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power-associative algebra (Definition)

A non-associative algebra is power-associative if, for any cyclic subalgebra $ B\subseteq A$, $ [B,B,B]=0$, where $ [-,-,-]$ is the associator.

If we inductively define the powers of an element $ a\in A$ by

  1. (when $ A$ is unital with $ 1\neq0$) $ a^0:=1$,
  2. $ a^1:=a$, and
  3. $ a^n:=a(a^{n-1})$ for $ n>1$,
then power-associativity of $ A$ means that $ [a^i,a^j,a^k]=0$ for any non-negative integers $ i,j$ and $ k$, since the associator is trilinear (linear in each of the three coordinates). This implies that $ a^ma^n=a^{m+n}$. In addition, $ (a^m)^n=a^{mn}$.

A theorem, due to A. Albert, states that any finite power-associative division algebra over the integers of characteristic not equal to 2, 3, or 5 is a field. This is a generalization of the Wedderburn's Theorem on finite division rings.

Bibliography

1
R. D. Schafer, An Introduction on Nonassociative Algebras, Dover, New York (1995).



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See Also: associator

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Cross-references: division rings, Wedderburn's theorem, field, characteristic, division algebra, finite, addition, implies, coordinates, integers, unital, powers, associator, cyclic, non-associative algebra
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This is version 10 of power-associative algebra, born on 2004-10-10, modified 2006-01-30.
Object id is 6350, canonical name is PowerAssociativeAlgebra.
Accessed 1652 times total.

Classification:
AMS MSC17A05 (Nonassociative rings and algebras :: General nonassociative rings :: Power-associative rings)

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