On representation theory of quantum $SL_q(2)$ groups at roots of unity

TL;DR

This paper classifies all irreducible representations of quantum $SL_q(2)$ groups at roots of unity, including even roots, describes tensor product structures, and explores the non-existence of Haar functional, with explicit computations.

Contribution

It extends the classification of irreducible representations of quantum $SL_q(2)$ to all roots of unity, including even degrees, and analyzes tensor products and Haar functional properties.

Findings

Complete classification of irreducible representations for all roots of unity.

Description of the diagonal part of tensor products of irreducibles.

Proof of non-existence of Haar functional for these quantum groups.

Abstract

Irreducible representations of quantum groups $SL_q(2)$ (in Woronowicz' approach) were classified in J.Wang, B.Parshall, Memoirs AMS 439 in the~case of $q$ being an~odd root of unity. Here we find the~irreducible representations for all roots of unity (also of an~even degree), as well as describe "the~diagonal part" of tensor product of any two irreducible representations. An~example of not completely reducible representation is given. Non--existence of Haar functional is proved. The~corresponding representations of universal enveloping algebras of Jimbo and Lusztig are provided. We also recall the~case of general~$q$. Our computations are done in explicit way.