zbMATH — the first resource for mathematics

Moments and reduction for symplectic groupoids. (English) Zbl 0659.58016
A Hamiltonian action of a Lie group G on a symplectic manifold M is generated by a momentum mapping J: \(M\to {\mathfrak g}^*\) which is equivariant with respect to the coadjoint representation. (\({\mathfrak g}^*\) is the space dual to the Lie algebra \({\mathfrak g}\) of G.) A reduction procedure of interest in Hamiltonian mechanics consists of forming the quotient \(M_{\mu}=J^{-1}(\mu)/G_{\mu}\) where \(\mu\) is an element of \({\mathfrak g}^*\) and \(G_{\mu}\) is its coadjoint isotropy group. The purpose of this paper is to extend this reduction procedure to actions by a symplectic groupoid. The reduction procedure, commonly thought of as inherently “symplectic”, is seen here to have a purely algebraic expression in the context of groupoid actions.
Reviewer: W.Satzer

37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70H30 Other variational principles in mechanics
Full Text: DOI
[1] Abraham, R. and Marsden, ], E., Foundations of mechanic s> 2nd ed. New York : Benjamin/ Cummings, 1978.
[2] Brown, R., Danesh-Naruie, G. and Hardy, J. P. L., Topological groupoids : II. Covering morphisms and G-spaces, Math. Nachr.s 74 (1976), 143-156. · Zbl 0281.22002
[3] Coste, A., Dazord, P. and Weinstein, A., Groupoides symple cliques. Publications du Department de Mathtmatiques de V University de Lyon I, 2/A (1987). · Zbl 0668.58017
[4] Drinfel’d, V. G., Hamiltonian structures on Lie groups, Lie bialgebras and the geo- metric meaning of the classical Yang-Baxter equations, Soviet Math. Dokl.9 27(1983), 68-71. · Zbl 0526.58017
[5] Ehresmann, C., Categories topologiques et categories diff£rentiables, Coll. de Geom. Differ. Globale, Bruxelles (1959), 137-150. · Zbl 0205.28202
[6] Guillemin, V. and Sternberg, S., The moment map and collective motion, Ann. of Phys., 127 (1980), 220-253. · Zbl 0453.58015
[7] Karasev, M. V., Analogues of objects of Lie group theory for nonlinear Poisson brackets. Math. USSR Izv.3 28 (1987), 497-527. · Zbl 0624.58007
[8] Kazhdan, D., Kostant, B. and Sternberg, S.,Hamiltonian group actions and dynamical systems of Calogero type, Comm. Pure and Appl. Math., 31 (1978), 481-507. · Zbl 0368.58008
[9] Lie, S., Theorie der Transformationsgruppen, Zweiter Abschnitt unter Mitwurking von Prof. Dr. Friedrich Engel, Teubner, Leibzig, 1890. · JFM 20.0368.01
[10] Marie, C.-M., Symplectic manifolds, dynamical groups, and Hamiltonian mechanics, Diff. Geometry and relativity, Reidel (1976), 243-269. · Zbl 0369.53042
[11] Marsden, J. and Ratiu, T., Reduction of Poisson manifolds, Lett in Math. Phys., 11 (1986), 161-169. · Zbl 0602.58016
[12] Marsden, J., Ratiu, T. and Weinstein, A., Semidirect products and reduction in mecha- nics, Trans. Amer. Math. Soc., 281 (1984), 147-177. · Zbl 0529.58011
[13] Marsden, J. and Weinstein, A., Reduction of symplectic manifolds with symmetry, Rep. Math. Phys., 5 (1974), 121-129. · Zbl 0327.58005
[14] Meyer, K. R., Symmetries and integrals in mechanics, in Dynamical Systems, M. M. Peixoto, ed., Academic Press, New York (1973), 259-272. · Zbl 0293.58009
[15] Renault, J., A groupoid approach to C*-algebras, Lecture Notes in Math., 793 (1980). · Zbl 0433.46049
[16] Semenov-Tian-Shansky, M. A., Dressing transformations and Poisson group actions, Publ. RIMS, Kyoto Univ., 21 (1985), 1237-1260. · Zbl 0673.58019
[17] Smale, S., Topology and mechanics. I, Invent. Math., 10 (1970), 305-331. · Zbl 0202.23201
[18] Weinstein, A., Lectures on symplectic manifolds, CBMS Regional Conf. Ser. in Math., 29 (1977). · Zbl 0406.53031
[19] The local structure of Poisson manifolds,/. Diff. Geom., 18 (1983), 523-557. · Zbl 0524.58011
[20] , Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc., 16(1987), 101-104. · Zbl 0618.58020
[21] , Poisson geometry of the principal series and nonlinearizable structures,/. Diff. Geom., 25 (1987), 55-73. · Zbl 0592.58024
[22] , Coisotropic calculus and Poisson groupoids, Subwitted to J. Math Soc Japan. · Zbl 0642.58025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.