Low dimensional cohomology of general conformal algebras gcN

Abstract

We compute the low dimensional cohomologies H̃q(gcN,C), Hq(gcN,C) of the infinite rank general Lie conformal algebras gcN with trivial coefficients for q⩽3, N=1 or q⩽2, N⩾2. We also prove that the cohomology of gcN with coefficients in its natural module is trivial, i.e., H*(gcN,C[∂]N)=0, and thus partially solve an open problem of Bakalov–Kac–Voronov [Commun. Math. Phys., 200, 561–598 (1999)].