Gelfand-Tsetlin bases of representations for super Yangian and quantum affine superalgebra

TL;DR

This paper provides explicit formulas for Gelfand-Tsetlin bases of super Yangian modules, proves their irreducibility, and characterizes finite-dimensional irreducible modules as tame or thin, extending results to quantum affine superalgebras.

Contribution

It introduces explicit actions of Drinfeld generators on Gelfand-Tsetlin bases and characterizes tame modules for super Yangians and quantum affine superalgebras.

Findings

Explicit formulas for generator actions on bases

Proof of irreducibility of these modules

Characterization of tame and thin modules

Abstract

We give explicit actions of Drinfeld generators on Gelfand-Tsetlin bases of super Yangian modules associated with skew Young diagrams. In particular, we give another proof that these representations are irreducible. We study irreducible tame $\mathrm Y(\mathfrak{gl}_{1|1})$-modules and show that a finite-dimensional irreducible $\mathrm Y(\mathfrak{gl}_{1|1})$-module is tame if and only if it is thin. We also give the analogous statements for quantum affine superalgebra of type A.