Geodesic
Hamiltonian reductions of free particles under polar actions of compact Lie groups
Teoretičeskaâ i matematičeskaâ fizika, Tome 155 (2008) no. 1, pp. 161-176 Cet article a éte moissonné depuis la source Math-Net.Ru

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We investigate classical and quantum Hamiltonian reductions of free geodesic systems of complete Riemannian manifolds. We describe the reduced systems under the assumption that the underlying compact symmetry group acts in a polar manner in the sense that there exist regularly embedded, closed, connected submanifolds intersecting all orbits orthogonally in the configuration space. Hyperpolar actions on Lie groups and on symmetric spaces lead to families of integrable systems of the spin Calogero–Sutherland type.
Keywords: Hamiltonian reduction, polar action, integrable system. 
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L. Feher; B. G. Pusztai. Hamiltonian reductions of free particles under polar actions of compact Lie groups. Teoretičeskaâ i matematičeskaâ fizika, Tome 155 (2008) no. 1, pp. 161-176. http://geodesic.mathdoc.fr/item/TMF_2008_155_1_a13/

[1] D. Kazhdan, B. Kostant, S. Sternberg, Comm. Pure Appl. Math., 31:4 (1978), 481–507 | DOI | MR | Zbl

[2] P. I. Etingof, I. B. Frenkel, A. A. Kirillov Jr., Duke Math. J., 80:1 (1995), 59–90 ; arXiv: hep-th/9407047 | DOI | MR | Zbl

[3] M. A. Olshanetsky, A. M. Perelomov, Phys. Rep., 71:5 (1981), 313–400 | DOI | MR

[4] M. A. Olshanetsky, A. M. Perelomov, Phys. Rep., 94:6 (1983), 313–404 | DOI | MR

[5] J. Gibbons, T. Hermsen, Physica D, 11:3 (1984), 337–348 | DOI | MR | Zbl

[6] N. Nekrasov, “Infinite-dimensional algebras many-body systems and gauge theories”, Moscow Seminar in Mathematical Physics, AMS Transl. Ser. 2, 191, eds. A. Yu. Morozov, M. A. Olshanetsky, Amer. Math. Soc., Providence, RI, 1999, 263–299 | MR | Zbl

[7] N. Reshetikhin, Lett. Math. Phys., 63:1 (2003), 55–71 ; arXiv: math.QA/0202245 | DOI | MR | Zbl

[8] D. Alekseevsky, A. Kriegl, M. Losik, P. W. Michor, Publ. Math. Debrecen, 62:3–4 (2003), 247–276 ; arXiv: math.DG/0102159 | MR | Zbl

[9] A. Oblomkov, Adv. Math., 186:1 (2004), 153–180 ; arXiv: math.RT/0202076 | DOI | MR | Zbl

[10] S. Hochgerner, Singular cotangent bundle reduction and spin Calogero–Moser systems, arXiv: math.SG/0411068 | MR

[11] L. Fehér, B. G. Pusztai, Nucl. Phys. B, 734:3 (2006), 304–325 ; arXiv: math-ph/0507062 | DOI | MR | Zbl

[12] L. Fehér, B. G. Pusztai, Nucl. Phys. B, 751:3 (2006), 436–458 ; arXiv: math-ph/0604073 | DOI | MR | Zbl

[13] L. Fehér, B. G. Pusztai, Lett. Math. Phys., 79:3 (2007), 263–277 ; arXiv: math-ph/0609085 | DOI | MR | Zbl

[14] R. Palais, C.-L. Terng, Trans. Amer. Math. Soc., 300:2 (1987), 771–789 | DOI | MR | Zbl

[15] L. Conlon, J. Differential Geom., 5 (1971), 135–147 | DOI | MR | Zbl

[16] J. Szenthe, Period. Math. Hungar., 15:4 (1984), 281–299 | DOI | MR | Zbl

[17] E. Heintze, R. Palais, C.-L. Terng, G. Thorbergsson, “Hyperpolar actions on symmetric spaces”, Geometry, Topology and Physics for Raoul Bott, Conf. Proc. Lecture Notes Geom. Topology, IV, ed. S.-T. Yau, International Press, Cambridge, MA, 1995, 214–245 | MR | Zbl

[18] A. Kollross, Polar actions on symmetric spaces, arXiv: math.DG/0506312 | MR

[19] S. Hochgerner, Spinning particles in a Yang–Mills field, arXiv: math.SG/0602062

[20] Gruppy Li i algebry Li – 1, Itogi nauki i tekhniki. Sovrem. problemy matem. Fundam. napr., 20, eds. A. L. Onischik, R. V. Gamkrelidze, VINITI, M., 1988 | MR | MR | Zbl

[21] J.-P. Ortega, T. S. Ratiu, Momentum Maps and Hamiltonian Reduction, Progr. Math., 222, Birkhäuser, Boston, MA, 2004 | MR | Zbl

[22] S. Tanimura, T. Iwai, J. Math. Phys., 41:4 (2000), 1814–1842 ; arXiv: math-ph/9907005 | DOI | MR | Zbl

[23] N. P. Landsman, Mathematical Topics Between Classical and Quantum Mechanics, Springer Monogr. Math., Springer, New York, 1998 | DOI | MR | Zbl

[24] S. Helgason, Groups and Geometric Analysis, Pure Appl. Math., 113, Academic Press, Orlando, FL, 1984 | MR | Zbl

[25] A. W. Knapp, Lie Groups Beyond an Introduction, Progr. Math., 140, Birkhäuser, Boston, MA, 2002 | MR | Zbl

[26] L. Fehér, B. G. Pusztai, On the self-adjointness of certain reduced Laplace–Beltrami operators, (to appear) arXiv: 0707.2708 | MR

[27] G. Kunstatter, Class. Quant. Grav., 9:6 (1992), 1469–1485 | DOI | MR | Zbl

[28] L. Fehér, B. G. Pusztai, Twisted spin Sutherland models from quantum Hamiltonian reduction, , submitted to J. Phys. A arXiv: 0711.4015 | MR