# Journal of Noncommutative Geometry

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**Volume 11, Issue 3, 2017, pp. 845–885**

**DOI: 10.4171/JNCG/11-3-3**

Published online: 2017-09-26

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

Theo Raedschelders^{[1]}and Michel Van den Bergh

^{[2]}(1) Free University of Brussels, Belgium

(2) University of Hasselt, Diepenbeek, Belgium

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.

*Keywords: *Hopf algebras, monoidal categories, quasi-hereditary algebras

Raedschelders Theo, Van den Bergh Michel: The representation theory of non-commutative $\mathcal O$(GL$_2)$. *J. Noncommut. Geom.* 11 (2017), 845-885. doi: 10.4171/JNCG/11-3-3