Serre functors for Lie algebras and superalgebras

Details Serre functors for Lie algebras and superalgebras Serre functors for Lie algebras and superalgebras

2010-08-06

arxiv.org/abs/1008.1166v2

Mathematics

Representation Theory

19 pages, to appear in Annales de l'Institut Fourier in 2011

Scientific paper

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 $\mathfrak{q}(2)$.

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