Cohomology of Lie-conformal algebras with derivations

Abstract:

      In this paper, we consider Lie conformal algebras with derivations. A pair consisting of a Lie conformal algebra and a distinguished derivation is called a LieCDer pair. We define a cohomology theory for LieCDer pair with coefficients in a representation. Furthermore, we study abelian extensions of a LieCDer pair as an application of cohomology theory. Finally, we consider homotopy derivations on 2-term conformal L$_{\infty}$-algebras and 2-derivations on conformal Lie 2-algebras. The category of 2-term conformal L$_{\infty}$-algebras with homotopy derivations is equivalent to the category of conformal Lie 2-algebras with 2-derivations.