Mixed Schur-Weyl-Sergeev duality for queer Lie superalgebras

TL;DR

This paper introduces walled Brauer superalgebras and establishes a duality with queer Lie superalgebras, providing a diagrammatic realization and a presentation in generators and relations.

Contribution

It defines new superalgebras called walled Brauer superalgebras and proves their duality with queer Lie superalgebras, including a diagrammatic realization and a presentation.

Findings

Supercentralizer algebra isomorphic to walled Brauer superalgebra for n ≥ r+s.

Constructed a diagrammatic realization of Sergeev superalgebra.

Provided a presentation of the superalgebra in generators and relations.

Abstract

We introduce a new family of superalgebras $\overrightarrow{B}_{r,s}$ for $r, s \ge 0$ such that $r+s>0$, which we call the walled Brauer superalgebras, and prove the mixed Scur-Weyl-Sergeev duality for queer Lie superalgebras. More precisely, let $\mathfrak{q}(n)$ be the queer Lie superalgebra, ${\mathbf V} =\mathbb{C}^{n|n}$ the natural representation of $\mathfrak{q}(n)$ and ${\mathbf W}$ the dual of ${\mathbf V}$. We prove that, if $n \ge r+s$, the superalgebra $\overrightarrow{B}_{r,s}$ is isomorphic to the supercentralizer algebra $_{\mathfrak{q}(n)}({\mathbf V}^{\otimes r} \otimes {\mathbf W}^{\otimes s})^{\op}$ of the $\mathfrak{q}(n)$-action on the mixed tensor space ${\mathbf V}^{\otimes r} \otimes {\mathbf W}^{\otimes s}$. As an ingredient for the proof of our main result, we construct a new diagrammatic realization $\overrightarrow{D}_{k}$ of the Sergeev superalgebra $Ser_{k}$. Finally, we give a presentation of $\overrightarrow{B}_{r,s}$ in terms of generators and relations.