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

Abstract

We study a mixed tensor product 3⊗m⊗3‾⊗n\mathbf{3}^{\otimes m} \otimes \mathbf{\overline{3}}^{\otimes n} of the three-dimensional fundamental representations of the Hopf algebra Uqsℓ(2∣1)U_{q} s\ell(2|1), whenever qq is not a root of unity. Formulas for the decomposition of tensor products of any simple and projective Uqsℓ(2∣1)U_{q} s\ell(2|1)-module with the generating modules 3\mathbf{3} and 3‾\mathbf{\overline{3}} are obtained. The centralizer of Uqsℓ(2∣1)U_{q} s\ell(2|1) on the chain is calculated. It is shown to be the quotient Xm,n\mathscr{X}_{m,n} of the quantum walled Brauer algebra. The structure of projective modules over Xm,n\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 Xm,n\mathscr{X}_{m,n}. This result forms a crucial step in decomposition of the mixed tensor product as a bimodule over Xm,n⊠Uqsℓ(2∣1)\mathscr{X}_{m,n}\boxtimes U_{q} s\ell(2|1). We give an explicit bimodule structure for all m,nm,n.Comment: 43 pages, 5 figure

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