Hopf PBW-deformations of a new type quantum group

Abstract

In this paper, we mainly focus on a new type quantum group $U_{q}(\mathfrak{sl}^{*}_2)$ and its Hopf PBW-deformations $U_{q}(\mathfrak{sl}^{*}_2,\kappa)$ in which $U_{q}(\mathfrak{sl}^{*}_2,0) = U_{q}(\mathfrak{sl}^{*}_2)$ and the classical Drinfeld-Jimbo quantum group $U_{q}(\mathfrak{sl}_2)$ is included. The category of finite dimensional $U_{q}(\mathfrak{sl}^{*}_2)$-modules is proved to be non-semisimple. We establish a uniform block decomposition of the category $U_{q}(\mathfrak{sl}^{*}_2,\kappa){\mbox -}{\rm \bf mod}_{\rm wt}$ of finite dimensional weight modules for each $U_{q}(\mathfrak{sl}^{*}_2,\kappa)$, and reduce the investigation on $U_{q}(\mathfrak{sl}^{*}_2,\kappa){\mbox -}{\rm \bf mod}_{\rm wt}$ to its principle block(s). We introduce the notion of primitive object in $U_{q}(\mathfrak{sl}^{*}_2,\kappa){\mbox -}{\rm \bf mod}_{\rm wt}$ which affords a new and elementary way to verify the semisimplicity of the category of finite dimensional $U_{q}(\mathfrak{sl}_2)$-modules. As the core of this present paper, a tensor equivalence between the principal block(s) of $U_{q}(\mathfrak{sl}^{*}_2,\kappa){\mbox -}{\rm \bf mod}_{\rm wt}$ and the category of finite dimensional representations of (deformed) preprojective algebras of Dynkin type $\A$ is obtained.