BGG category for the quantum Schrödinger algebra

In 1996, a q-deformation of the universal enveloping algebra of the Schrodinger Lie algebra was introduced in Dobrev et al. [J. Phys. A 29 (1996) 5909–5918.]. This algebra is called the quantum Schrodinger algebra. In this paper, we study the Bernstein-Gelfand-Gelfand (BGG) category for the quantum Schrodinger algebra , where q is a nonzero complex number which is not a root of unity. If the central charge , using the module over the quantum Weyl algebra , we show that there is an equivalence between the full subcategory consisting of modules with the central charge and the BGG category for the quantum group . In the case that , we study the subcategory consisting of finite dimensional -modules of type 1 with zero action of Z. We directly construct an equivalence functor from to the category of finite dimensional representations of an infinite quiver with some quadratic relations. As a corollary, we show that the category of finite dimensional -modules is wild.