Multipliers and Unicentral Diassociative Algebras

TL;DR

This paper extends Lie-theoretic concepts to diassociative algebras, establishing foundational properties like uniqueness of covers, characterizing multipliers via cohomology, and exploring centers and extensions.

Contribution

It introduces diassociative analogues of key Lie algebra results, including criteria for centers and a theory of unicentral diassociative algebras.

Findings

Covers of diassociative algebras are unique.

Multiplier characterized by second cohomology group.

Criteria for center mappings in covers.

Abstract

The objective of this paper is to develop diassociative analogues of Lie-theoretic results from Peggy Batten's 1993 dissertation. We first prove that covers of diassociative algebras are unique. Second, we show that the multiplier of a diassociative algebra is characterized by the second cohomology group with coefficients in the field. Third, we establish criteria for when the center of a cover maps onto the center of the algebra. Along the way, we obtain a collection of exact sequences, characterizations, and a brief theory of unicentral diassociative algebras and stem extensions. This paper is part of an ongoing project to advance extension theory in the context of several Loday algebras.