Cohomology of $\mathfrak{osp}(1|2)$ acting on linear differential operators on the supercircle $S^{1|1}

Abstract

We compute the first cohomology spaces $H^1(\mathfrak{osp}(1|2);{\mathfrak{D}}_{\lambda ,\mu })$ ($\lambda ,\mu \in \mathbb{R}$) of the Lie superalgebra $\mathfrak{osp}(1|2)$ with coefficients in the superspace ${\mathfrak{D}}_{\lambda ,\mu }$ of linear differential operators acting on weighted densities on the supercircle $S^{1|1}$. The structure of these spaces was conjectured in \cite{gmo}. In fact, we prove here that the situation is a little bit more complicated. (To appear in LMP.)

References in corpus (2)

• Differential operators on supercircle: conformally equivariant quantization and symbol calculus
• Cohomology of the Lie Superalgebra of Contact Vector Fields on ${\mathbb{R}}^{1|1}$ and Deformations of the Superspace of Symbols

Cited by in corpus (9)

• Cohomology of the Lie Superalgebra of Contact Vector Fields on ${\mathbb{R}}^{1|1}$ and Deformations of the Superspace of Symbols
• Deformation of sl(2) and osp(1|2)-Modules of Symbols
• Differential Operators on the Weighted Densities on the Supercircle $S^{1|n}$
• Cohomology of $\mathfrak{osp}(1|2)$ acting on the space of bilinear differential operators on the superspace ${\mathbb{R}}^{1|1}$
• Cohomology of $\mathfrak{sl}(2)$ acting on the space of $n$-ary differential operators on $\mathbb{R}$
• The spaces ${\mathrm{H}}^n(\mathfrak{osp}(1|2),M)$ for some weight modules $M$
• Symetries of modules of differential operators on $S^{1|1}$
• Cohomology of $\mathfrak{aff}(1)$ and $\mathfrak{aff}(1|1)$ acting on the space of $n$-ary differential operators on the superspace ${\mathbb{R}}^{1|1}$
• Deformation of $\mathfrak{osp}(2|2)$-Modules of Symbols