Some homological properties of category \mathcal O for Lie superalgebras

2020/10/14 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.2010.06852

openalex publication_date 2020/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For classical Lie superalgebras of type I, we provide necessary and sufficient conditions for a Verma supermodule Δ(λ) to be such that every non-zero homomorphism from another Verma supermodule to Δ(λ) is injective. This is applied to describe the socle of the cokernel of an inclusion of Verma supermodules over the periplectic Lie superalgebras \mathfrakpe(n) and, furthermore, to reduce the problem of description of Ext¹_{\mathcal O}(L(μ),Δ(λ)) for \mathfrakpe(n) to the similar problem for the Lie algebra \mathfrakgl(n). Additionally, we study the projective and injective dimensions of structural supermodules in parabolic category \mathcal O^{\mathfrak p} for classical Lie superalgebras. In particular, we completely determine these dimensions for structural supermodules over the periplectic Lie superalgebra \mathfrakpe(n) and the ortho-symplectic Lie superalgebra \mathfrakosp(2|2n).

Citations

• Gorenstein homological algebra for rngs and Lie superalgebras

Related

• Complexity and module varieties for classical Lie superalgebras
• Category O for Lie superalgebras
• The primitive spectrum and category O for the periplectic Lie superalgebra
• Classification of supersymmetries
• Typical blocks of Lie superalgebras in prime characteristic