BGG Category for the Quantum Schrödinger Algebra

Abstract

In this paper, we study the BGG category O for the quantum Schrödinger algebra U q (s), where q is a nonzero complex number which is not a root of unity. If the central chargeż = 0, using the module Bż over the quantum Weyl algebra H q , we show that there is an equivalence between the full subcategory O[ż] consisting of modules with the central chargeż and the BGG category O (sl2) for the quantum group U q (sl 2 ). In the case thatż = 0, we study the subcategory A consisting of finite dimensional U q (s)-modules of type 1 with zero action of Z. Motivated by the ideas in [DLMZ, Mak], we directly construct an equivalent functor from A 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 U q (s)-modules is wild.