Hopf PBW-deformations of a new type quantum group

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