Algebra gl(\lambda) inside the algebra of differential operators on the real line

Abstract

The Lie algebra gl(\lambda) with \lambda \in \mathbb C, introduced by B.L.Feigin, can be embedded into the Lie algebra of differential operators on the real line. We give an explicit formula of the embedding of gl(\lambda) into the algebra D_{\lambda} of differential operators on the space of tensor densities of degree \lambda on \mathbb R. Our main tool is the notion of projectively equivariant symbol of a differential operator.