Singular reduction of Dirac structures

Jotz, M.; Ratiu, T.; Śniatycki, J.

doi:10.1090/S0002-9947-2011-05220-7

Singular reduction of Dirac structures
HTML articles powered by AMS MathViewer

by M. Jotz, T. S. Ratiu and J. Śniatycki PDF

Trans. Amer. Math. Soc. 363 (2011), 2967-3013 Request permission

Abstract:

The regular reduction of a Dirac manifold acted upon freely and properly by a Lie group is generalized to a nonfree action. For this, several facts about $G$-invariant vector fields and one-forms are shown.

References

Judith M. Arms, Richard H. Cushman, and Mark J. Gotay, A universal reduction procedure for Hamiltonian group actions, The geometry of Hamiltonian systems (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 22, Springer, New York, 1991, pp. 33–51. MR 1123275, DOI 10.1007/978-1-4613-9725-0_{4}
N. Aronszajn. Subcartesian and subriemannian spaces. Notices Amer. Math. Soc., 14:111, 1967.
Edward Bierstone, Lifting isotopies from orbit spaces, Topology 14 (1975), no. 3, 245–252. MR 375356, DOI 10.1016/0040-9383(75)90005-1
G. Blankenstein. Implicit Hamiltonian Systems: Symmetry and Interconnection. Ph.D.Thesis, University of Twente, 2000.
Guido Blankenstein and Tudor S. Ratiu, Singular reduction of implicit Hamiltonian systems, Rep. Math. Phys. 53 (2004), no. 2, 211–260. MR 2068644, DOI 10.1016/S0034-4877(04)90013-4
G. Blankenstein and A. J. van der Schaft, Symmetry and reduction in implicit generalized Hamiltonian systems, Rep. Math. Phys. 47 (2001), no. 1, 57–100. MR 1823009, DOI 10.1016/S0034-4877(01)90006-0
K. Buchner, M. Heller, P. Multarzyński, and W. Sasin. Literature on differential spaces. Acta Cosmologica, 19:111–129, 1993.
Henrique Bursztyn, Gil R. Cavalcanti, and Marco Gualtieri, Reduction of Courant algebroids and generalized complex structures, Adv. Math. 211 (2007), no. 2, 726–765. MR 2323543, DOI 10.1016/j.aim.2006.09.008
Theodore James Courant, Dirac manifolds, Trans. Amer. Math. Soc. 319 (1990), no. 2, 631–661. MR 998124, DOI 10.1090/S0002-9947-1990-0998124-1
Ted Courant and Alan Weinstein, Beyond Poisson structures, Action hamiltoniennes de groupes. Troisième théorème de Lie (Lyon, 1986) Travaux en Cours, vol. 27, Hermann, Paris, 1988, pp. 39–49. MR 951168
Richard Cushman and Jȩdrzej Śniatycki, Differential structure of orbit spaces, Canad. J. Math. 53 (2001), no. 4, 715–755. MR 1848504, DOI 10.4153/CJM-2001-029-1
Richard Cushman, Reduction, Brouwer’s Hamiltonian, and the critical inclination, Celestial Mech. 31 (1983), no. 4, 401–429. MR 737806, DOI 10.1007/BF01230294
J.J. Duistermaat. Dynamical systems with symmetry. Available at http://www.math.uu.nl/ people/duis/.
J. J. Duistermaat and J. A. C. Kolk, Lie groups, Universitext, Springer-Verlag, Berlin, 2000. MR 1738431, DOI 10.1007/978-3-642-56936-4
Rui Loja Fernandes, Juan-Pablo Ortega, and Tudor S. Ratiu, The momentum map in Poisson geometry, Amer. J. Math. 131 (2009), no. 5, 1261–1310. MR 2555841, DOI 10.1353/ajm.0.0068
M. Jotz and T.S. Ratiu. Dirac and nonholonomic reduction. Preprint, arXiv:0806.1261v2, 2008.
M. Jotz and T. S. Ratiu, Induced Dirac structure on isotropy type manifolds., arXiv:1008.2280v1, to appear in “Transformation groups” (2010).
—, Optimal Dirac reduction., arXiv:1008.2283v1 (2010).
M. Jotz, T.S. Ratiu, and M. Zambon, Invariant frames for vector bundles and applications., Preprint (2010).
Paulette Libermann and Charles-Michel Marle, Symplectic geometry and analytical mechanics, Mathematics and its Applications, vol. 35, D. Reidel Publishing Co., Dordrecht, 1987. Translated from the French by Bertram Eugene Schwarzbach. MR 882548, DOI 10.1007/978-94-009-3807-6
T. Lusala and J. Śniatycki. Stratified subcartesian spaces. arXiv:0805.4807v1, To appear in Canad. Math. Bull.
Jerrold E. Marsden and Tudor Ratiu, Reduction of Poisson manifolds, Lett. Math. Phys. 11 (1986), no. 2, 161–169. MR 836071, DOI 10.1007/BF00398428
Jerrold Marsden and Alan Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Mathematical Phys. 5 (1974), no. 1, 121–130. MR 402819, DOI 10.1016/0034-4877(74)90021-4
Hideyuki Matsumura, Commutative algebra, 2nd ed., Mathematics Lecture Note Series, vol. 56, Benjamin/Cummings Publishing Co., Inc., Reading, Mass., 1980. MR 575344
Juan-Pablo Ortega and Tudor S. Ratiu, Momentum maps and Hamiltonian reduction, Progress in Mathematics, vol. 222, Birkhäuser Boston, Inc., Boston, MA, 2004. MR 2021152, DOI 10.1007/978-1-4757-3811-7
Richard S. Palais, On the existence of slices for actions of non-compact Lie groups, Ann. of Math. (2) 73 (1961), 295–323. MR 126506, DOI 10.2307/1970335
Markus J. Pflaum, Analytic and geometric study of stratified spaces, Lecture Notes in Mathematics, vol. 1768, Springer-Verlag, Berlin, 2001. MR 1869601
Gerald W. Schwarz, Smooth functions invariant under the action of a compact Lie group, Topology 14 (1975), 63–68. MR 370643, DOI 10.1016/0040-9383(75)90036-1
Roman Sikorski, Abstract covariant derivative, Colloq. Math. 18 (1967), 251–272. MR 222799, DOI 10.4064/cm-18-1-251-272
Roman Sikorski, Differential modules, Colloq. Math. 24 (1971/72), 45–79. MR 482794, DOI 10.4064/cm-24-1-45-79
Roman Sikorski, Wstęp do geometrii różniczkowej, Biblioteka Matematyczna, Tom 42. [Mathematics Library, Vol. 42], Państwowe Wydawnictwo Naukowe, Warsaw, 1972 (Polish). MR 0467544
Jȩdrzej Śniatycki, Integral curves of derivations on locally semi-algebraic differential spaces, Discrete Contin. Dyn. Syst. suppl. (2003), 827–833. Dynamical systems and differential equations (Wilmington, NC, 2002). MR 2018191
Jȩdrzej Śniatycki, Orbits of families of vector fields on subcartesian spaces, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 7, 2257–2296 (English, with English and French summaries). MR 2044173, DOI 10.5802/aif.2006
P. Stefan, Accessibility and foliations with singularities, Bull. Amer. Math. Soc. 80 (1974), 1142–1145. MR 353362, DOI 10.1090/S0002-9904-1974-13648-7
P. Stefan, Accessible sets, orbits, and foliations with singularities, Proc. London Math. Soc. (3) 29 (1974), 699–713. MR 362395, DOI 10.1112/plms/s3-29.4.699
P. Stefan, Integrability of systems of vector fields, J. London Math. Soc. (2) 21 (1980), no. 3, 544–556. MR 577729, DOI 10.1112/jlms/s2-21.3.544
Héctor J. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc. 180 (1973), 171–188. MR 321133, DOI 10.1090/S0002-9947-1973-0321133-2
Izu Vaisman, Lectures on the geometry of Poisson manifolds, Progress in Mathematics, vol. 118, Birkhäuser Verlag, Basel, 1994. MR 1269545, DOI 10.1007/978-3-0348-8495-2
P. Walczak, A theorem on diffeomorphisms in the category of differential spaces, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 21 (1973), 325–329 (English, with Russian summary). MR 317348
Hermann Weyl, The classical groups, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Their invariants and representations; Fifteenth printing; Princeton Paperbacks. MR 1488158