Roots of Unity: Representations of Quantum Groups

TL;DR

This paper explores how to construct representations of quantum groups at roots of unity from lower-rank quantum groups, highlighting special cases with maximal dimension and parameters.

Contribution

It introduces a method to build quantum group representations at roots of unity from lower-rank cases, revealing new maximal and parameter-rich irreducible representations.

Findings

Representations of U_q(g_n) can be derived from those of U_q(g_{n-1}) at roots of unity.

Maximal dimension irreducible representations are identified.

Special cases with the most free parameters are characterized.

Abstract

Representations of Quantum Groups U_q (g_n), g_n any semi simple Lie algebra of rank n, are constructed from arbitrary representations of rank n-1 quantum groups for q a root of unity. Representations which have the maximal dimension and number of free parameters for irreducible representations arise as special cases.