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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 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

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

where $\cdot$ 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

^ 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. Bibcode:1950PNAS...36..137C. 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 (2): 651–691. arXiv:math/9811035. doi:10.1006/jabr.2000.8514.
^ 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, vol. 28, Providence, RI: American Mathematical Society, ISBN 978-0-8218-3287-5, MR 1999383
Garibaldi, Skip (2009). Cohomological invariants: exceptional groups and Spin groups. Memoirs of the American Mathematical Society. Vol. 200. doi:10.1090/memo/0937. ISBN 978-0-8218-4404-5.
Jordan, Pascual; Neumann, John von; Wigner, Eugene (1934), "On an Algebraic Generalization of the Quantum Mechanical Formalism", Annals of Mathematics, 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, vol. 44, With a preface by J. Tits, Providence, RI: American Mathematical Society, ISBN 978-0-8218-0904-4, 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