Representation Type Of The Category (?) Of Highest Weight Modules For The Quantum Schr(?)dinger Algebra

In this paper,we study the BGG category(?) for the quantum Schr?dinger algebra Uq(s),where q is a nonzero complex number which is not a root of unity.If the central charge z ?0,using the module B(?) over the quantum Weyl algebra Hq,we show that there is an equivalence between the full subcategory (?)[(?)]consisting of modules with the central charge z and the BGG category (?) q(sl2)for the quantum group Uq(sl2).In the case that (?)=0,we study the subcategory A consisting of finite dimensional Uq(s)-modules of type 1 with zero action of Z.Motivated by the ideas in[9,10],we directly construct an equivalent functor from A to the category of finite dimensional representations of an infinite quiver.As a corollary,we show that the category of finite dimensional modules for the quantum Schr?dinger algebra Uq(s) is wild.