Computation of cohomology of finite Lie conformal algebras

Abstract

By using the energy operator introduced by B. Bakalov, A. De Sole, and V.\,G. Kac, we propose an algorithm for computing the cohomology of finitely generated free Lie conformal algebras with a Virasoro element. This computational method enables us to determine cohomologies with coefficients in their conformal modules, including the trivial module, irreducible finitely free modules, and adjoint modules. As concrete applications, we compute by \textit{Mathematica} both basic and reduced cohomologies for the $\mathcal{W}(b)$, Schr\"odinger-Virasoro and extended Schr\"odinger-Virasoro Lie conformal algebras.