Multipliers and Covers of Perfect Diassociative Algebras

TL;DR

This paper explores the structure of perfect diassociative algebras, establishing their connection to universal central extensions, and introduces a special cover with properties linked to their multipliers, using spectral sequences.

Contribution

It introduces a special cover for perfect diassociative algebras and demonstrates its properties, advancing the extension theory for Loday algebras.

Findings

Perfect diassociative algebras are strongly tied to universal central extensions.

A special cover with specific properties is constructed for perfect diassociative algebras.

The cover of a perfect diassociative algebra is itself perfect with a trivial multiplier.

Abstract

The paper concerns perfect diassociative algebras and their implications to the theory of central extensions. It is first established that perfect diassociative algebras have strong ties with universal central extensions. Then, using a known characterization of the multiplier in terms of a free presentation, we obtain a special cover for perfect diassociative algebras, as well as some of its properties. The subsequent results connect and build on the previous topics. For the final theorem, we invoke an extended Hochschild-Serre type spectral sequence to show that, for a perfect diassociative algebra, its cover is perfect and has trivial multiplier. This paper is part of an ongoing project to advance extension theory in the context of several Loday algebras.