Associator-cyclic bimodules (Q1905308)

This paper is concerned with the representation theory of associator-cyclic algebras. An algebra \(A\) is said to be associator-cyclic if the following identity holds: \((x,y,z) = (y,z,x)\), where \((x,y,z) = (xy)z-x(yz)\). Any alternative algebra is associator-cyclic. Moreover, it was proved by the author in [Algebra Logic 32, No. 3, 133-142 (1993; Zbl 0823.17046)] that any associator-cyclic algebra which has no non-zero nilpotent ideal is alternative. As usual an \(A\)-bimodule \(M\) is called associator-cyclic provided the split null extension \(A\oplus M\) is such. In this paper the author proves that an associator-cyclic bimodule \(M\) over an associator-cyclic algebra \(A\) is alternative in the following two cases: 1. \(M\) is irreducible, \(A\) is a composition algebra of characteristic not 2; 2. \(M\) is faithful irreducible, the characteristic of \(A\) is neither 2 nor 3.