References

J. Śniatycki

doi:10.1017/CBO9781139136990.010

References

Published online by Cambridge University Press: 05 June 2013

J. Śniatycki

Show author details

J. Śniatycki: Affiliation:
University of Calgary

Book contents

Get access

Summary

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'

Type: Chapter
Information: Differential Geometry of Singular Spaces and Reduction of Symmetry , pp. 228 - 232

DOI: https://doi.org/10.1017/CBO9781139136990.010 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abraham, R. and J., Robbin (1967), Transversal Mappings and Flows, Academic Press, New York.

Aprodu, M. and J., Nagel (2010), Koszul Cohomology and Algebraic Geometry, University Lecture Series, vol. 52, American Mathematical Society, Providence, RI.

Arms, J.M. (1996), ‘Reduction of Poisson algebras at non-zero momentum values’, J. Geom. Phys. 21 81–95.Google Scholar

Arms, J.M., J.E., Marsden and V., Moncrief (1981), ‘Symmetry and bifurcations of momentum mappings’, Commun. Math. Phys. 78 455–478.Google Scholar

Arms, J.M., M., Gotay and G., Jennings (1990), ‘Geometric and algebraic reduction for singular momentum mappings’, Adv. Math. 79 43–103.Google Scholar

Arms, J.M., R., Cushman and M.J., Gotay (1991), ‘A universal reduction procedure for Hamiltonian group actions’. In The Geometry of Hamiltonian Systems, T., Ratiu (ed.), MSRI Publication 20, Springer, Berlin, pp. 33–51.

Aronszajn, N. (1967), ‘Subcartesian and subRiemannian spaces’, Not. Amer. Math. Soc., 14 111.Google Scholar

Auslander, L. and B., Kostant (1971), ‘Polarizations and unitary representations of solvable Lie groups’, Invent. Math. 14 255–354.Google Scholar

Bates, L. (2007), ‘A smooth invariant not a smooth function of polynomial invariants’, Proc. Amer. Math. Soc. 135 3039–3040.Google Scholar

Bates, L. and J., Śniatycki (1993), ‘Non-holonomic reduction’, Rep. Math. Phys. 32 99–115.Google Scholar

Bates, L., R., Cushman, M., Hamilton and J., Śniatycki (2009), ‘Quantization of singular reduction’, Rev. Math. Phys. 21 315–371.Google Scholar

Bierstone, E. (1975), ‘Lifting isotopies from orbit spaces’, Topology 14 245–252.Google Scholar

Bierstone, E. (1980), The structure of orbit spaces and the singularities of equivariant mappings, Monografias de Matemática, vol. 35, Instituto de Matemática Pura e Applicada, Rio de Janeiro.

Binz, E., J., Šniatycki and H., Fischer (2006), Geometry of Classical Fields, Dover Publications, Mineola, NY.

Blankenstein, G. (2000), Implicit Hamiltonian Systems: Symmetry and Interconnection, PhD thesis, University of Twente.

Blankenstein, G. and T., Ratiu (2004), ‘Singular reduction of implicit Hamiltonian systems’, Rep. Math. Phys. 53 211–260.Google Scholar

Blankenstein, G. and A.J., van der Schaft (2001), ‘Symmetry and reduction in implicit generalized Hamiltonian systems’, Rep. Math. Phys. 47 57–100.Google Scholar

Bleuler, K. (1950), ‘Eine neue Methode zur Behandlung der longitudinalen und skalaren Photonen’, Helv. Phys. Acta 23 567.Google Scholar

Buchner, K., M., Heller, P., Multarzyński and W., Sasin (1993), ‘Literature on differential spaces’, Acta Cosmol. 19 111–129.Google Scholar

Bursztyn, H., G.R., Cavalcanti and M., Gualtieri (2007), ‘Reduction of Courant algebroids and generalized complex structures’, Adv. Math. 211 726–765.Google Scholar

Cegiełka, K. (1974), ‘Existence of smooth partition of unity and of scalar product in a differential space’, Demonstratio Math. 6 493–504.Google Scholar

Courant, T.J. (1990), ‘Dirac manifolds’, Trans. Amer. Math. Soc. 319 631–661.Google Scholar

Courant, T. and A., Weinstein (1988), ‘Beyond Poisson structures’. In Seminaire Sud-Rhodanien de Geometrie VIII, Travaux en Cours 27, Hermann, Paris, pp. 39–49.

Cushman, R. (1983), ‘Reduction, Brouwer's Hamiltonian and the critical inclination’, Celest. Mech. 31 401–429.Google Scholar

Cushman, R.H. and L.M., Bates (1997), Global Aspects of Classical Integrable Systems, Birkhäuser, Basel.

Cushman, R. and J., Šniatycki (2001), ‘Differential structure of orbit spaces’, Canad. J. Math. 53 235–248.Google Scholar

Cushman, R., J.J., Duistermaat and J., Šniatycki (2010), Geometry of Nonholonomically Constrained Systems, World Scientific, Singapore.

Dirac, P.A.M. (1950), ‘Generalized Hamiltonian dynamics’, Canad. J. Math. 2 129–148.Google Scholar

Dirac, P.A.M. (1964), Lectures on Quantum Mechanics, Belfer Graduate School of Science Monograph Series, Yeshiva University, New York.

Dorfman, I. (1993), Dirac Structures and Integrability of Nonlinear Evolution Equations, Wiley, Chichester.

Duistermaat, J.J. (2004), Lectures on dynamical systems with symmetry given at Utrecht Spring School on Lie Groups, unpublished.

Duistermaat, J.J. and J.A.C., Kolk (2000), Lie Groups, Springer, Berlin.

Duval, C., J., Elhadad, M.J., Gotay and G.M., Tuynman (1990), ‘Nonunimodularity and the quantization of pseudo-rigid body’. In Hamiltonian Systems, Transformation Groups and Spectral Transform Methods, J., Harnad and J.E., Marsden (eds.), CRM, Montréal, pp. 149–160.

Duval, C., J., Elhadad, M.J., Gotay, J., Šniatycki and G.M., Tuynman (1991), ‘Quantization and bosonic BRST theory’, Ann. Phys. 206 1–26.Google Scholar

Epstein, M. and J., Šniatycki (2006), ‘Fractal mechanics’, Physica D 220 54–68.Google Scholar

Guillemin, V. and S., Sternberg (1982), ‘Geometric quantization and multiplicities of group representations’, Invent. Math. 67 515–538.Google Scholar

Guillemin, V. and S., Sternberg (1984), Symplectic Techniques in Physics, Cambridge University Press, Cambridge.

Gupta, S.N. (1950), ‘Theory of longitudinal photons in quantum electrodynamics’, Proc. Phys. Soc. A63 681.Google Scholar

Hall, B. and W., Kirwin (2007), ‘Unitarity in quantization commutes with reduction’, Commun. Math. Phys. 275(2) 401–422.Google Scholar

Huebschmann, J. (2006), ‘Kähler quantization and reduction’, J. Reine Angew. Math. 591 75–109.Google Scholar

Jotz, M. and T., Ratiu (2012), ‘Dirac structures, nonholonomic systems and reduction’, Rep. Math. Phys. 69 5–56.Google Scholar

Jotz, M., T., Ratiu and J., Šniatycki (2011), ‘Singular reduction of Dirac structures’, Trans. Amer. Math. Soc. 363 2967–3013.Google Scholar

Kelley, J.L. (1955), General Topology, Van Nostrand, New York (reprinted by Springer, New York, 1975).

Kimura, T. (1993), ‘Generalized classical BRST cohomology and reduction of Poisson manifolds’, Commun. Math. Phys. 151 155–182.Google Scholar

Kirillov, A.A. (1962), ‘Unitary representations of nilpotent Lie groups’, Russian Math. Surveys 17 53–104.Google Scholar

Kobayashi, S. and K., Nomizu (1963), Foundations of Differential Geometry, vol. 1, Wiley, New York.

Koon, W.S. and J.E., Marsden (1998), ‘The Poisson reduction of nonholonomic mechanical systems’, Rep. Math. Phys. 42 101–134.Google Scholar

Kostant, B. (1966), ‘Orbits, symplectic structures, and representation theory’. In Proceedings of US—Japan Seminar in Differential Geometry, Kyoto, 1965, Nippon Hyoronsha, Tokyo.

Kostant, B. (1970), ‘Quantization and unitary representations’. In Lectures in Modern Analysis and Applications III, R.M., Dudley, J., Feldman, B., Kostant, R.P., Lang-lands, E.M., Stein and C.T., Taam (eds.), Lecture Notes in Mathematics, vol. 170, Springer, Berlin, pp. 87–207.

Li, H. (2008), ‘Singular unitarity in “quantization commutes with reduction”’, J. Geom. Phys. 58 –742.Google Scholar

Libermann, P. and C.-M., Marle (1987), Symplectic Geometry and Analytical Mechanics (translated from French by B.E., Schwarzbach), Mathematics and Its Applications, vol. 35, Reidel, Dordrecht.

Lusala, T. and J., Śniatycki (2011), ‘Stratified subcartesian spaces’, Canad. Math. Bull. 54 693–705.Google Scholar

Lusala, T. and J., Śniatycki (to appear), ‘Slicet heorem for differentials paces and reduction by stages’, Demonstratio Math., in press.

Lusala, T., J., Śniatycki and J., Watts (2010), ‘Regular points of a subcartesian space’, Canad. Math. Bull. 53 340–346.Google Scholar

Marsden, J.E. and A., Weinstein (1974), ‘Reduction of s ymplectic manifolds with symmetry’, Rep. Math. Phys. 5 121–130.Google Scholar

Marsden, J.E., G., Misiołek, J.-P., Ortega, M., Perlmutter and T., Ratiu (2007), Hamiltonian Reduction by Stages, Lecture Notes in Mathematics, vol. 1913, Springer, Berlin.

Marshall, C.D. (1975a), ‘Calculus on subcartesian spaces’, J. Differential Geom. 10 551–573.Google Scholar

Marshall, C.D. (1975b), ‘The de Rham cohomology on subcartesian spaces’, J. Differential Geom. 10 575–588.Google Scholar

Mather, J.N. (1973), ‘Stratifications and mappings’. In Dynamical Systems, M.M., Peixoto (ed.), Academic Press, New York, pp. 195–232.

Meyer, K.R. (1973), ‘Symmetries and integrals in mechanics’. In Dynamical Systems, M.M., Peixoto (ed.), Academic Press, New York, pp. 259–273.

Noether, E. (1918), ‘Invariante Variationsprobleme’, Kgl. Ges. Wiss. Nachr. Göttingen Math. Physik 2 235–257.Google Scholar

Ortega, J.-P. and T.S., Ratiu (2002), ‘The optimal momentum map’. In Geometry, Mechanics and Dynamics: Volume in Honor of the 60th Birthday of J. E. Marsden, P., Newton, P., Holmes and A., Weinstein (eds.), Springer, Berlin, pp. 329–362.

Ortega, J.-P. and T.S., Ratiu (2004), Momentum Maps and Hamiltonian Reduction, Birkhäuser, Boston.

Palais, R. (1961), ‘On the existence of slices for actions of noncompact Lie groups’, Ann. of Math. 73 295–323.Google Scholar

Pasternak-Winiarski, Z. (1984), ‘On some diff erentials tructures defined by actions of groups’. In Proceedings of the Conference on Differential Geometry and Its Applications, Part 1 (Nove Mesto na Morave, 1983), Charles University, Prague, pp. 105–115.

Pflaum, M.J. (2001), Analytic and Geometric Study of Stratified Spaces, Springer, Berlin.

Pukanszky, L. (1971), ‘Unitary representations of solvable groups’, Ann. Sci. Ecole Normale Sup. 4 457–608.Google Scholar

Sasin, W. (1986), ‘On some exterior algebra of differential forms over a differential space’, Demonstratio Math. 19 1063–1075.Google Scholar

Satake, I. (1957), ‘The Gauss–Bonnet theorem for V-manifolds’, J. Math. Soc. Japan 9 464–492.Google Scholar

van der Schaft, A.J. and B.M., Maschke (1994), ‘On the Hamiltonian formulation of nonholonomic mechanical systems’, Rep. Math. Phys. 34 225–233.Google Scholar

Schwarz, G. (1975), ‘Smooth functions invariant under the action of compact Lie groups’, Topology 14 63–68.Google Scholar

Sikorski, R. (1967), ‘Abstract covariant derivative’, Colloq. Math. 18 252–272.Google Scholar

Sikorski, R. (1972), Wstȩp do Geometrii Różniczkowej, PWN, Warsaw.

Sjamaar, R. (1995), ‘Holomorphic slices, symplectic reduction and multiplicities of representations’, Ann. of Math. 141 87–129.Google Scholar

Śniatycki, J. (1980), Geometric Quantization and Quantum Mechanics, Applied Mathematical Science, vol. 30, Springer, New York.

Śniatycki, J. (1983), ‘Constraints and quantization’. In Non-Linear Partial Differential Operators and Quantization Procedures: Proceedings of a Workshop Held at Clausthal, Federal Republic of Germany, 1981, S., Andersson and H., Doebner (eds.), Lecture Notes in Mathematics, vol. 1037, Springer, Berlin, pp. 301–334.

Śniatycki, J. (2003a), ‘Integral curves of derivations on locally semi-algebraic differential spaces’. In Dynamical Systems and Differential Equations (Proceedings of the Fourth International Conference on Dynamical Systems and Differential Equations, May 24–27, 2002, Wilmington, NC, USA), W. Feng, S. Hu and X. Lu (eds.), American Institute of Mathematical Sciences Press, Springfield, MO, pp. 825–831.

Śniatycki, J. (2003b), ‘Orbits of families of vector fields on subcartesian spaces’, Ann. Inst. Fourier (Grenoble) 53 2257–2296.Google Scholar

Śniatycki, J. (2005), ‘Poisson algebras in reduction of symmetries’, Rep. Math. Phys. 56 53–73.Google Scholar

Śniatycki, J. (2011), ‘Reduction of symmetries of Dirac structures’, J. Fixed Point Theory Appl. 10 339–358.Google Scholar

Śniatycki, J. (2012), ‘Commutation of geometric quantization and algebraic reduction’. In Mathematical Aspects of Quantization: Center for Mathematics at Notre Dame: Summer School and Conference on Mathematical Aspects of Quantization, May 31– June 10, 2011, Notre Dame University, Notre Dame, Indiana, S., Evens, M., Gekhman, B.C., Hall, X., Liu, C., Paulini (eds.), Contemporary Mathematics, vol. 583, American Mathematical Society, Providence, R.I., pp. 295–308.

Śniatycki, J. and A., Weinstein (1983), ‘Reduction and quantization for singular momentum mappings’, Lett. Math. Phys. 7 155–161.Google Scholar

Souriau, J.-M. (1966), ‘Quantification géométrique’, Commun. Math. Phys. 1 374–398.Google Scholar

Stefan, P. (1974), ‘Accessible sets, orbits and foliations with singularities’, Proc. London Math. Soc. 29 699–713.Google Scholar

Sussmann, H. (1973), ‘Orbits of families of vector fields and integrability of distributions’, Trans. A mer. Math. Soc. 180 171–188.Google Scholar

Thom, R. (1955–1956), ‘La stabilité topologique des applications différentiables’, Ann. Inst. Fourier (Grenoble) 6 43–87.Google Scholar

Walczak, P. (1973), ‘At heorem on diffeomorphisms in the category of differential spaces’, Bull. Acad. Polon. Sci., Ser. Sci. Math. Astronom. Phys. 21 325–329.Google Scholar

Warner, F. (1971), Foundations of Differentiable Manifolds and Lie Groups, Scott, Foresman and Company, Glenview, IL.

Watts, J. (2006), The Calculus on Subcartesian Spaces, MSc thesis, Department of Mathematics, University of Calgary.

Watts, J. (2012), Diffeologies, Differential Spaces, and Symplectic Geometry, PhD thesis, Department of Mathematics, University of Toronto.

Ważewski, T. (1934), ‘Sur un problème de caractère intègral relatif a l'équation ∂z/∂x + Q (x, y)∂z/∂y = 0’, Mathematica 8 103–116.Google Scholar

Weyl, H. (1946), The Classical Groups, second edition, Princeton University Press, Princeton, NJ.

Whitney, H. (1955), ‘On singularities of mappings of Euclidean spaces. I. Mappings of the plane into the plane’, Ann. of Math. 62 374–410.Google Scholar

Wilbour, D.C. (1993), Poisson Algebras and Singular Reduction of Constrained Hamiltonian Systems, PhD thesis, University of Washington.

Woodhouse, N.M.J. (1992), Geometric Quantization, second edition, Clarendon Press, Oxford.

Yoshimura, H. and J. E., Marsden (2007), ‘Reduction of Dirac structures and Hamilton-Pontryagin principle’, Rep. Math. Phys. 60 381–426.Google Scholar

Book contents

References

Summary

Access options

References

Save book to Kindle

Save book to Dropbox

Save book to Google Drive