A non-associative algebra is alternative if
(left alternative laws) , and
(right alternative laws) ,
for any , where is the associator on .
Let be alternative and suppose . From the fact that , we can deduce that the associator is anti-commutative, when one of the three coordinates is held fixed. That is, for any ,
Put more succinctly,
where , the symmetric group on three letters, and is the sign (http://planetmath.org/SignatureOfAPermutation) of .
Alternativity can be defined for a general ring : it is a non-associative ring such that for any , and . Equivalently, an alternative ring is an alternative algebra over .
|Date of creation||2013-03-22 14:43:24|
|Last modified on||2013-03-22 14:43:24|
|Last modified by||CWoo (3771)|
|Defines||Artin’s theorem on alternative algebras|
|Defines||left alternative law|
|Defines||right alternative law|