The quantized walled Brauer algebra and mixed tensor space

TL;DR

This paper introduces a multi-parameter deformation of the walled Brauer algebra, constructs an integral basis, and proves a form of Schur--Weyl duality for mixed tensor space involving quantum groups.

Contribution

It develops a new multi-parameter deformation of the walled Brauer algebra, constructs an integral basis, and establishes a Schur--Weyl duality for mixed tensor space with quantum groups.

Findings

Constructed an integral basis of the deformed algebra using oriented tangles.

Proved the kernel of the algebra action is free over the ground ring.

Established one side of Schur--Weyl duality for 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 $\mathfrak{B}_{r,s}^n(q)$.