alternative algebra

A non-associative 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 anti-commutative, when one of the three coordinates is held fixed. That is, for any $a,b,c\in A$,

  1. $[a,b,c]=-[b,a,c]$

  2. $[a,b,c]=-[a,c,b]$

  3. $[a,b,c]=-[c,b,a]$

  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 $\pi$.

• 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 anti-commutative identities 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 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 non-associative 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	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