Abstract
An action σ: G × P→P of a Poisson Lie group G on a Poisson manifold P is called a Poisson action if σ is a Poisson map. It is believed that Poisson actions should be used to understand the “hidden symmetries” of certain integrable systems [STS2]. If the Poisson Lie group G has the zero Poisson structure, then σ being a Poisson action is equivalent to each transformation σ g : P→ P for g ∈ G preserving the Poisson structure on P. In this case, if the orbit space G \ P is a smooth manifold, it has a reduced Poisson structure such that the projection map P→G \ P is a Poisson map. If P is symplectic and if the action σ is generated by an equivariant momentum mapping J: P→ g*, the reduction procedure of Meyer [Me] and Marsden and Weinstein [Ms-We] gives a way of describing the symplectic leaves of G \ P as the quotients P µ := G µ \J −1(µ), where µ∈ g* and G µ ⊂ G is the coadjoint isotropy subgroup of µ.
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References
Abraham, R., Marsden, J.E., “Foundations of mechanics,” 2nd ed., Benjamin/Cummings, New York, 1978.
Bourbaki, N., “Groupes et algèbres de Lie,” Chapitres 2 et 3, Hermann, Paris, 1972.
Coste, A., Dazord, P., Weinstein, A., Groupoides symplectiques,(notes d’un cours de A. Weinstein), Publ. Dept. Math. (1987), Université Claude Bernard Lyon I.
Dazord, P., Sondaz, D., Variétés de Poisson-algébroides de Lie,Séminaire Sud-Rhodanien (1988–1/B), Publ. Univ. Claude Bernard-Lyon 1.
Dazord, P., Sondaz, D.,Groupes de Poisson affines,Proceedings of the Seminaire Sud-Rhodanien de Geometrie (1989) (to appear), Springer-MSRI series.
Drinfel’d, V.G., Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of the classical Yang-Baxter equations, Soviet Math. Dokl. 27 (1) (1983), 68–71.
Kosmann-Schwarzbach, Y., Poisson-Drinfel’d groups,Topics in soliton theory and exactly solvable nonlinear equations, M. Ablowitz, B. Fuchsteiner and M. Kruskal, eds. (1987), 191–215, World Scientific, Singapore.
Kosmann-Schwarzbach, Y., Magri, F., Poisson-Lie groups and complete integrability, part 1, Drinfel’d bialgebras, dual extensions and their canonical representations, Annales Inst. Henri Poincaré, Série A (Physique Théorique) 49 (4) (1988), 433–460.
Lu, J.H., Weinstein, A., Poisson Lie groups, dressing transformations, and Bruhat decompositions, Journal of Differential Geometry 31 (1990), 501–526.
Lu, J.H., Multiplicative and affine Poisson structures on Lie groups, PhD thesis (1990), University of California, Berkeley.
Marsden, J., Weinstein, A., Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974), 121–129.
Meyer, K.R., Symmetries and integrals in mechanics, in “Dynamical Systems,” M.N. Peixoto, ed., Academic Press, New York, 1973, pp. 259–272.
Semenov-Tian-Shansky, M.A., What is a classical r-matrix ?, Funct. Anal. Appl. 17 (4) (1983), 259–272.
Semenov-Tian-Shansky, M.A., Dressing transformations and Poisson group actions, Publ. RIMS, Kyoto University 21 (1985), 1237–1260.
Spivak, M., “A comprehensive introduction to differential geometry,” Vol. 1, Publish or Perish, 1970.
Weinstein, A., The local structure of Poisson manifolds, J. Diff. Geometry 18 (1983), 523–557.
Weinstein, A., Poisson geometry of the principal series and nonlinearizable structures, J. Diff. Geometry 23 (1987), 55–73.
Weinstein, A., Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan 40 (4) (1988), 705–727.
Weinstein, A., Some remarks on dressing transformations, J. Fac. Sci. Univ. Tokyo. Sect. 1A, Math. 36 (1988), 163–167.
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© 1991 Springer-Verlag New York, Inc.
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Lu, JH. (1991). Momentum Mappings And Reduction of Poisson Actions. In: Dazord, P., Weinstein, A. (eds) Symplectic Geometry, Groupoids, and Integrable Systems. Mathematical Sciences Research Institute Publications, vol 20. Springer, New York, NY. https://doi.org/10.1007/978-1-4613-9719-9_15
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DOI: https://doi.org/10.1007/978-1-4613-9719-9_15
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