The half-quantum group U^>=0

TL;DR

This paper investigates the algebraic structure and representation theory of the half quantum group U^>=0, revealing its non-quasi-cocommutativity and classifying its simple and projective modules as well as Yetter-Drinfel'd modules.

Contribution

It provides a detailed analysis of the properties and representations of the half quantum group U^>=0, including its non-quasi-cocommutativity and classification of modules.

Findings

U^>=0 is not quasi-cocommutative.

Classification of simple and projective modules in the weight category.

Description of simple Yetter-Drinfel'd U^>=0-weight modules.

Abstract

Let U^>=0 denote the half quantum group for a fixed simple Lie algebra. We examine some properties and representation of U^>=0. We prove that the Hopf algebra U^>=0 is not quasi-cocommutative, and hence the category of left U^>=0-module is not a braided monoidal category. In the weight module category W, we describe all the simple objects and the projective objects. We also describe all simple Yetter-Drinfel'd U^>=0-weight modules.