# The representation theory of non-commutative $\mathcal O$(GL$_2)$

### Theo Raedschelders

Free University of Brussels, Belgium### Michel Van den Bergh

University of Hasselt, Diepenbeek, Belgium

## Abstract

In our companion paper "The Manin Hopf algebra of a Koszul Artin–Schelter regular algebra is quasi-hereditary" we used the Tannaka–Krein formalism to study the universal coacting Hopf algebra $\underline {\mathrm {aut}}(A)$ for a Koszul Artin–Schelter regular algebra $A$. In this paper we study in detail the case $A=k[x,y]$. In particular we give a more precise description of the standard and costandard representations of $\underline {\mathrm {aut}}(A)$ as a coalgebra and we show that the latter can be obtained by induction from a Borel quotient algebra. Finally we give a combinatorial characterization of the simple $\underline {\mathrm {aut}}(A)$-representations as tensor products of $\underline {\mathrm {end}}(A)$-representations and their duals.

## Cite this article

Theo Raedschelders, Michel Van den Bergh, The representation theory of non-commutative $\mathcal O$(GL$_2)$. J. Noncommut. Geom. 11 (2017), no. 3, pp. 845–885

DOI 10.4171/JNCG/11-3-3