alternative algebra
A nonassociative algebra $A$ is alternative if

1.
(left alternative laws) $[a,a,b]=0$, and

2.
(right alternative laws) $[b,a,a]=0$,
for any $a,b\in A$, where $[,,]$ is the associator^{} on $A$.
Remarks

•
Let $A$ be alternative and suppose $\mathrm{char}(A)\ne 2$. From the fact that $[a+b,a+b,c]=0$, we can deduce that the associator $[,,]$ is anticommutative, when one of the three coordinates is held fixed. That is, for any $a,b,c\in A$,

(a)
$[a,b,c]=[b,a,c]$

(b)
$[a,b,c]=[a,c,b]$

(c)
$[a,b,c]=[c,b,a]$
Put more succinctly,
$$[{a}_{1},{a}_{2},{a}_{3}]=\mathrm{sgn}(\pi )[{a}_{\pi (1)},{a}_{\pi (2)},{a}_{\pi (3)}],$$ where $\pi \in {S}_{3}$, the symmetric group on three letters, and $\mathrm{sgn}(\pi )$ is the sign (http://planetmath.org/SignatureOfAPermutation) of $\pi $.

(a)

•
An alternative algebra is a flexible algebra, provided that the algebra^{} is not Boolean (http://planetmath.org/BooleanLattice) (characteristic (http://planetmath.org/Characteristic) $\ne 2$). To see this, replace $c$ in the first anticommutative identities^{} above with $a$ and the result follows.

•
Artin’s Theorem: If a nonassociative algebra $A$ is not Boolean, then $A$ is alternative iff every subalgebra^{} of $A$ generated by two elements is associative. The proof is clear from the above discussion.

•
A commutative^{} alternative algebra $A$ is a Jordan algebra^{}. This is true since ${a}^{2}(ba)={a}^{2}(ab)=(ab){a}^{2}=((ab)a)a=(a(ab))a=({a}^{2}b)a$ shows that the Jordan identity is satisfied.

•
Alternativity can be defined for a general ring $R$: it is a nonassociative ring such that for any $a,b\in R$, $(aa)b=a(ab)$ and $(ab)b=a(bb)$. Equivalently, an alternative ring is an alternative algebra over $\mathbb{Z}$.
Title  alternative algebra 
Canonical name  AlternativeAlgebra 
Date of creation  20130322 14:43:24 
Last modified on  20130322 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 