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The quantized walled Brauer algebra and mixed tensor space. (English) Zbl 1368.17017
Summary: In this paper we investigate a multi-parameter deformation $$\mathfrak B_{r,s}^n(a,\lambda,\delta)$$ of the walled Brauer algebra which was previously introduced by R. Leduc [“A two-parameter version of the centralizer algebra of the mixed tensor representation of the general linear group and quantum general linear group.” Thesis, University of Wisconsin-Madison (1994)]. We construct an integral basis of $$\mathfrak B_{r,s}^n(a,\lambda,\delta)$$ consisting of oriented tangles which is in bijection with walled Brauer diagrams. Moreover, we study a natural action of $$\mathfrak B_{r,s}^n(q)=\mathfrak B_{r,s}^n(q^{-1}-q,q^n,[n]_q)$$ on mixed tensor space and prove that the kernel is free over the ground ring $$R$$ of rank independent of $$R$$. As an application, we prove one side of Schur-Weyl duality for mixed tensor space: the image of $$\mathfrak B_{r,s}^n(q)$$ in the $$R$$-endomorphism ring of mixed tensor space is, for all choices of $$R$$ and the parameter $$q$$, the endomorphism algebra of the action of the (specialized via the Lusztig integral form) quantized enveloping algebra $$U$$ of the general linear Lie algebra $$\mathfrak{gl}_n$$ on mixed tensor space. Thus, the $$U$$-invariants in the ring of $$R$$-linear endomorphisms of mixed tensor space are generated by the action of $$\mathfrak B_{r,s}^n(q)$$.

##### MSC:
 17B37 Quantum groups (quantized enveloping algebras) and related deformations 20C08 Hecke algebras and their representations 20G43 Schur and $$q$$-Schur algebras
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