A class of non-weight modules of 𝑈𝑝(𝖘𝖑2) and Clebsch–Gordan type formulas

Abstract

Abstract In this paper, we construct a class of new modules for the quantum group Uq\u2062(s\u2062l2)U_{q}(\\mathfrak{sl}_{2}) which are free of rank 1 when restricted to C\u2062[K±1]\\mathbb{C}[K^{\\pm 1}]. The irreducibility of these modules and submodule structure for reducible ones are determined. It is proved that any C\u2062[K±1]\\mathbb{C}[K^{\\pm 1}]-free Uq\u2062(s\u2062l2)U_{q}(\\mathfrak{sl}_{2})-module of rank 1 is isomorphic to one of the modules we constructed, and their isomorphism classes are obtained. We also investigate the tensor products of the C\u2062[K±1]\\mathbb{C}[K^{\\pm 1}]-free modules with finite-dimensional simple modules over Uq\u2062(s\u2062l2)U_{q}(\\mathfrak{sl}_{2}), and for the generic cases, we obtain direct sum decomposition formulas for them, which are similar to the well-known Clebsch–Gordan formula for tensor products between finite-dimensional weight modules over Uq\u2062(s\u2062l2)U_{q}(\\mathfrak{sl}_{2}).