipped with the binary operation
- <math>x \circ y = \frac12 (x \cdot y + y \cdot x),</math>
where <math>\cdot</math> 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.
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
Notes
References
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Cite book
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
- Шаблон:Citation
Further reading
- ↑ Springer & Veldkamp (2000) 5.8, p.153
- ↑ Springer & Veldkamp (2000) 7.2
- ↑ Шаблон:Cite journal
- ↑ Knus et al (1998) p.517
- ↑ Шаблон:Cite journal
- ↑ Garibaldi, Merkurjev, Serre (2003), p.50
- ↑ Garibaldi (2009), p.20