Low dimensional cohomology of general conformal algebras $gc_N$

Abstract

We compute the low dimensional cohomologies $\tilde H^q(gc_N,C)$, H^q(gc_N,\C) of the infinite rank general Lie conformal algebras $gc_N$ with trivial coefficients for $q\le3, N=1$ or $q\le2, N\ge2$. We also prove that the cohomology of $gc_N$ with coefficients in its natural module is trivial, i.e., H^*(gc_N,\C[\ptl]^N)=0; thus partially solve an open problem of Bakalov-Kac-Voronov in [{\it Comm. Math. Phys.,} {\bf200} (1999), 561-598].Comment: 18 page