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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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