power-associative algebra

Let A be a non-associative algebra. A subalgebraMathworldPlanetmathPlanetmathPlanetmathPlanetmath B of A is said to be cyclic if it is generated by one element.

A non-associative algebra is power-associative if, [B,B,B]=0 for any cyclic subalgebra B of A, where [-,-,-] is the associatorMathworldPlanetmath.

If we inductively define the powers of an element aA by

  1. 1.

    (when A is unital with 10) a0:=1,

  2. 2.

    a1:=a, and

  3. 3.

    an:=a(an-1) for n>1,

then power-associativity of A means that [ai,aj,ak]=0 for any non-negative integers i,j and k, since the associator is trilinear (linear in each of the three coordinates). This implies that aman=am+n. In addition, (am)n=amn.

A theorem, due to A. Albert, states that any finite power-associative division algebraMathworldPlanetmath over the integers of characteristic not equal to 2, 3, or 5 is a field. This is a generalizationPlanetmathPlanetmath of the Wedderburn’s Theorem on finite division rings.


Title power-associative algebra
Canonical name PowerassociativeAlgebra
Date of creation 2013-03-22 14:43:27
Last modified on 2013-03-22 14:43:27
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 15
Author CWoo (3771)
Entry type Definition
Classification msc 17A05
Synonym di-associative
Synonym diassociative
Related topic Associator