Universal central extensions of superdialgebras of matrices

TL;DR

This paper determines the universal central extensions of Leibniz superalgebras of matrices over superdialgebras, including associative, superalgebra, and dialgebra cases, using a novel approach involving non-abelian tensor squares.

Contribution

It introduces a new method using non-abelian tensor squares to find universal central extensions of Leibniz superalgebras of matrices over superdialgebras, completing previous open cases.

Findings

Unified description of universal central extensions for various superdialgebras.

Application of non-abelian tensor squares in Leibniz superalgebra theory.

Resolution of the extension problem for matrix superalgebras over superdialgebras.

Abstract

We complete the problem of finding the universal central extension in the category of Leibniz superalgebras of $\mathfrak{sl}(m, n, D)$ when $m+n \geq 3$ and $D$ is a superdialgebra, solving in particular the problem when $D$ is an associative algebra, superalgebra or dialgebra. To accomplish this task we use a different method than the standard studied in the literature. We introduce and use the non-abelian tensor square of Leibniz superalgebras and its relations with the universal central extension.