The representation theory of non-commutative 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 (aut)under bar(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 (aut)under bar(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 (aut)under bar(A)-representations as tensor products of (end)under bar(A)-representations and their duals.