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.

中文翻译：

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量子薛定谔代数的 BGG 范畴

1996 年，一个q-在 Dobrev 中介绍了薛定谔李代数的通用包络代数的变形等人. [J.物理。一种29(1996) 5909–5918。]。这个代数被称为量子薛定谔代数。在本文中，我们研究了 Bernstein-Gelfand-Gelfand (BGG) 类别$\数学{O}$对于量子薛定谔代数$U_q(\mathfrak{s})$， 在哪里q是一个非零复数，它不是单位根。如果中央收费$\dot z\neq 0$, 使用模块$B_{\dot z}$在量子外尔代数上$H_q$，我们证明了完整的子类别之间存在等价$\mathcal{O}[\dot Z]$由具有中心电荷的模块组成$\点z$和 BGG 类别$\mathcal{O}^{(\mathfrak{sl}_2)}$对于量子群$U_q(\mathfrak{sl}_2)$. 在这种情况下$\点 z = 0$, 我们研究子类别$\mathcal{A}$由有限维组成$U_q(\mathfrak{s})$- 类型 1 的模块，零动作Z. 我们直接构造一个等价函子$\mathcal{A}$到具有一些二次关系的无限箭袋的有限维表示的范畴。作为推论，我们证明了有限维的范畴$U_q(\mathfrak{s})$-modules 是狂野的。