BGG CATEGORY FOR THE QUANTUM SCHRÖDINGER ALGEBRA

Abstract

AbstractIn 1996, aq-deformation of the universal enveloping algebra of the Schrödinger Lie algebra was introduced in Dobrevet al. [J. Phys.A29(1996) 5909–5918.]. This algebra is called the quantum Schrödinger algebra. In this paper, we study the Bernstein-Gelfand-Gelfand (BGG) category$\mathcal{O}$for the quantum Schrödinger algebra$U_q(\mathfrak{s})$, whereqis a nonzero complex number which is not a root of unity. If the central charge$\dot z\neq 0$, using the module$B_{\dot z}$over the quantum Weyl algebra$H_q$, we show that there is an equivalence between the full subcategory$\mathcal{O}[\dot Z]$consisting of modules with the central charge$\dot z$and the BGG category$\mathcal{O}^{(\mathfrak{sl}_2)}$for the quantum group$U_q(\mathfrak{sl}_2)$. In the case that$\dot z = 0$, we study the subcategory$\mathcal{A}$consisting of finite dimensional$U_q(\mathfrak{s})$-modules of type 1 with zero action ofZ. We directly construct an equivalence functor from$\mathcal{A}$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$U_q(\mathfrak{s})$-modules is wild.