Space of Second-Order Linear Differential Operators as a Module over the Lie Algebra of Vector Fields
✍ Scribed by C. Duval; V.Yu. Ovsienko
- Publisher
- Elsevier Science
- Year
- 1997
- Tongue
- English
- Weight
- 271 KB
- Volume
- 132
- Category
- Article
- ISSN
- 0001-8708
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✦ Synopsis
The space of linear differential operators on a smooth manifold M has a natural one-parameter family of Diff(M )-(and Vect(M )-) module structures, defined by their action on the space of tensor densities. It is shown that, in the case of secondorder differential operators, the Vect(M)-module structures are equivalent, for any degree of tensor densities except for three critical values; [0, 1 2 , 1]. A second-order analogue of the Lie derivative appears as an intertwining operator between the spaces of second-order differential operators on tensor densities. 1997 Academic Press 1. INTRODUCTION: MAIN PROBLEM Let M be an oriented manifold of dimension n. Consider the space D k (M ) of k th order linear differential operators on M. In local coordinates, such an operator is given by
where i = Â x i and a i 1 } } } i l k , , # C (M) with l=0, 1, ..., k. (From now on we suppose a summation over repeated indices.)
The group Diff(M ) of all diffeomorphisms of M and the Lie algebra Vect(M) of all smooth vector fields naturally act on the space D k (M ). Let G # Diff(M ), then the action is defined by G(A) :=G* &1 AG*.