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

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.