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.Springer & Veldkamp (2000) 5.8, p.153 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

:$x \circ y = \frac12 (x \cdot y + y \cdot x),$

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.Springer & Veldkamp (2000) 7.2 (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'').Knus et al (1998) p.517Garibaldi, Petersson, Racine (2024), pp. 599, 600

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.

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, ''f''₃, ''f''₅, of degrees 0, 3, 5.Garibaldi, Merkurjev, Serre (2003), p.50 The cohomological invariants with 3-torsion coefficients have a basis 1, ''g''₃ of degrees 0, 3.Garibaldi (2009), p.20 The invariants ''f''₃ and ''g''₃ 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

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*{{Citation , last1=Springer , first1=Tonny A. , author1-link=T. A. Springer , last2=Veldkamp , first2=Ferdinand D. , title=Octonions, Jordan algebras and exceptional groups , orig-year=1963 , url=https://books.google.com/books?isbn=3540663371 , publisher=Springer-Verlag , location=Berlin, New York , series=Springer Monographs in Mathematics , isbn=978-3-540-66337-9 , year=2000 , mr=1763974

Further reading

''Albert algebra''
at Encyclopedia of Mathematics.

Non-associative algebras