Mathematics – Representation Theory
Scientific paper
2006-10-29
Mathematics
Representation Theory
8 pages
Scientific paper
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 $U_{q}(sl_{2})$.
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