Representations of Alternative Algebras
R. D. Schafer · Transactions of the American Mathematical Society · 1952
of the identities (1) yields the fact that the associator (2) (x, y, z) = (xy)z -x(yz)"alternates" : it changes sign under an odd permutation of the letters x, y, z, but remains unchanged under an even permutation.The most economical statement of this fact isfor all x, y, z in 31.If the characteristic of F is not two, then the identities (3) define an alternative algebra.Let 23 be a vector space over F. Following Eilenberg [3, §2](2), we define a representation of 31 as a pair of linear mappings x->SX, x-+Tx of 31 into the algebra of all linear transformations on 23, satisfying \^) [-Í u *JzJ = wjí O :cOz = ■*■ zx ' J-z-l z ~ l/J i, L z\ for all x, z in 31, where [X, Y] denotes the commutator XY-YX.We write (S, T) for the representation.The representation space 23 in which (5, T) acts is made into an alternative module by defining vx = vSx, xv -vTx for v in 23, x in 31.Assumption (4) becomes (5) (x, v, z) = -(v, x, z) = (z, x, v) = -(z, v, x)