The quantized walled Brauer algebra and mixed tensor space

Abstract

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 Leduc (\cite{leduc}). 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 $\mathbf{U}$ of the general linear Lie algebra ${\mathfrak{gl}}_n$ on mixed tensor space. Thus, the $\mathbf{U}$-invariants in the ring of $R$-linear endomorphisms of mixed tensor space are generated by the action of ${\mathfrak{B}}_{r,s}^n(q)$.

34 pages