BGG category for the quantum Schrödinger algebra

Abstract

In this paper, we study the BGG category $\mathcal{O}$ for the quantum Schr{ö}dinger algebra $U_q(\mathfrak{s})$, where $q$ is a nonzero complex number which is not a root of unity. If the central charge $\dot{z}\ne 0$, 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 $Z$. Motivated by the ideas in \cite{DLMZ, Mak}, we directly construct an equivalent 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.

18 pages

References in corpus (2)

• Quantized symplectic oscillator algebras of rank one
• Center and representations of infinitesimal Hecke algebras of sl_2