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flexible algebra
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(Definition)
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A non-associative algebra  is flexible if
![$ [\ a,b,a\ ]=0$](http://images.planetmath.org:8080/cache/objects/6351/l2h/img2.png "$ [\ a,b,a\ ]=0$") for all  , where ![$ [\ , , ]$](http://images.planetmath.org:8080/cache/objects/6351/l2h/img4.png "$ [\ , , ]$") is the associator on  . In other words, we have
 for all  . Any associative algebra is clearly flexible. Furthermore, any alternative algebra with characteristic  is flexible.
Given an element  in a flexible algebra  , define the left power of  iteratively as follows:
 ,-
 .
Similarly, we can define the right power of  as:
 ,-
 .
Then, we can show that
 for all positive integers  . As a result, in a flexible algebra, one can define the (multiplicative) power of an element  as  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 11 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 2456 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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