On irreducible weight representations of a new deformation Uq (sl2) of U (sl2)

Starting from a Hecke R-matrix, Jing and Zhang constructed a new deformation Uq(sl2) of U(sl2) and studied its finite dimensional representations in [Pacific J. Math., 171 (1995), 437-454]. In this note, more irreducible representations for this algebra are constructed. At first, by using methods in non commutative algebraic geometry the points of the spectrum of the category of representations over this new deformation are studied. The construction recovers all finite dimensional irreducible representations classified by Jing and Zhang, and yields new families of infinite dimensional irreducible weight representations.

Spectral theory of abelian categories was first initiated by Gabriel in. In particular, Gabriel defined the injective spectrum of any noetherian Grothendieck category. The injective spectrum consists of isomorphism classes of in decomposable injective objects inthe category endowed with the Zariski topology. If R is a commutative noetherian ring, then the injective spectrum of the category of all R-modules is homeomorphic to the prime spectrum of R. This homeomorphism is a part (and the main step in the argument) of the Gabriel’s reconstruction Theorem, according to which any noetherian commutative scheme can be uniquely reconstructed up to isomorphism from the category of quasi-coherent sheaves on it.