Affine Super Yangian and Weyl groupoid

TL;DR

This paper introduces affine Super Yangian algebras for affine superalgebras, establishes their presentations and isomorphisms, and defines the Weyl groupoid acting on these structures.

Contribution

It defines affine Super Yangian for affine superalgebras, proves isomorphism of different presentations, and introduces the Weyl groupoid acting on these algebras.

Findings

Proved isomorphism between Drinfeld and minimalistic presentations.

Constructed isomorphisms between Yangians for different root systems.

Defined and described the Weyl groupoid and its action on super Yangians.

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.