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 CDer(A) of a differential algebra A, the gauge LCAd G(A,M) associated to a differential algebra A and its module M, the current LCAd F^ of a Lie algebroid F, and the LCAd structure of the space Omega(V) of Kahler differentials over a Poisson vertex algebra (PVA) 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 V with coefficients in a module M and the corresponding cohomology of the LCAd Omega(V) with coefficients in the same module.