Algebra with associative powers
A linear algebra over a field F each element of which generates an associative subalgebra. The set of all algebras with associative powers over a given field F forms a variety of algebras which, if the characteristic of the field F is zero, is defined by the system of identities \label{1} (x,x,x) = (x^2,x,x) = 0 where the associator (a,b,c) = (ab)c - a(bc). If F is an infinite field of prime characteristic p, then the variety of algebras with associative powers cannot be defined by any finite system of identities, but an independent, infinite system of identities which defines it is known [3]. If a commutative algebra A with associative powers of characteristic other than 2 has an idempotent e \neq 0, then A can be decomposed according to Peirce into a direct sum of vector subspaces: \label{2} A = A_0(e) \oplus A_{\frac12}(e) \oplus A_1(e) where A_\lambda(e) = \{ a \in A : ea = \lambda a \}, \lambda = 0,\frac12,1. Here A_0(e) and A_1(e) are subalgebras, A_0(e) A_1(e) = 0, A_{\frac12}(e)A_{\frac12}(e) \subseteq A_0(e) + A_1(e), A_\lambda(e) A_{\frac12}(e) \subseteq A_{\frac12}(e) + A_{1-\lambda}(e) for \lambda = 0,1. The Pierce decomposition (2) plays a fundamental role in the structure theory of algebras with associative powers.
References
[1] | A.A. Albert, "Power-associative rings" Trans. Amer. Math. Soc. , 64 (1948) pp. 552–593 |
[2] | A.T. Gainov, "Identity relations for binary Lie rings" Uspekhi Mat. Nauk , 12 : 3 (1957) pp. 141–146 (In Russian) |
[3] | A.T. Gainov, "Power-associative algebras over a finite-characteristic field" Algebra and Logic , 9 : 1 (1970) pp. 5–19 Algebra i Logika , 9 : 1 (1970) pp. 9–33 |
Comments
An algebra with associative powers is also called a power-associative algebra. The fact that the set of algebras with associative powers over a field of non-zero characteristic forms a variety defined by (1) (x,x,x) = (x^2,x,x) = 0 was proved in [a1].
References
[a1] | A.A. Albert, "On the power associativity of rings" Summa Brasiliensis Math. , 2 (1948) pp. 21–33 |
Algebra with associative powers. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Algebra_with_associative_powers&oldid=16404