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We present a general theorem which computes the cohomology of a homological vector field on global sections of vector bundles over smooth affine supervarieties. The hypotheses and results have the clear flavor of a localization theorem.

 

Abstract Let 𝒲 n {{\mathcal{W}}_{n}} be the Lie algebra of polynomial vector fields.We classify simple weight 𝒲 n {{\mathcal{W}}_{n}} -modules M with finite weight multiplicities. We prove that every such nontrivial module M is either a tensor module or the unique simple submodule in a tensor module associatedwith the de Rham complex on ℂ n {\mathbb{C}^{n}} . 

We describe all blocks of the category of finite-dimensional q(3)-supermodules by providing their extension quivers. We also obtain two general results about the representation of q(n): we show that the Ext quiver of the standard block of q(n) is obtained from the principal block of q(n-1) by identifying certain vertices of the quiver and prove a virtual BGG-reciprocity for q(n). The latter result is used to compute the radical filtrations of q(3) projective covers. 

We prove an analogue of the Kac–Wakimoto conjecture for the periplectic Lie superalgebra [Formula: see text], stating that any simple module lying in a block of non-maximal atypicality has superdimension zero.