Affine Super Yangian and Weyl groupoid

Abstract

We define affine Super Yangian $Y_{\hbar}(\hat{sl}(m|n), \Pi) $ for affine special linear superalgebra $\hat{sl}(m|n)$ and arbitrary system of simple roots $\Pi$ in terms of minimalistic system of generators. We also consider Drinfeld presentation for affine super Yangian in the case of arbitrary simple root system $\Pi$ and prove that these two presentations (Drinfeld and minimalistic) of $Y_{\hbar}(\hat{sl}(m|n), \Pi)$ are isomorphic as associative superalgebras. We also construct isomorphism of affine super Yangians $Y_{\hbar}(\hat{sl}(m|n), \Pi)$ and $Y_{\hbar}(\hat{sl}(m|n), \Pi')$ for different simple root systems $\Pi$ and $\Pi'$. After them we also define Weyl groupoid as a set of morphisms in category with objects, which are super Yanginas $Y_{\hbar}(\hat{sl}(m|n), \Pi)$, where $\Pi$ is simple root system. We describe Weyl groupoid in terms of generators and describe action of these generators on super Yangians. We describe isomorphisms between $Y_{\hbar}(\hat{sl}(m|n), \Pi)$ and $Y_{\hbar}(\hat{sl}(m|n), \Pi')$ as elements of Weyl groupoid.