Serre functors for Lie algebras and superalgebras (Q442119)

Understanding the category of finite dimensional modules is the first step towards understanding the representation theory of Lie superalgebras and then the analogue of the \(BBG\) category \(\mathcal{O}\) (associated with a fixed triangular decomposition of a semisimple finite-dimensional complex Lie algebra).NEWLINENEWLINEThe authors extend results of \textit{J. N. Bernstein} and \textit{S. I. Gelfand} [Compos. Math. 41, 245--285 (1980; Zbl 0445.17006)] and \textit{D. Miličić} and \textit{W. Soergel} [Comment. Math. Helv. 72, No. 4, 503--520 (1997; Zbl 0956.17004)], and propose a new realization, using Harish-Chandra bimodules, of the Serre functor for the BBG category \(\mathcal{O}\) and show that this realization carries over to classical Lie superalgebras in many cases. Then, it is proved that the category \(\mathcal{O}\) and its parabolic generalization for classical Lie superalgebras are categories with full projective functors. As an application, the authors obtain that in many cases the endomorphism algebra of the basic projective-injective module in (parabolic) category \(\mathcal{O}\) for classical Lie superalgebra is symmetric. In particular, the algebras describing blocks of the category of finite dimensional modules are symmetric. The letter algebras for the superalgebra \(\mathfrak{q}(2)\) are computed.