# Bimodule structure of the mixed tensor product over $U_{q} s\ell(2|1)$ and quantum walled Brauer algebra

## TL;DR

This paper analyzes the bimodule structure of mixed tensor products over the quantum superalgebra U_q sl(2|1) and relates it to the quantum walled Brauer algebra, providing explicit decompositions and module structures.

## Contribution

It introduces explicit formulas for tensor product decompositions and describes the bimodule structure over the quantum walled Brauer algebra and U_q sl(2|1).

## Key findings

- Decomposition formulas for tensor products with simple and projective modules.
- Identification of the centralizer as a quotient of the quantum walled Brauer algebra.
- Explicit structure of projective modules over the algebra.

## Abstract

We study a mixed tensor product $\mathbf{3}^{\otimes m} \otimes \mathbf{\overline{3}}^{\otimes n}$ of the three-dimensional fundamental representations of the Hopf algebra $U_{q} s\ell(2|1)$, whenever $q$ is not a root of unity. Formulas for the decomposition of tensor products of any simple and projective $U_{q} s\ell(2|1)$-module with the generating modules $\mathbf{3}$ and $\mathbf{\overline{3}}$ are obtained. The centralizer of $U_{q} s\ell(2|1)$ on the chain is calculated. It is shown to be the quotient $\mathscr{X}_{m,n}$ of the quantum walled Brauer algebra. The structure of projective modules over $\mathscr{X}_{m,n}$ is written down explicitly. It is known that the walled Brauer algebras form an infinite tower. We have calculated the corresponding restriction functors on simple and projective modules over $\mathscr{X}_{m,n}$. This result forms a crucial step in decomposition of the mixed tensor product as a bimodule over $\mathscr{X}_{m,n}\boxtimes U_{q} s\ell(2|1)$. We give an explicit bimodule structure for all $m,n$.