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alternative algebra
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(Definition)
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A non-associative algebra $A$ is alternative if
- $[\ a,a,b\ ]=0$ , and
- $[\ b,a,a\ ]=0$ ,
for any $a,b\in A$ , where $[\ , , ]$ is the associator on $A$ .
Remarks
- Let $A$ be alternative and suppose $\operatorname{char}(A)\neq2$ . 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 $a,b,c\in A$ ,
- $[\ a,b,c\ ]=-[\ b,a,c\ ]$
- $[\ a,b,c\ ]=-[\ a,c,b\ ]$
- $[\ a,b,c\ ]=-[\ c,b,a\ ]$
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 of $\pi$ .
- An alternative algebra is a flexible algebra, provided that the algebra is not Boolean (characteristic $\neq2$ ). 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 $a^2(ba)=a^2(ab)=(ab)a^2=((ab)a)a=(a(ab))a=(a^2b)a$ shows that the Jordan identity is satisfied.
- Alternativity can be defined for a general ring $R$ : it is a ring such that for any $a,b\in R$ , $(aa)b=a(ab)$ and $(ab)b=a(bb)$ .
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"alternative algebra" is owned by CWoo.
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Cross-references: ring, Jordan identity, Jordan algebra, commutative, clear, proof, associative, generated by, iff, Boolean, theorem, identities, algebra, flexible algebra, symmetric group on three letters, fixed, coordinates, associator, non-associative algebra
There are 6 references to this entry.
This is version 6 of alternative algebra, born on 2004-10-10, modified 2006-02-24.
Object id is 6349, canonical name is AlternativeAlgebra.
Accessed 6910 times total.
Classification:
| AMS MSC: | 17D05 (Nonassociative rings and algebras :: Other nonassociative rings and algebras :: Alternative rings) |
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