The $Z_2 \times Z_2$-graded general linear Lie superalgebra

TL;DR

This paper introduces a new realization of the $Z_2 imes Z_2$-graded Lie superalgebra $rak{gl}(m_1,m_2|n_1,n_2)$ within an extension of the enveloping algebra of the classical $Z_2$-graded Lie superalgebra $rak{gl}(m|n)$, enabling the lifting of representations.

Contribution

It provides a novel algebraic realization of the $Z_2 imes Z_2$-graded Lie superalgebra and characterizes how representations of $rak{gl}(m|n)$ extend to this new structure.

Findings

Representations of $rak{gl}(m|n)$ lift to $rak{gl}(m_1,m_2|n_1,n_2)$ with sign differences.

The realization is achieved inside an algebraic extension of the enveloping algebra.

Matrix elements of lifted representations differ only by a sign, characterized concisely.

Abstract

We present a novel realisation of the $\mathbb{Z}_2\times\mathbb{Z}_2$-graded Lie superalgebra $\mathfrak{gl}(m_1,m_2|n_1,n_2)$ inside an algebraic extension of the enveloping algebra of the $\mathbb{Z}_2$-graded Lie superalgebra $\mathfrak{gl}(m|n)$, with $m=m_1+m_2$ and $n=n_1+n_2$. A consequence of this realisation is that the representations of $\mathfrak{gl}(m|n)$ "lift up" to representations of $\mathfrak{gl}(m_1,m_2|n_1,n_2)$, with matrix elements differing only by a sign, which we are able to characterise concisely.