Bimodule structure of the mixed tensor product over Uqsℓ(2|1) and quantum walled Brauer algebra

We study a mixed tensor product 3⊗m⊗3‾⊗n of the three-dimensional fundamental representations of the Hopf algebra Uqsℓ(2|1), whenever q is not a root of unity. Formulas for the decomposition of tensor products of any simple and projective Uqsℓ(2|1)-module with the generating modules 3 and 3‾ are obtained. The centralizer of Uqsℓ(2|1) on the mixed tensor product is calculated. It is shown to be the quotient Xm,n of the quantum walled Brauer algebra qwBm,n. The structure of projective modules over Xm,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 Xm,n. This result forms a crucial step in decomposition of the mixed tensor product as a bimodule over Xm,n⊠Uqsℓ(2|1). We give an explicit bimodule structure for all m,n.