Beck modules and alternative algebras

Abstract

We set out the general theory of ``Beck modules'' in a variety of algebras and describe them as modules over suitable ``universal enveloping'' unital associative algebras. We develop a theory of ``noncommutative partial differentiation'' to pass from the equations of the variety to relations in a universal enveloping algebra. We pay particular attention to the case of alternative algebras, defined by a restricted associative law, and determine the Poincaré polynomial of the universal enveloping algebra in the homogeneous case.

arXiv comment21 pages. Version accepted by Communications in Algebra, with improvements following referee report