SIGMA 5 (2009), 063, 7 pages      arXiv:0906.2988      https://doi.org/10.3842/SIGMA.2009.063
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Non-Hamiltonian Actions and Lie-Algebra Cohomology of Vector Fields

Roberto Ferreiro Pérez ^a and Jaime Muñoz Masqué ^b
a) Departamento de Economía Financiera y Contabilidad I, Facultad de Ciencias Económicas y Empresariales, UCM, Campus de Somosaguas, 28223-Pozuelo de Alarcón, Spain
^b) Instituto de Física Aplicada, CSIC, C/ Serrano 144, 28006-Madrid, Spain

Received April 03, 2009, in final form June 08, 2009; Published online June 16, 2009

Abstract
Two examples of Diff⁺S¹-invariant closed two-forms obtained from forms on jet bundles, which does not admit equivariant moment maps are presented. The corresponding cohomological obstruction is computed and shown to coincide with a nontrivial Lie algebra cohomology class on H²(X(S¹)).

Key words: Gel'fand-Fuks cohomology; moment mapping; jet bundle.

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References

1. Ferreiro Pérez R., Equivariant characteristic forms in the bundle of connections, J. Geom. Phys. 54 (2005), 197-212, math-ph/0307022.
2. Ferreiro Pérez R., Local cohomology and the variational bicomplex, Int. J. Geom. Methods Mod. Phys. 5 (2008), 587-604.
3. Ferreiro Pérez R., Muñoz Masqué J., Pontryagin forms on (4k–2)-manifolds and symplectic structures on the spaces of Riemannian metrics, math.DG/0507076.
4. Fuks D.B., Cohomology of infinite-dimensional Lie algebras, Contemporary Soviet Mathematics, Consultants Bureau, New York, 1986.
5. Gel'fand I.M., Fuks D.B., Cohomologies of the Lie algebra of vector fields on the circle, Funkcional. Anal. i Prilozen. 2 (1968), no. 4, 92-93.
6. Hamilton R., The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), 65-222.
7. McDuff D., Salamon D., Introduction to symplectic topology, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995, 1995.
8. Pohjanpelto J., Anderson I.M., Infinite-dimensional Lie algebra cohomology and the cohomology of invariant Euler-Lagrange complexes: a preliminary report, in Differential Geometry and Applications (Brno, 1995), Masaryk Univ., Brno, 1996, 427-448.