Simple modules over finite quantum groups and their Drinfel'd doubles

TL;DR

This paper extends the classification of simple modules over finite quantum groups and their Drinfel'd doubles, revealing a unified parametrization pattern linked to triangular decomposition.

Contribution

It broadens the known description of simple modules to a larger class of Hopf algebras, connecting parametrization with triangular decomposition.

Findings

Simple modules over extended classes of quantum groups are parametrized in a unified pattern.

The description links the module classification to the triangular decomposition structure.

The results include quantum Frobenius Kernels as special cases.

Abstract

By finite quantum groups we mean Lusztig's finite-dimensional pointed Hopf algebras called quantum Frobenius Kernels [9, 10], and their natural generalizations due to Andruskiewitsch and Schneider [2, 3]. For a Hopf algebra $H$ in a special class of the latter generalizations, which arises from a pair of quantum linear spaces, Krop and Radford [8] described the simple modules over $H$ and over the Drinfel'd double $D(H)$, showing that they fall into a simple pattern of parametrization. We extend the description to a wider class of Hopf algebras which includes the quantum Frobenius Kernels, renewing the parametrization pattern so as to connect directly to the so-called triangular decomposition.