# Affine walled Brauer-Clifford superalgebras

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Abstract: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. 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, 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. Finally, we prove that the degenerate affine (resp., cyclotomic) walled Brauer-Clifford superalgebras defined by Comes-Kujawa are isomorphic to our affine (resp., cyclotomic) walled Brauer-Clifford superalgebras. This is a joint work with Mengmeng Gao, Hebing Rui and Linliang Song.