Multipliers of Nilpotent Diassociative Algebras

TL;DR

This paper extends cohomological sequences for nilpotent diassociative algebras, providing new bounds and computations for their multipliers, and draws parallels with associative and Lie algebra cases.

Contribution

It introduces new cohomological extensions and dimension bounds for nilpotent diassociative algebra multipliers, advancing extension theory in Loday algebra contexts.

Findings

Extended five-term cohomological sequence for nilpotent diassociative algebras.

Derived new dimension bounds for the multipliers.

Computed multipliers for specific associative and diassociative algebras.

Abstract

The paper concerns nilpotent associative dialgebras and their corresponding diassociative Schur multipliers. Using Lie (and group) theory as a guide, we first extend a classic five-term cohomological sequence under alternative conditions in the nilpotent setting. This main result is then applied to obtain a new proof for a previous extension of the same sequence. It also yields a different extension of the sequence that involves terms in the upper central series. Furthermore, we use the main result to obtain a collection of dimension bounds on the multiplier of a nilpotent diassociative algebra. These differ notably from the Lie case. Since diassociative algebras generalize associative algebras, we obtain an associative analogue of the results herein. We conclude by computing both the associative and diassociative multipliers of an associative algebra. This paper is part of an ongoing project to advance extension theory in the context of several Loday algebras.