Affine walled Brauer-Clifford superalgebras

TL;DR

This paper introduces affine walled Brauer-Clifford superalgebras over arbitrary domains, proves their freeness and structure, and establishes isomorphisms with previously defined superalgebras, advancing the algebraic understanding of these structures.

Contribution

It constructs and analyzes affine walled Brauer-Clifford superalgebras, proves their freeness, and relates them to existing superalgebras, providing new structural insights.

Findings

$BC_{r, t}^{ m aff}$ is free over $R$ with infinite rank.

Finite-dimensional irreducible modules factor through cyclotomic quotients.

$BC_{k, r, t}$ is free over $R$ if and only if admissible.

Abstract

In this paper, a notion of affine walled Brauer-Clifford superalgebras $BC_{r, t}^{\rm aff} $ is introduced over an arbitrary integral domain $R$ containing $2^{-1}$. These superalgebras can be considered as affinization of walled Brauer superalgebras in \cite{JK}. By constructing infinite many homomorphisms from $BC_{r, t}^{\rm aff}$ to a class of level two walled Brauer-Clifford superagebras over $\mathbb C$, we prove that $BC_{r, t}^{\rm aff} $ is free over $R$ with infinite rank. We explain that any finite dimensional irreducible $BC_{r, t}^{\rm aff} $-module over an algebraically closed field $F$ of characteristic not $2$ factors through a cyclotomic quotient of $BC_{r, t}^{\rm aff} $, called a cyclotomic (or level $k$) walled Brauer-Clifford superalgebra $ BC_{k, r, t}$. Using a previous method on cyclotomic walled Brauer algebras in \cite{RSu1}, we prove that $BC_{k, r, t}$ is free over $R$ with super rank $(k^{r+t}2^{r+t-1} (r+t)!, k^{r+t}2^{r+t-1} (r+t)!)$ if and only if it is admissible in the sense of Definition~6.4. Finally, we prove that the degenerate affine (resp., cyclotomic) walled Brauer-Clifford superalgebras defined by Comes-Kujawa in \cite{CK} are isomorphic to our affine (resp., cyclotomic) walled Brauer-Clifford superalgebras.