JournalsjncgVol. 11, No. 3pp. 845–885

The representation theory of non-commutative O\mathcal O(GL2)_2)

  • Theo Raedschelders

    Free University of Brussels, Belgium
  • Michel Van den Bergh

    University of Hasselt, Diepenbeek, Belgium
The representation theory of non-commutative $\mathcal O$(GL$_2)$ cover
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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(A)\underline {\mathrm {aut}}(A) for a Koszul Artin–Schelter regular algebra AA. In this paper we study in detail the case A=k[x,y]A=k[x,y]. In particular we give a more precise description of the standard and costandard representations of aut(A)\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 aut(A)\underline {\mathrm {aut}}(A)-representations as tensor products of end(A)\underline {\mathrm {end}}(A)-representations and their duals.

Cite this article

Theo Raedschelders, Michel Van den Bergh, The representation theory of non-commutative O\mathcal O(GL2)_2). J. Noncommut. Geom. 11 (2017), no. 3, pp. 845–885

DOI 10.4171/JNCG/11-3-3