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 3 3 bm b3 3 3 bn of the three-dimensional fundamental representations of the Hopf algebra U q s ℓp2|1q, whenever q is not a root of unity. Formulas for the decomposition of tensor products of any simple and projective U q s ℓp2|1q-module with the generating modules 3 3 3 and 3 3 3 are obtained. The centralizer of U q s ℓp2|1q on the mixed tensor product is calculated. It is shown to be the quotient X m,n of the quantum walled Brauer algebra qwB m,n . The structure of projective modules over 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 X m,n . This result forms a crucial step in decomposition of the mixed tensor product as a bimodule over X m,n b U q s ℓp2|1q.We give an explicit bimodule structure for all m , n.