Structurable algebra

In abstract algebra, a structurable algebra is a certain kind of unital involutive non-associative algebra over a field. For example, all Jordan algebras are structurable algebras (with the trivial involution), as is any alternative algebra with involution, or any central simple algebra with involution. An involution here means a linear anti-homomorphism whose square is the identity.^[1]

Assume A is a unital non-associative algebra over a field, and ${\displaystyle x\mapsto {\bar {x}}}$ is an involution. If we define ${\displaystyle V_{x,y}z:=(x{\bar {y}})z+(z{\bar {y}})x-(z{\bar {x}})y}$, and ${\displaystyle [x,y]=xy-yx}$, then we say A is a structurable algebra if:^[2]

${\displaystyle [V_{x,y},V_{z,w}]=V_{V_{x,y}z,w}-V_{z,V_{y,x}w}.}$

Structurable algebras were introduced by Allison in 1978.^[3] The Kantor–Koecher–Tits construction produces a Lie algebra from any Jordan algebra, and this construction can be generalized so that a Lie algebra can be produced from an structurable algebra. Moreover, Allison proved over fields of characteristic zero that a structurable algebra is central simple if and only if the corresponding Lie algebra is central simple.^[1]

Another example of a structurable algebra is a 56-dimensional non-associative algebra originally studied by Brown in 1963, which can be constructed out of an Albert algebra.^[4] When the base field is algebraically closed over characteristic not 2 or 3, the automorphism group of such an algebra has identity component equal to the simply connected exceptional algebraic group of type E₆.^[5]

References

1. R.D. Schafer (1985). "On Structurable algebras". Journal of Algebra. Vol. 92. pp. 400–412.
2. Skip Garibaldi (2001). "Structurable Algebras and Groups of Type E_6 and E_7". Journal of Algebra. Vol. 236. pp. 651–691.
3. Garibaldi, p.658
4. R. B. Brown (1963). "A new type of nonassociative algebra". Vol. 50. Proc. Natl. Acad. Sci. U.S. A. pp. 947–949. JSTOR 71948.
5. Garibaldi, p.660