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

TL;DR

This paper provides a detailed analysis of the representation theory of the noncommutative algebra ()GL_2, focusing on standard, costandard, and simple representations using Tannaka-Krein formalism.

Contribution

It offers a precise description of representations of aut(A) for A=k[x,y], including their construction and combinatorial classification, extending previous work on universal coacting Hopf algebras.

Findings

Standard and costandard representations characterized as coalgebra structures.

Costandard representations obtained via induction from a Borel quotient.

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

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.