On irreducible weight representations of a new deformation U_q(sl_2) of U(sl_2)

Xin Tang

doi:10.4303/jglta/s070202

Abstract

Starting from a Hecke $R-$matrix, Jing and Zhang constructed a new deformation $U_{q}(sl_{2})$ of $U(sl_{2})$, and studied its finite dimensional representations in \cite{JZ}. Especically, this algebra is proved to be just a bialgebra, and all finite dimensional irreducible representations are constructed in \cite{JZ}. In addition, an example is given to show that not every finite dimensional representation of this algebra is completely reducible. In this note, we take a step further by constructing more irreducible representations for this algebra. We first construct points of the spectrum of the category of representations over this new deformation by using methods in noncommutative algebraic geometry. Then applied to the study of representations, our construction recovers all finite dimensional irreducible representations as constructed in \cite{JZ}, and yields new families of infinite dimensional irreducible weight representations of this new deformation $U_{q}(sl_{2})$.

Highlights

Spectral theory of abelian categories was first started by Gabriel in [4]
Gabriel defined the injective spectrum of any noetherian grothendieck category
The injective spectrum consists of isomorphism classes of indecomposible injective objects in the category

Summary

Introduction

Spectral theory of abelian categories was first started by Gabriel in [4]. Gabriel defined the injective spectrum of any noetherian grothendieck category. Deformation of U (sl2), Irreducible Representations, Hyperbolic Algebras. As a specific application of spectral theory to representation theory, points of the spectrum have been constructed for a large family of algebras, which are called Hyperbolic algebras in [13]. We first construct families of points for the spectrum of the category of representations of this deformation.

Results

Conclusion