JournalsjemsVol. 17, No. 2pp. 405–431

Category O\mathcal O for quantum groups

  • Henning Haahr Andersen

    University of Aarhus, Denmark
  • Volodymyr Mazorchuk

    Uppsala Universitet, Sweden
Category $\mathcal O$ for quantum groups cover
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Abstract

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

Of particular interest is the case when qq 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 O\mathcal O and for finite dimensional UqU_q-modules we are able to determine all irreducible characters as well as the characters of all indecomposable tilting modules in Oq\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 O\mathcal O.

Cite this article

Henning Haahr Andersen, Volodymyr Mazorchuk, Category O\mathcal O for quantum groups. J. Eur. Math. Soc. 17 (2015), no. 2, pp. 405–431

DOI 10.4171/JEMS/506