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Concept
Alternative algebra
Summary
In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have
for all x and y in the algebra.
Every associative algebra is obviously alternative, but so too are some strictly non-associative algebras such as the octonions.
Alternative algebras are so named because they are the algebras for which the associator is alternating. The associator is a trilinear map given by
By definition, a multilinear map is alternating if it vanishes whenever two of its arguments are equal. The left and right alternative identities for an algebra are equivalent to
Both of these identities together imply that
for all and . This is equivalent to the flexible identity
The associator of an alternative algebra is therefore alternating. Conversely, any algebra whose associator is alternating is clearly alternative. By symmetry, any algebra which satisfies any two of:
left alternative identity:
right alternative identity:
flexible identity:
is alternative and therefore satisfies all three identities.
An alternating associator is always totally skew-symmetric. That is,
for any permutation . The converse holds so long as the characteristic of the base field is not 2.
Every associative algebra is alternative.
The octonions form a non-associative alternative algebra, a normed division algebra of dimension 8 over the real numbers.
More generally, any octonion algebra is alternative.
The sedenions and all higher Cayley–Dickson algebras lose alternativity.
Artin's theorem states that in an alternative algebra the subalgebra generated by any two elements is associative. Conversely, any algebra for which this is true is clearly alternative. It follows that expressions involving only two variables can be written unambiguously without parentheses in an alternative algebra. A generalization of Artin's theorem states that whenever three elements in an alternative algebra associate (i.e., ), the subalgebra generated by those elements is associative.
Official source
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