2008/06/23 by Irfan Bagci, Bagci, Irfan, Jonathan R. Kujawa +3
Mathematics · #17B10 #17B56 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #math.RT #msc:17B10 #msc:17B56
paper · pdf · doi:10.48550/arxiv.0806.3740
30 pages
arxiv created 2008/06/23 · openalex publication_date 2008/06/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Boe, Kujawa and Nakano recently investigated relative cohomology for classical Lie superalgebras and developed a theory of support varieties. The dimensions of these support varieties give a geometric interpretation of the combinatorial notions of defect and atypicality due to Kac, Wakimoto, and Serganova. In this paper we calculate the cohomology ring of the Cartan type Lie superalgebra W(n) relative to the degree zero component W(n)0 and show that this ring is a finitely generated polynomial ring. This allows one to define support varieties for finite dimensional W(n)-supermodules which are completely reducible over W(n)0. We calculate the support varieties of all simple supermodules in this category. Remarkably our computations coincide with the prior notion of atypicality for Cartan type superalgebras due to Serganova. We also present new results on the realizability of support varieties which hold for both classical and Cartan type superalgebras.