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

Abstract

We study a mixed tensor product $3^{\otimes m}\otimes {\overline{3}}^{\otimes n}$ 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 $3$ and $\overline{3}$ are obtained. The centralizer of $U_{q}s\ell (2|1)$ on the chain is calculated. It is shown to be the quotient ${\mathcal{X}}_{m,n}$ of the quantum walled Brauer algebra. The structure of projective modules over ${\mathcal{X}}_{m,n}$ 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 ${\mathcal{X}}_{m,n}$. This result forms a crucial step in decomposition of the mixed tensor product as a bimodule over ${\mathcal{X}}_{m,n}⊠U_{q}s\ell (2|1)$. We give an explicit bimodule structure for all $m,n$.

43 pages, 5 figures