Coproducts for affine super-Yangian and Weyl groupoid action

Abstract

For affine special linear superalgebra $\widehat{sl}(m|n, \Pi)$ defined by an arbitrary system of simple roots $\Pi$ we define the affine super Yangian $Y_{\hbar}(\widehat{sl}(m|n, \Pi))$ as Hopf superalgebra which is a quantization of superbialgebra $\widehat{sl}(m|n, \Pi)[t]$ and describe super Yangian in terms of minimalistic system of generators. We consider Drinfeld presentation for $Y^D_{\hbar}(\widehat{sl}(m|n, \Pi))$ and prove that these two presentations are isomorphic as associative superalgebras. We induce by means of this isomorphism a co-multiplication on the Drinfeld presentation $Y^D_{\hbar}(\widehat{sl}(m|n, \Pi))$ of the super Yangian. We introduce the action of Weyl groupoid by isomorphisms on super Yangians as an extension of its action on universal enveloping algebra and deformation of action on univesal enveloping superalgebra of current Lie superalgebra and prove that such extension exists and unique. As a consequence of this construction we obtain that super Yangians $Y_{\hbar}(\widehat{sl}(m|n, \Pi_1))$ and $Y_{\hbar}(\widehat{sl}(m|n, \Pi_2))$, defined by different simple root systems $ \Pi_1$ and $ \Pi_2$ are isomorphic as Hopf superalgebras.