Albert algebra explained

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 and studied by, is the set of 3×3 self-adjoint matrices over the octonions, equipped with the binary operation

x \circ y = (12)/((x) ⋅ y + y ⋅ x),

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.

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

References

• • • Book: Garibaldi, Skip. Skip Garibaldi . Cohomological invariants: exceptional groups and Spin groups. . 2009. 200. 937. 978-0-8218-4404-5. 10.1090/memo/0937.

Further reading

• • Petersson . Holger P.. Structure theorems for Jordan algebras of degree three over fields of arbitrary characteristic. Communications in Algebra. 32. 2004. 3. 1019–1049. 10.1081/AGB-120027965. 10.1.1.496.2136. 34280968.
• Albert algebra at Encyclopedia of Mathematics.

Notes and References

1. Springer & Veldkamp (2000) 5.8, p.153
2. Springer & Veldkamp (2000) 7.2
3. Chevalley C, Schafer RD . The Exceptional Simple Lie Algebras F(4) and E(6) . Proc. Natl. Acad. Sci. U.S.A. . 36 . 2 . 137–41 . February 1950 . 16588959 . 1063148 . 10.1073/pnas.36.2.137. 1950PNAS...36..137C. free .
4. Knus et al (1998) p.517
5. Structurable Algebras and Groups of Type E_6 and E_7. Skip Garibaldi. Skip Garibaldi. Journal of Algebra. 236. 2001. 2. 651–691. 10.1006/jabr.2000.8514. math/9811035.
6. Garibaldi, Merkurjev, Serre (2003), p.50
7. Garibaldi (2009), p.20