Quantum groups of GL(2) representation type

TL;DR

This paper classifies certain cosemisimple Hopf algebras related to GL(2), introduces a new family generalizing quantum similitude groups, and describes their corepresentation categories.

Contribution

It provides a classification of Hopf algebras with corepresentation rings like GL(2) and introduces a new family generalizing quantum similitude groups.

Findings

Classification of cosemisimple Hopf algebras matching GL(2)

Introduction of a new family of Hopf algebras

Description of corepresentation categories

Abstract

We classify the cosemisimple Hopf algebras whose corepresentation semi-ring is isomorphic to that of GL(2). This leads us to define a new family of Hopf algebras which generalize the quantum similitude group of a non-degenerate bilinear form. A detailed study of these Hopf algebras gives us an isomorphic classification and the description of their corepresentation categories.