Local Lie algebra structure and momentum mapping (Q1081162)

A local Lie algebra structure in the space of real-valued functions on a manifold is a more general concept than symplectic structures, Poisson structures, or contact structures. A manifold with this structure is also called a Jacobi manifold. In this paper, an idea of momentum mappings for a Lie group action which preserves the local Lie algebra structure is considered by analogy to momentum mappings of symplectic actions. To these momentum mappings, Noether's theorem holds good under one more assumption. By using the Schouten bracket, group actions preserving a local Lie algebra structure, momentum mappings, and especially equivariance of momentum mappings are studied. However, it is not successful in studying existence of our momentum mappings; we have several results about equivariance of our momentum mappings. These are direct generalizations of those of symplectic or Poisson cases by the key lemma that every Casimir function is invariant under the group action if the group action has momentum mappings.