Beck modules and alternative algebras

TL;DR

This paper develops a general theory of Beck modules across various algebraic structures, focusing on alternative algebras, and introduces noncommutative partial differentiation to relate algebraic equations to universal enveloping algebras.

Contribution

It extends the theory of Beck modules to alternative algebras and introduces noncommutative partial differentiation for constructing universal enveloping algebras.

Findings

Describes Beck modules as modules over universal enveloping algebras.

Develops noncommutative partial differentiation techniques.

Determines the Poincaré polynomial for the universal enveloping algebra in the homogeneous case.

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 Poincar\'e polynomial of the universal enveloping algebra in the homogeneous case.