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 Jordan, Neumann & Wigner (1934) and studied by Albert (1934), 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.[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
- ↑ Springer & Veldkamp (2000) 5.8, p.153
- ↑ Springer & Veldkamp (2000) 7.2
- ↑ 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. doi:10.1073/pnas.36.2.137. PMC 1063148
. PMID 16588959. - ↑ Knus et al (1998) p.517
- ↑ Skip Garibaldi (2001). "Structurable Algebras and Groups of Type E_6 and E_7". Journal of Algebra. 236. pp. 651–691.
- ↑ Garibaldi, Merkurjev, Serre (2003), p.50
- ↑ Garibaldi (2009), p.20
References
- Albert, A. Adrian (1934), "On a Certain Algebra of Quantum Mechanics", Annals of Mathematics, Second Series, 35 (1): 65–73, doi:10.2307/1968118, ISSN 0003-486X, JSTOR 1968118
- Garibaldi, Skip; Merkurjev, Alexander; Serre, Jean-Pierre (2003), Cohomological invariants in Galois cohomology, University Lecture Series, 28, Providence, RI: American Mathematical Society, ISBN 0-8218-3287-5, MR 1999383
- Garibaldi, Skip (2009). "Cohomological invariants: exceptional groups and Spin groups". Memoirs of the American Mathematical Society. 200 (937). ISBN 0-8218-4404-0.
- Jordan, P.; Neumann, J. von; Wigner, E. (1934), "On an Algebraic Generalization of the Quantum Mechanical Formalism", Annals of Mathematics, Princeton, 35 (1): 29–64, doi:10.2307/1968117, JSTOR 1968117
- Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre (1998), The book of involutions, Colloquium Publications, 44, With a preface by J. Tits, Providence, RI: American Mathematical Society, ISBN 0-8218-0904-0, Zbl 0955.16001
- McCrimmon, Kevin (2004), A taste of Jordan algebras, Universitext, Berlin, New York: Springer-Verlag, doi:10.1007/b97489, ISBN 978-0-387-95447-9, MR 2014924
- Springer, Tonny A.; Veldkamp, Ferdinand D. (2000) [1963], Octonions, Jordan algebras and exceptional groups, Springer Monographs in Mathematics, Berlin, New York: Springer-Verlag, ISBN 978-3-540-66337-9, MR 1763974
Further reading
- Petersson, Holger P.; Racine, Michel L. (1994), "Albert algebras", in Kaup, Wilhelm, 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. doi:10.1081/AGB-120027965.
- Albert algebra at Encyclopedia of Mathematics.