Editing Alternative algebra (section)

==The associator==

Alternative algebras are so named because they are the algebras for which the [[associator]] is [[alternating form|alternating]]. The associator is a [[trilinear map]] given by
:<math>[x,y,z] = (xy)z - x(yz)</math>.
By definition, a [[multilinear map]] is alternating if it [[Vanish_(mathematics)|vanishes]] whenever two of its arguments are equal. The left and right alternative identities for an algebra are equivalent to<ref name=Sch27>Schafer (1995) p. 27</ref>
:<math>[x,x,y] = 0</math>
:<math>[y,x,x] = 0</math>
Both of these identities together imply that:
:<math>[x,y,x]=[x,x,x]+[x,y,x]+</math>
:<math>-[x,x+y,x+y] =</math>
:<math>= [x,x+y,-y] =</math>
:<math>= [x,x,-y] - [x,y,y] = 0</math>
for all <math>x</math> and <math>y</math>. This is equivalent to the ''[[flexible identity]]''<ref name=Sch28>Schafer (1995) p. 28</ref>
:<math>(xy)x = x(yx).</math>
The associator of an alternative algebra is therefore alternating. [[Converse (logic)|Conversely]], any algebra whose associator is alternating is clearly alternative. By symmetry, any algebra which satisfies any two of:
*left alternative identity: <math>x(xy) = (xx)y</math>
*right alternative identity: <math>(yx)x = y(xx)</math>
*flexible identity: <math>(xy)x = x(yx).</math>
is alternative and therefore satisfies all three identities.

An alternating associator is always totally skew-symmetric. That is,
:<math>[x_{\sigma(1)}, x_{\sigma(2)}, x_{\sigma(3)}] = \sgn(\sigma)[x_1,x_2,x_3]</math>
for any [[permutation]] <math>\sigma</math>. The converse holds so long as the [[characteristic (algebra)|characteristic]] of the base [[field (mathematics)|field]] is not 2.