Graded representations of current Lie superalgebras $\mathfrak{sl}(1|2)[t]$

TL;DR

This paper investigates finite-dimensional graded representations of the current Lie superalgebra sl(1|2)[t], introducing super POPs for basis parametrization, deriving character formulas, and establishing module isomorphisms.

Contribution

It introduces super POPs as a new combinatorial tool, derives graded character formulas, and proves isomorphisms between Chari-Venkatesh modules and fusion of Kac modules.

Findings

Derived graded character formula for local Weyl modules.

Constructed short exact sequences of Chari-Venkatesh modules.

Proved isomorphism between Chari-Venkatesh modules and fusion of Kac modules.

Abstract

This paper is the study of finite-dimensional graded representations of current lie superalgebras $\mathfrak{sl}(1|2)[t]$. We define the notion of super POPs, a combinatorial tool to provide another parametrization of the basis of the local Weyl module given in [2]. We derive the graded character formula of local Weyl module for $\mathfrak{sl}(1|2)[t]$. Furthermore, we construct a short exact sequence of Chari-Venkatesh modules for $\mathfrak{sl}(1|2)[t]$. As a consequence, we prove that Chari-Venkatesh modules are isomorphic to the fusion of generalized Kac modules.