Filtered Lie conformal algebras whose associated graded algebras are isomorphic to that of general conformal algebra $gc_1$

Abstract

Let $G$ be a filtered Lie conformal algebra whose associated graded conformal algebra is isomorphic to that of general conformal algebra $gc_1$. In this paper, we prove that $G\cong gc_1$ or ${\rm gr\,}gc_1$ (the associated graded conformal algebra of $gc_1$), by making use of some results on the second cohomology groups of the conformal algebra \fg with coefficients in its module $M_{b,0}$ of rank 1, where \fg=\Vir\ltimes M_{a,0} is the semi-direct sum of the Virasoro conformal algebra \Vir with its module $M_{a,0}$. Furthermore, we prove that ${\rm gr\,}gc_1$ does not have a nontrivial representation on a finite \C[\partial]-module, this provides an example of a finitely freely generated simple Lie conformal algebra of linear growth that cannot be embedded into the general conformal algebra $gc_N$ for any $N$