On Level Zero Representations of Quantized Affine Algebras

TL;DR

This paper investigates the structure of level zero modules over quantized affine algebras, proving conjectures about tensor product cyclicity and exploring properties of extremal vector modules.

Contribution

It proves the conjecture on tensor product cyclicity and characterizes modules generated by extremal vectors in the context of quantized affine algebras.

Findings

Proof of the cyclicity conjecture for tensor products

Properties of modules generated by extremal vectors established

Universal extremal weight modules are irreducible and related to finite-dimensional modules

Abstract

We study the properties of level zero modules over quantized affine algebras. The proof of the conjecture on the cyclicity of tensor products by Akasaka and the present author is given. Several properties of modules generated by extremal vectors are proved. The weights of a module generated by an extremal vector are contained in the convex hull of the Weyl group orbit of the extremalweight. The universal extremal weight module with level zero fundamental weight as an extremal weight is irreducible, and isomorphic to the affinization of an irreducible finite-dimensional module.