Every Akivis Algebra is Linear

Abstract

An Akivis algebra is a vector space V endowed with a skew-symmetric bilinear product [x,y] and a trilinear product A(x,y,z) that satisfy the identity

$$\begin{gathered} [[x,y],z] + [[y,z],x] + [[z,x],y] \hfill \\ = {\mathcal{A}}(x,y,z) + {\mathcal{A}}(y,z,x) + {\mathcal{A}}(z,x,y) - {\mathcal{A}}(y,x,z) - {\mathcal{A}}(x,z,y) - {\mathcal{A}}(z,y,x). \hfill \\ \end{gathered}$$

These algebras were introduced in 1976 by M.A. Akivis as local algebras of three-webs. For any (nonassociative) algebra B one may obtain an Akivis algebra Ak (B) by considering in B the usual commutator [x,y] = xy − yx and associator A(x,y,z) = (xy)z − x(yz). Akivis posed the problem whether every Akivis algebra is isomorphic to a subalgebra of Ak (B) for a certain B. We prove that this problem has a positive answer.