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]