BGG category for quantum Schrödinger algebra

The quantum Schrödinger algebra is a quantized universal enveloping algebra of the (centrally extended) Schrödinger algebra. It can be realized as the smash product of the quantum group [Formula: see text] and its a module algebra. For generic [Formula: see text], in this paper we study the BGG category [Formula: see text] for the quantum Schrödinger algebra. Then we use the result to determine its center, which is a polynomial ring in two variables.