Drinfeld category and the classification of singular Gelfand-Tsetlin gl_n-modules

TL;DR

This paper establishes a complete classification of irreducible Gelfand-Tsetlin modules with 1-singularity by introducing a new Drinfeld category and proving a uniqueness theorem, thereby confirming the modules constructed previously are exhaustive.

Contribution

It introduces the Drinfeld category for Gelfand-Tsetlin modules and proves a uniqueness theorem that completes their classification with 1-singularity.

Findings

Complete classification of irreducible Gelfand-Tsetlin modules with 1-singularity

Introduction of the Drinfeld category related to Yangian Y(gl_n)

Verification that previously constructed modules exhaust all cases

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 \cite{FGR2}. In particular, we show that the modules constructed in \cite{FGR2} 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(gl_n) and define a functor from the category of non-critical Gelfand-Tsetlin modules to the Drinfeld category.