Category $\mathcal{O}$ for quantum groups

Abstract

In this paper we study the BGG-categories ${\mathcal{O}}_q$ associated to quantum groups. We prove that many properties of the ordinary BGG-category $\mathcal{O}$ for a semisimple complex Lie algebra carry over to the quantum case.

Of particular interest is the case when $q$ is a complex root of unity. Here we prove a tensor decomposition for both simple modules, projective modules, and indecomposable tilting modules. Using the known Kazhdan-Lusztig conjectures for $\mathcal{O}$ and for finite dimensional $U_q$-modules we are able to determine all irreducible characters as well as the characters of all indecomposable tilting modules in ${\mathcal{O}}_q$.

As a consequence of these results we are able to recover also a known result, namely that the generic quantum case behaves like the classical category $\mathcal{O}$.