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flexible algebra
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(Definition)
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A non-associative algebra $A$ is flexible if $[\ a,b,a\ ]=0$ for all $a,b\in A$ where $[\ , , ]$ is the associator on $A$ In other words, we have $(ab)a=a(ba)$ for all $a,b\in A$ Any associative algebra is clearly flexible. Furthermore, any alternative algebra with
characteristic $\neq 2$ is flexible.
Given an element $a$ in a flexible algebra $A$ define the left power of $a$ iteratively as follows:
- $L^1(a)=a$
- $L^n(a)=a\cdot L^{n-1}(a)$
Similarly, we can define the right power of $a$ as:
- $R^1(a)=a$
- $R^n(a)=R^{n-1}(a)\cdot a$
Then, we can show that $L^{n}(a)=R^{n}(a)$ for all positive integers $n$ As a result, in a flexible algebra, one can define the (multiplicative) power of an element $a$ as $a^n$ unambiguously.
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"flexible algebra" is owned by CWoo.
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Cross-references: power, multiplicative, integers, positive, characteristic, alternative algebra, algebra, associative, associator, non-associative algebra
There are 10 references to this entry.
This is version 8 of flexible algebra, born on 2004-10-10, modified 2008-10-07.
Object id is 6351, canonical name is FlexibleAlgebra.
Accessed 3354 times total.
Classification:
| AMS MSC: | 17A20 (Nonassociative rings and algebras :: General nonassociative rings :: Flexible algebras) |
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Pending Errata and Addenda
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