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BGG CATEGORY FOR THE QUANTUM SCHRÖDINGER ALGEBRA
Published online by Cambridge University Press: 20 April 2020
Abstract
In 1996, a q-deformation of the universal enveloping algebra of the Schrödinger Lie algebra was introduced in Dobrev et al. [J. Phys. A 29 (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 
$U_q(\mathfrak{s})$, where q is 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 of Z. 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.
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