Global InformationStructurable algebra information
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 is an involution. If we define , and , then we say A is a structurable algebra if:[2]
![{\displaystyle [x,y]=xy-yx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42b4220c8122ebd2a21c517ca80639581679cfa6)
![{\displaystyle [V_{x,y},V_{z,w}]=V_{V_{x,y}z,w}-V_{z,V_{y,x}w}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4d1641c880d6bb97c39a9966640b9bf81669dc99)
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]
- ^ a b R.D. Schafer (1985). "On Structurable algebras". Journal of Algebra. Vol. 92. pp. 400–412.
- ^ Skip Garibaldi (2001). "Structurable Algebras and Groups of Type E_6 and E_7". Journal of Algebra. Vol. 236. pp. 651–691.
- ^ Garibaldi, p.658
- ^ R. B. Brown (1963). "A new type of nonassociative algebra". Vol. 50. Proc. Natl. Acad. Sci. U.S. A. pp. 947–949. JSTOR 71948.
- ^ Garibaldi, p.660