Operator pencils on the algebra of densities

Abstract

In this paper we continue to study equivariant pencil liftings and differential operators on the algebra of densities. We emphasize the role that the geometry of the extended manifold plays. Firstly we consider basic examples. We give a projective line of diff($M$)-equivariant pencil liftings for first order operators, and the canonical second order self-adjoint lifting. Secondly we study pencil liftings equivariant with respect to volume preserving transformations. This helps to understand the role of self-adjointness for the canonical pencils. Then we introduce the Duval-Lecomte-Ovsienko (DLO)-pencil lifting which is derived from the full symbol calculus of projective quantisation. We use the DLO-pencil lifting to describe all regular proj-equivariant pencil liftings. In particular the comparison of these pencils with the canonical pencil for second order operators leads to objects related to the Schwarzian. Within this paper the question of whether the pencil lifting factors through a full symbol map naturally arises.