Hopf PBW-deformations of a new type quantum group Uq(𝔰𝔩2∗) and deformed preprojective algebras

We classify all the Hopf PBW-deformations of a new type quantum group [Formula: see text] from which the classical Drinfeld–Jimbo quantum group [Formula: see text] can arise as an almost unique nontrivial one. Different from the [Formula: see text] case, the category of finite-dimensional [Formula: see text]-modules is non-semisimple. We establish a block decomposition theorem for the category [Formula: see text] of finite-dimensional weight modules of [Formula: see text]. On the level of tensor category, we show that [Formula: see text] (respectively, the category [Formula: see text] of finite-dimensional [Formula: see text]-modules) can be realized via (respectively, deformed) preprojective algebras of Dynkin type [Formula: see text].