The representation theory of noncommutative $\mathcal{O}({\text{GL}}_2)$

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(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.

36 pages

References in corpus (1)

• On quantum groups associated to non-Noetherian regular algebras of dimension 2

Cited by in corpus (2)

• On quantum groups associated to a pair of preregular forms
• Calabi-Yau property under monoidal Morita-Takeuchi equivalence