Associative Algebras with Small Derived Ideal

TL;DR

This paper classifies and characterizes extra special associative algebras, revealing their structure, multipliers, and capability, and explores related algebraic concepts like unicentral and diassociative algebras.

Contribution

It provides a classification of extra special associative algebras and links their structure to Leibniz algebras, including analysis of their multipliers and capability.

Findings

Classified extra special associative algebras.

Connected their structure to Leibniz algebras.

Determined their (Schur) multipliers and capability.

Abstract

The paper concerns extra special associative algebras, an analogue of the Heisenberg Lie algebra. In particular, we say that an associative algebra is extra special if its center is equal to its derived ideal and the center is 1-dimensional. In this paper, we classify extra special associative algebras by proving that their structure is equivalent to that of extra special Leibniz algebras. We then characterize their (Schur) multipliers via dimension and completely determine their capability. We connect this with the related notion of unicentral algebras and discuss the problem of classifying extra special diassociative algebras.