Associating Geometry to the Lie Superalgebra $\mathfrak{sl}(1|1)$ and to the Color Lie Algebra $\mathfrak{sl}^c_2(\Bbbk)$

TL;DR

This paper extends the geometric correspondence between modules and Lie algebra structures to the superalgebra rak{sl}(1|1) and a related color Lie algebra, enriching the understanding of their representation theory.

Contribution

It establishes new geometric correspondences for modules over the homogenized universal enveloping algebras of rak{sl}(1|1) and a color Lie algebra linked to rak{sl}_2, analogous to known results for rak{sl}_2.

Findings

Established line module correspondences for rak{sl}(1|1)

Extended geometric module descriptions to a color Lie algebra

Provided new insights into the representation theory of super and color Lie algebras

Abstract

In the 1990s, in work of Le Bruyn and Smith and in work of Le Bruyn and Van den Bergh, it was proved that point modules and line modules over the homogenization of the universal enveloping algebra of a finite-dimensional Lie algebra describe useful data associated to the Lie algebra. In particular, in the case of the Lie algebra $\mathfrak{sl}_2(\mathbb{C})$, there is a correspondence between Verma modules and certain line modules that associates a pair $(\mathfrak{h},\,\phi)$, where $\mathfrak{h}$ is a two-dimensional Lie subalgebra of $\mathfrak{sl}_2(\mathbb{C})$ and $\phi \in \mathfrak{h}^*$ satisfies $\phi([\mathfrak{h}, \, \mathfrak{h}]) = 0$, to a particular type of line module. In this article, we prove analogous results for the Lie superalgebra $\mathfrak{sl}(1|1)$ and for a color Lie algebra associated to the Lie algebra $\mathfrak{sl}_2$.