The Quantized Walled Brauer Algebra and Mixed Tensor Space

In this paper we investigate a multi-parameter deformation of the walled Brauer algebra which was previously introduced by Leduc (1994). We construct an integral basis of consisting of oriented tangles which is in bijection with walled Brauer diagrams. Moreover, we study a natural action of 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 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 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 .