Hopf algebras having a dense big cell

TL;DR

This paper explores conditions under which Hopf algebras can classify simple comodules via an analogue of the Borel-Weil construction, focusing on those with a dense big cell and examining specific examples like universal cosovereign Hopf algebras.

Contribution

It establishes that Hopf algebras with a dense big cell satisfy the criteria for classifying simple comodules using Borel-Weil type methods, extending previous work to new classes.

Findings

Hopf algebras with a dense big cell satisfy Borel-Weil classification criteria.

Universal cosovereign Hopf algebras have a free group as their weight group.

The method generalizes the approach used for quantum groups like GL and SL.

Abstract

We discuss some axioms that ensure that a Hopf algebra has its simple comodules classified using an analogue of the Borel-Weil construction. More precisely we show that a Hopf algebra having a dense big cell satisfies the above requirement. This method has its roots in the work of Parshall and Wang in the case of q-deformed quantum groups GL and SL. Here we examine the example of universal cosovereign Hopf algebras, for which the weight group is the free group on two generators.