# Affine quantum super Schur-Weyl duality

## TL;DR

This paper extends the classical Schur-Weyl duality to the affine quantum super case, establishing a categorical equivalence between affine Iwahori-Hecke algebra modules and affine quantum Lie superalgebra modules.

## Contribution

It completes the chain of dualities by constructing an affine quantum super Schur-Weyl duality, relating affine Iwahori-Hecke algebras and affine quantum Lie superalgebras.

## Key findings

- Constructed a functor establishing the duality
- Proved the functor is an equivalence of categories
- Extended duality to the general affine super case

## Abstract

The Schur-Weyl duality, which started as the study of the commuting actions of the symmetric group $S_d$ and $\mathrm{GL}(n,\mathbb{C})$ on $V^{\otimes d}$ where $V=\mathbb{C}^n$, was extended by Drinfeld and Jimbo to the context of the finite Iwahori-Hecke algebra $H_d(q^2)$ and quantum algebras $U_q(\mathrm{gl}(n))$, on using universal $R$-matrices, which solve the Yang-Baxter equation. There were two extensions of this duality in the Hecke-quantum case: to the affine case, by Chari and Pressley, and to the super case, by Moon and by Mitsuhashi. We complete this chain of works by completing the cube, dealing with the general affine super case, relating the commuting actions of the affine Iwahori-Hecke algebra $H^a_d(q^2)$ and of the affine quantum Lie superalgebra $U_{q,a}^\sigma(\mathrm{sl}(m,n))$ using the presentation by Yamane in terms of generators and relations, acting on the $d$th tensor power of the superspace $V=\mathbb{C}^{m+n}$. Thus we construct a functor and show it is an equivalence of categories of $H_d^a(q^2)$ and $U_{q,a}^\sigma(\mathrm{sl}(m,n))$-modules when $d<n'=m+n$.