Representations of Super Yangians with Gelfand-Tsetlin bases

TL;DR

This paper explores the structure of super Yangian representations, focusing on simple quotients of tensor modules and conditions for tameness, extending known results from the non-super case.

Contribution

It generalizes the classification of finite-dimensional simple modules and tameness conditions from Yangians of gl_m to super Yangians with Gelfand-Tsetlin bases.

Findings

Characterization of simple quotients of tensor modules.

Necessary and sufficient conditions for modules to be tame.

Extension of Nazarov and Tarasov's results to super Yangians.

Abstract

The evaluation homomorphisms from the super Yangian $\Ymn$ to the universal enveloping algebra $\U(\gl_{m|n})$ allows one to regard the covariant tensor module of $\gl_{m|n}$ as $\Ymn$ modules. We study simple quotients of the submodules generated by a tensor product of highest weight vectors inside the tensor products of covariant evaluation modules. In the case $n=0$, this recover all finite-dimensional simple modules of $\Y(\mathfrak{gl}_m)$. We give a necessary and sufficient condition for such modules to be tame, which generalizes the earlier work of Nazarov and Tarasov for $\Y(\mathfrak{gl}_m)$ to the super case.