Equivariant Operators Between Some Modules of the Lie Algebra of Vector Fields

Abstract

The space of differential operators of order ≤ k, from the differential forms of degree p of a smooth manifold M into the functions of M, is a module over the Lie algebra of vector fields of M, when it is equipped with the natural Lie derivative. In this paper, we compute all equivariant i.e., intertwining operators and conclude that the preceding modules of differential operators are never isomorphic. We also answer a question of Lecomte, who observed that the restriction of some homotopy operator – introduced in Lecomte [Lecomte, P. (1994). On some sequence of graded Lie algebras associated to manifolds. Ann. Glob. Ana. Geo. 12:183–192] – to is equivariant for small values of k and p.

Key Words:

• Lie algebra of vector fields
• Modules of differential operators
• Intertwining operators

Mathematics Subject Classification (2000):

• 17B66
• 16D99
• 58J99

Acknowledgments

This work was supported by MCESR Grant MEN/CUL/99/007. The author thanks Lecomte and Mathonet for helpful comments.

Notes

^#Communicated by C. Cibils.