Cohomology and support varieties for Lie superalgebras
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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. The outcome of our computations naturally divides the simple supermodules into two families. Remarkably, this partition coincides with the one based on Serganova’s prior notion of atypicality for Cartan-type superalgebras. In this way the support variety construction gives a geometric interpretation of atypicality. We also present new results on the realizability of support varieties, which hold for both classical and Cartan-type superalgebras.