Alternative algebra

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. The sedenions, on the other hand, are not alternative.

Alternative algebras are so named because they are precisely 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 the associator is totally skew-symmetric. That is,

for any permutation σ. It follows that

for all x and y. 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:

is alternative and therefore satisfies all three identities.

An alternating associator is always totally skew-symmetric. The converse holds so long as the characteristic of the base field is not 2.

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 ${\displaystyle x,y,z}$ in an alternative algebra associate (i.e., ${\displaystyle [x,y,z]=0}$), the subalgebra generated by those elements is associative.

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