Serre functors for Lie algebras and superalgebras
Abstract
We propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BGG category $\mathcal{O}$ associated to a semi-simple complex finite dimensional Lie algebra. We further show that our realization carries over to classical Lie superalgebras in many cases. Along the way we prove that category $\mathcal{O}$ and its parabolic generalizations for classical Lie superalgebras are categories with full projective functors. As an application we prove that in many cases the endomorphism algebra of the basic projective-injective module in (parabolic) category $\mathcal{O}$ for classical Lie superalgebras is symmetric. As a special case we obtain that in these cases the algebras describing blocks of the category of finite dimensional modules are symmetric. We also compute the latter algebras for the superalgebra $\mathfrak{q}(2)$.
- Publication:
-
arXiv e-prints
- Pub Date:
- August 2010
- DOI:
- 10.48550/arXiv.1008.1166
- arXiv:
- arXiv:1008.1166
- Bibcode:
- 2010arXiv1008.1166M
- Keywords:
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- Mathematics - Representation Theory;
- 17B10
- E-Print:
- 19 pages, to appear in Annales de l'Institut Fourier in 2011