Remarks on the Second Homology Groups of Queer Lie Superalgebras

TL;DR

This paper determines the second homology group of the special queer Lie superalgebra $rak{sq}_n(R)$ by establishing an isomorphism with a special linear Lie superalgebra, advancing understanding of its algebraic structure.

Contribution

It provides a complete characterization of the second homology group of $rak{sq}_n(R)$ using an isomorphism with $rak{sl}_n(R ensor Q_1)$, a novel approach in the field.

Findings

Explicit description of the second homology group of $rak{sq}_n(R)$

Establishment of an isomorphism between $rak{sl}_n(R ensor Q_1)$ and $rak{sq}_n(R)$

Advancement in the algebraic understanding of queer Lie superalgebras

Abstract

The aim of this note is to completely determine the second homology group of the special queer Lie superalgebra $\mathfrak{sq}_n(R)$ coordinatized by a unital associative superalgebra $R$, which will be achieved via an isomorphism between the special linear Lie superalgebra $\mathfrak{sl}_{n}(R\otimes Q_1)$ and the special queer Lie superalgebra $\mathfrak{sq}_n(R)$.