In this paper we study the BGG-categories associated to quantum groups. We prove that many properties of the ordinary BGG-category for a semisimple complex Lie algebra carry over to the quantum case.
Of particular interest is the case when 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 and for finite dimensional -modules we are able to determine all irreducible characters as well as the characters of all indecomposable tilting modules in .
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 .
Cite this article
Henning Haahr Andersen, Volodymyr Mazorchuk, Category for quantum groups. J. Eur. Math. Soc. 17 (2015), no. 2, pp. 405–431DOI 10.4171/JEMS/506