Multipliers and Unicentral Triassociative Algebras

TL;DR

This paper extends the concept of Schur multipliers to triassociative algebras, providing new theoretical tools and criteria for understanding their centers, covers, and cohomological properties.

Contribution

It introduces an analogue of the Schur multiplier for triassociative algebras and develops criteria for centers of covers, along with characterizations and cohomological sequences.

Findings

Established criteria for center mappings in covers

Proved the uniqueness of the cover

Derived exact cohomological sequences

Abstract

We introduce an analogue of the famous Schur multiplier in the context of associative trialgebras, or triassociative algebras. The latter were first studied by Loday and Ronco in 2001, and are characterized by three operations and eleven relations. The paper highlights an extension-theoretic crossroads of multipliers, covers, and unicentral triassociative algebras. Using theory from previous algebraic contexts as a guide, we develop criteria for when the center of the cover maps onto the center of the algebra. Along the way, we obtain the uniqueness of the cover, two different characterizations of the multiplier, and several exact cohomological sequences.