Finite dimensional representations of quantum affine algebras at roots of unity

TL;DR

This paper explicitly describes the spectrum map of finite-dimensional irreducible representations of quantum loop algebras at roots of unity, revealing a Poisson algebraic group structure and identifying the image as principal adeles.

Contribution

It provides an explicit description of the canonical map from the spectrum of the quantum algebra to its center, linking representation theory with adelic groups at roots of unity.

Findings

Spectrum of the center forms a Poisson proalgebraic group.

The spectrum map's image is the subgroup of principal adeles.

The structure connects quantum algebra representations with adelic groups.

Abstract

We describe explicitly the canonical map $\chi:$ Spec $\ue(\a{g})\ \rightarrow \ $Spec $\ze$, where $\ue(\a{g})$ is a quantum loop algebra at an odd root of unity $\ve$. Here $\ze$ is the center of $\ue(\a{g})$ and Spec $R$ stands for the set of all finite--dimensional irreducible representations of an algebra $R$. We show that Spec $\ze$ is a Poisson proalgebraic group which is essentially the group of points of $G$ over the regular adeles concentrated at $0$ and $\infty$. Our main result is that the image under $\chi$ of Spec $\ue(\a{g})$ is the subgroup of principal adeles.