On Scalar Dependent Algebras

Abstract

The intent of this paper is to study a class of algebras which do not necessarily obey the association law but instead obey a law which bears a marked resemblance to associativity. For lack of a better name we call this class the class of scalar dependent algebras. Specifically, an algebra A over a field F is called scalar dependent if there is a map g: A × A × A → F such that (xy)z = g(x, y, z)x(yz), for all x, y, z in A. To obtain our results we shall assume throughout that A is a scalar dependent algebra with an identity element e over a field of characteristic not 2 satisfying(I) (x, x, x) = 0.As usual, the associator (x,y,z) is defined by (x,y,z) = (xy)z — x(yz). An example is given to show that (I) is not implied by scalar dependency.