Cohomology of Lie Conformal Algebroids

Abstract

We study Lie conformal algebroids (LCAd) and their representations using the language of \(\lambda \)-brackets and Lie conformal algebras. We describe several general constructions, such as the LCAd of conformal derivations \({{\,\textrm{CDer}\,}}(A)\) of a differential algebra A, the gauge LCAd \(\mathcal {G}(M,E)\) associated to a differential algebra A, its module M, and an LCAd E, the current LCAd \(\widehat{F}\) of a Lie algebroid F, and the LCAd structure of the space \(\Omega (\mathcal {V})\) of Kähler differentials over a Poisson vertex algebra (PVA) \(\mathcal {V}\). We develop the cohomology theories of LCAd and we relate them to the corresponding cohomology theories of PVA. In particular, we find an isomorphism between the cohomology of a PVA \(\mathcal {V}\) with coefficients in a module M and the corresponding cohomology of the LCAd \(\Omega (\mathcal {V})\) with coefficients in the same module.