# 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 $a,b\in A$, where $[\ ,,]$ is the associator on $A$.

Remarks

• Let $A$ be alternative and suppose $\operatorname{char}(A)\neq 2$. 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)

$[\ a,b,c\ ]=-[\ b,a,c\ ]$

2. (b)

$[\ a,b,c\ ]=-[\ a,c,b\ ]$

3. (c)

$[\ a,b,c\ ]=-[\ c,b,a\ ]$

Put more succinctly,

 $[\ a_{1},a_{2},a_{3}\ ]=\operatorname{sgn}(\pi)[\ a_{\pi(1)},a_{\pi(2)},a_{\pi% (3)}\ ],$

where $\pi\in S_{3}$, the symmetric group on three letters, and $\operatorname{sgn}(\pi)$ 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) $\neq 2$). 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