  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(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(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(A)-representations as tensor
products of end(A)-representations and their duals.
