Drinfeld category and the classification of singular Gelfand–Tsetlin $\displaystyle \mathfrak{gl}_n$-modules

Abstract

Abstract We prove a uniqueness theorem for irreducible non-critical Gelfand–Tsetlin modules. The uniqueness result leads to a complete classification of the irreducible Gelfand–Tsetlin modules with $1$-singularity. An explicit construction of such modules was given in Futorny et al. [7]. In particular, we show that the modules constructed in Futorny et al. [7] exhaust all irreducible Gelfand–Tsetlin modules with $1$-singularity. To prove the result, we introduce a new category of modules (called Drinfeld category) related to the Drinfeld generators of the Yangian $Y({\mathfrak{gl}}_n)$ and define a functor from the category of non-critical Gelfand–Tsetlin modules to the Drinfeld category.