Filtered Lie conformal algebras whose associated graded algebras are isomorphic to that of general conformal algebra gc1

Let G be a filtered Lie conformal algebra whose associated graded conformal algebra is isomorphic to that of general conformal algebra gc1. In this paper, we prove that G≅gc1 or (the associated graded conformal algebra of gc1), by making use of some results on the second cohomology groups of the conformal algebra g with coefficients in its module Mb,0 of rank 1, where g=Vir⋉Ma,0 is the semi-direct sum of the Virasoro conformal algebra Vir with its module Ma,0. Furthermore, we prove that does not have a nontrivial representation on a finite C[∂]-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 gcN for any N.