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Albert algebra

## Summary

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

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

where ${\displaystyle \cdot }$ 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] (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.

## Notes

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. ^ 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 (2): 651–691. arXiv:math/9811035. doi:10.1006/jabr.2000.8514.
6. ^ Garibaldi, Merkurjev, Serre (2003), p.50
7. ^ Garibaldi (2009), p.20

## References

• Petersson, Holger P.; Racine, Michel L. (1994), "Albert algebras", in Kaup, Wilhelm (ed.), 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. CiteSeerX 10.1.1.496.2136. doi:10.1081/AGB-120027965. S2CID 34280968.
• Albert algebra at Encyclopedia of Mathematics.