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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
*x(xy) = (xx)y
*(yx)x = y(xx)
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 associator
Alternative algebras are so named because they are the algebras for which the associator is alternating. The associator is a trilinear map given by
:[x,y,z] = (xy)z - x(yz).
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
:[x,x,y] = 0
:[y,x,x] = 0.
Both of these identities together imply that
:[x,y,x] = [x, x, x] + [x, y, x] - [x, x+y, x+y] = [x, x+y, -y] = [x, x, -y] - [x, y, y] = 0
for all x and y

Official source

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