Serre functors for Lie superalgebras and tensoring with S^top(\mathfrakg_1)

2025/05/06 by Chih-Whi Chen, Chen, Chih-Whi, Volodymyr Mazorchuk +1
Mathematics · #17B10 #17B55 #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2505.03197

openalex publication_date 2025/05/06 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28

Abstract

We show that the action of the Serre functor on the subcategory of projective-injective modules in a parabolic BGG category \mathcal O of a quasi-reductive finite dimensional Lie superalgebra is given by tensoring with the top component of the symmetric power of the odd part of our superalgebra. As an application, we determine, for all strange Lie suepralgebras, when the subcategory of projective injective modules in the parabolic category \mathcal O is symmetric.

Citations

• Annihilator ideals and blocks of Whittaker modules over quasireductive Lie superalgebras
• Symmetric structure for the endomorphism algebra of projective-injective module in parabolic category
• On the Ghost Centre of Lie Superalgebras
• Semiinfinite cohomology of associative algebras and bar duality

Related

• Tensoring with infinite-dimensional modules in \scr O₀
• Serre presentations of Lie superalgebras
• Semisimple algebraic tensor categories
• Whittaker categories, properly stratified categories and Fock space categorification for Lie superalgebras
• Notes on the Duflo-Serganova functor in positive characteristic