Demazure modules of level two and prime representations of quantum affine $\mathfrak{sl}_{n+1}$

TL;DR

This paper investigates the classical limits of quantum affine algebra representations, showing they specialize to Demazure modules of level two, and establishes a connection between prime representations and Demazure modules.

Contribution

It demonstrates that prime representations of quantum affine rak{sl}_{n+1} specialize to Demazure modules of level two, linking quantum and classical representation theories.

Findings

Prime representations specialize to Demazure modules after twisting.

Any level two Demazure module is a limit of tensor products of prime representations.

The results connect quantum affine representations with classical Demazure modules.

Abstract

We study the classical limit of a family of irreducible representations of the quantum affine algebra associated to $\mathfrak{sl}_{n+1}$. After a suitable twist, the limit is a module for $\mathfrak{sl}_{n+1}[t]$, i.e., for the maximal standard parabolic subalgebra of the affine Lie algebra. Our first result is about the family of prime representations introduced in the context of a monoidal categorification of cluster algebras. We show that these representations specialize (after twisting), to $\mathfrak{sl}_{n+1}[t]$--stable, prime Demazure modules in level two integrable highest weight representations of the classical affine Lie algebra. More generally, we prove that any level two Demazure module is the limit of the tensor product of the corresponding irreducible prime representations of quantum affine $\mathfrak{sl}_{n+1}$.