# The regular representation of $U_v(\mathfrak{gl}_{m|n})$

## TL;DR

This paper constructs a super representation of the quantum superalgebra $U_v(\mathfrak{gl}_{m|n})$ using quantum differential operators, providing an explicit basis and multiplication formulas for the algebra.

## Contribution

It introduces a new super representation of $U_v(\mathfrak{gl}_{m|n})$ via quantum differential operators and explicitly describes its basis and multiplication rules.

## Key findings

- Constructed a super representation on polynomial superalgebra
- Extended to a formal power series algebra containing the regular representation
- Provided explicit basis and multiplication formulas for $U_v(\mathfrak{gl}_{m|n})$

## Abstract

Using quantum differential operators, we construct a super representation of $U_v(\mathfrak{gl}_{m|n})$ on a certain polynomial superalgebra. We then extend the representation to its formal power series algebra which contains a $U_v(\mathfrak{gl}_{m|n})$-submodule isomorphic to the regular representation of $U_v(\mathfrak{gl}_{m|n})$. In this way, we obtain a presentation of $U_v(\mathfrak{gl}_{m|n})$ by a basis together with explicit multiplication formulas of the basis elements by generators.