Representations of pointed Hopf algebras over S_3

TL;DR

This paper classifies simple modules over the two finite-dimensional pointed Hopf algebras with group S_3, analyzes their module categories, and shows these algebras are not of finite representation type.

Contribution

It determines all simple modules, Gabriel quivers, and projective covers for these Hopf algebras, extending understanding of their representation theory.

Findings

Two Hopf algebras classified with explicit module structures

Gabriel quivers computed, showing infinite representation type

Modules over related complex pointed Hopf algebras analyzed

Abstract

The classification of finite-dimensional pointed Hopf algebras with group S_3 was finished in "The Nichols algebra of a semisimple Yetter-Drinfeld module", arXiv:0803.2430v1 [math.QA], by Andruskiewitsch, Heckenberger and Schneider: there are exactly two of them, the bosonization of a Nichols algebra of dimension 12 and a non-trivial lifting. Here we determine all simple modules over any of these Hopf algebras. We also find the Gabriel quivers, the projective covers of the simple modules, and prove that they are not of finite representation type. To this end, we first investigate the modules over some complex pointed Hopf algebras defined in the papers "Examples of liftings of Nichols algebras over racks", by Andruskiewitsch and Gra\~na and "Finite dimensional pointed Hopf algebras over S_4", arXiv:0904.2558v2 [math.QA], by G. Garcia and the author, whose restriction to the group of group-likes is a direct sum of 1-dimensional modules.