Alternative Algebra - The Associator

The Associator

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 so-called 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. The converse holds so long as the characteristic of the base field is not 2.