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power-associative algebra
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(Definition)
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Let $A$ be a non-associative algebra. A subalgebra $B$ of $A$ is said to be cyclic if it is generated by one element.
A non-associative algebra is power-associative if, $[B,B,B]=0$ for any cyclic subalgebra $B$ of $A$ , where $[-,-,-]$ is the associator.
If we inductively define the powers of an element $a\in A$ by
- (when $A$ is unital with $1\neq0$ ) $a^0:=1$ ,
- $a^1:=a$ , and
- $a^n:=a(a^{n-1})$ 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.
- 1
- R. D. Schafer, An Introduction on Nonassociative Algebras, Dover, New York (1995).
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"power-associative algebra" is owned by CWoo.
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See Also: associator
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di-associative, diassociative |
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Cross-references: division rings, Wedderburn's theorem, field, characteristic, division algebra, finite, theorem, addition, implies, coordinates, integers, unital, powers, associator, generated by, cyclic, non-associative algebra
There is 1 reference to this entry.
This is version 12 of power-associative algebra, born on 2004-10-10, modified 2008-12-11.
Object id is 6350, canonical name is PowerAssociativeAlgebra.
Accessed 2415 times total.
Classification:
| AMS MSC: | 17A05 (Nonassociative rings and algebras :: General nonassociative rings :: Power-associative rings) |
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Pending Errata and Addenda
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