Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 5 (2009), 063, 7 pages      arXiv:0906.2988      http://dx.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.

pdf (200 kb)   ps (154 kb)   tex (9 kb)

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.

Previous article   Next article   Contents of Volume 5 (2009)