Complete reducibility theorems for modules over pointed Hopf algebras

Nicolás Andruskiewitsch,David Radford,Hans-Jürgen Schneider

doi:10.1016/j.jalgebra.2010.06.002

Abstract

We investigate the representation theory of a large class of pointed Hopf algebras, extending results of Lusztig and others. We classify all simple modules in a suitable category and determine the weight multiplicities; we establish a complete reducibility theorem in this category.

Highlights

The main achievements of the finite-dimensional representation theory of finite-dimensional complex semisimple Lie algebras include (a) the parametrization of the simple finite-dimensional representations by their highest weights,(b) the complete reducibility of any finite-dimensional representation, (c) the determination of the weight multiplicities of a finite-dimensional highest weight representation
The goal of the present paper is to study the representation theory of these Hopf algebras and of their natural generalizations with arbitrary symmetrizable Cartan matrices
We prove a general structure result, Theorem 2.1, analogous to the classical Levi decomposition for Lie algebras

Summary

Introduction

The main achievements of the finite-dimensional representation theory of finite-dimensional complex semisimple Lie algebras include (a) the parametrization of the simple finite-dimensional representations by their highest weights,. The representation theory of the q-analogue Uq(g), where g is a symmetrizable Kac-Moody algebra, was developed in [L2], where analogues of the highest weight modules from [K] were introduced and a complete reducibility theorem was proved [L2, 6.2.2] using a quantum version of the Casimir operator. In [AS3] a family of pointed Hopf algebras was introduced having a Cartan matrix of finite type as one of the inputs This family contains the q-analogues Uq(g) and their multiparametric variants; they are close to them but one parameter of deformation for each connected component and more general linking relations are allowed. Combined with the main results of [AS3, AA], our theory gives in Theorem 5.4 a characterization of the pointed Hopf algebras U with finite Cartan matrix and free abelian group of finite rank Γ by axiomatic properties.

Nichols algebras and linking

A Levi-type theorem for pointed Hopf algebras

Perfect linkings and reduced data

Representation theory of U

A characterization of quantum groups