alternative algebra

A non-associative algebra A is alternative if

  1. 1.

    (left alternative laws) [a,a,b]=0, and

  2. 2.

    (right alternative laws) [b,a,a]=0,

for any a,bA, where [,,] is the associatorMathworldPlanetmath on A.


  • Let A be alternative and suppose char(A)2. From the fact that [a+b,a+b,c]=0, we can deduce that the associator [,,] is anti-commutative, when one of the three coordinates is held fixed. That is, for any a,b,cA,

    1. (a)


    2. (b)


    3. (c)


    Put more succinctly,


    where πS3, the symmetric group on three letters, and sgn(π) is the sign ( of π.

  • An alternative algebra is a flexible algebra, provided that the algebraMathworldPlanetmathPlanetmathPlanetmath is not Boolean ( (characteristic ( 2). To see this, replace c in the first anti-commutative identitiesPlanetmathPlanetmathPlanetmath above with a and the result follows.

  • Artin’s Theorem: If a non-associative algebra A is not Boolean, then A is alternative iff every subalgebraMathworldPlanetmathPlanetmathPlanetmath of A generated by two elements is associative. The proof is clear from the above discussion.

  • A commutativePlanetmathPlanetmathPlanetmath alternative algebra A is a Jordan algebraMathworldPlanetmathPlanetmath. This is true since a2(ba)=a2(ab)=(ab)a2=((ab)a)a=(a(ab))a=(a2b)a shows that the Jordan identity is satisfied.

  • Alternativity can be defined for a general ring R: it is a non-associative ring such that for any a,bR, (aa)b=a(ab) and (ab)b=a(bb). Equivalently, an alternative ring is an alternative algebra over .

Title alternative algebra
Canonical name AlternativeAlgebra
Date of creation 2013-03-22 14:43:24
Last modified on 2013-03-22 14:43:24
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 11
Author CWoo (3771)
Entry type Definition
Classification msc 17D05
Related topic Associator
Related topic FlexibleAlgebra
Defines Artin’s theorem on alternative algebras
Defines alternative ring
Defines left alternative law
Defines right alternative law