Journal of Differential Geometry

Homological reduction of constrained Poisson algebras

Jim Stasheff

Full-text: Open access

Article information

Source
J. Differential Geom. Volume 45, Number 1 (1997), 221-240.

Dates
First available in Project Euclid: 26 June 2008

Permanent link to this document
https://projecteuclid.org/euclid.jdg/1214459757

Digital Object Identifier
doi:10.4310/jdg/1214459757

Mathematical Reviews number (MathSciNet)
MR1443334

Zentralblatt MATH identifier
0874.58020

Subjects
Primary: 17B55: Homological methods in Lie (super)algebras
Secondary: 17B81: Applications to physics 58F05

Citation

Stasheff, Jim. Homological reduction of constrained Poisson algebras. J. Differential Geom. 45 (1997), no. 1, 221--240. doi:10.4310/jdg/1214459757. https://projecteuclid.org/euclid.jdg/1214459757


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References

  • [1] J.M. Arms, M. J. Gotay and G. Jennings, (Geometric and algebraic) reduction for singular momentum maps, Adv. Math. 79 (1990) 43-103.
  • [2] I.A. Batalin and E.S. Fradkin, A generalized canonical formalism and quantization of reducible gauge theories, Phys. Lett. 122B (1983) 157-164.
  • [3] I.A. Batalin and G.S. Vilkovisky, Existence theorem for gauge algebra, J. Math. Phys. 26 (1985) 172-184.
  • [4] I.A. Batalin and G.S. Vilkovisky, Quantization of gauge theories with linearly dependent generators, Phys. Rev. D 28 (1983) 2567-2582.
  • [5] I.A. Batalin and G.S. Vilkovisky, Relativistic S-matrix of dynamical systems with boson and fermion constraints, Phys. Lett. 6 9 B (1977) 309-312.
  • [6] C. Becchi, A. Rouet and R. Stora, Renormalization of the abelian Higgs-Kibble model, Comm. Math. Phys. 42 (1975) 127-162.
  • [7] A. Borel, Sur la cohomologie des espaces fibres principaux et des espaces homogenes de groupes de Lie compacts, Ann. of Math. 57 (1953) 115-207.
  • [8] A.D. Browning and D. McMullen, The Batalin, Fradkin, Vilkovisky formalism for higher order theories, J. Math. Phys. 28 (1987) 438-444.
  • [9] C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948) 85-124.
  • [10] P.A.M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School Monograph Ser. 2, 1964.
  • [11] M. Dubois-Violette, Systemes dynamiques constraints: l'approche homologique, Ann. Inst. Fourier (Grenoble) 37 (1987) 45-57.
  • [12] E.S. Fradkin and T.E. Fradkina, Quantization of relativistic system with boson and fermion first and second-constraints, Phys. Lett. 7 2 B (1978) 343-348.
  • [13] J. Fisch, M. Henneaux, J. Stasheff and C. Teitelboim, Existence, uniqueness and cohomology of the classical BRST change with ghosts of ghosts, Comm. Math. Phys. 120 (1989) 379-407.
  • [14] E.S. Fradkin and G.S. Vilkovisky, Quantization of relativistic systems with constraints, Phys. Lett. 5 5 B (1975) 224-226.
  • [15] M.J. Gotay, J. Isenberg, J.E. Marsden, R. Montgomery, J. Sniatycki and P. Yasskin, Constraints and momentum mappings in relativistic field theory, Preprint.
  • [16] V.K.A.M. Gugenheim, On a perturbation theory for the homology of a loop space, J. Pure Appl. Algebra 25 (1982) 197-205.
  • [17] V.K.A.M. Gugenheim, L. Lambe and J. Stasheff, Algebraic aspects of Chen's twisting cochain, Illinois J. Math. 34 (1990) 485-502.
  • [18] V.K.A.M. Gugenheim and J.P. May, On the theory and application of torsion products, Mem. Amer. Math. Soc. 142 (1974).
  • [19] V.K.A.M. Gugenheim and J.D. Stasheff, On perturbations and Aoo-structures, Bull. Soc. Math. Belg. 38 (1986) 237-245.
  • [20] V. Guillemin and S. Sternberg, Sympletic Techniques in Physics, Cambridge University Press, Cambridge, 1984.
  • [21] J. L. Heitsch and S. E. Hurder, Geometry of foliations, J. Differential Geom. 20 (1984) 291-309.
  • [22] M. Henneaux, Hamiltonian form of the path integral for theories with a gauge freedom, Phys. Rev. 126 (1985) 1-66.
  • [23] M. Henneaux and Claudio Teitelboim, Quantization of gauge systems, Princeton Univ. Press, Princeton, NJ, 1992.
  • [24] J. Huebschmann, Poisson cohomology and quantization, J. Reine Angew. Math. 408 (1990) 57-113.
  • [25] J. Huebschmann, On the quantization of Poisson algebras, Symplectic Geom. Math. Phys., actes du colloque en l'honneur de J.M. Souriau, (P. Donato, C. Duval, J. Elhadad, G. M. Tuynman, eds.), Progr. Math. Vol. 99, Birkhaeuser, Basel, 1991, 204-233.
  • [26] J. Huebschmann, Extensions of Lie Rinehart algebras. I. The Chern-Weil construction, Preprint, 1989
  • [27] J. Huebschmann, Extensions of Lie-Rinehart algebras. II. The spectral sequence and the Weil algebra, Preprint, 1989
  • [28] J. Huebschmann, Graded Lie-Rinehart algebras, graded Poisson algebras, and BRST-quantization. I. The finitely generated case, Preprint, 1990.
  • [29] J. Huebschmann, Perturbation thoery and small methods for the chains of certain induced fibre spaces, Habilitationsschrift, Universität Heidelberg, 1984.
  • [30] J. Huebschmann, The cohomology of F^ q. The additive structure, J. Pure Appl. Algebra 45 (1987) 73-91.
  • [31] J. Huebschmann, Perturbation theory and free resolutions for nilpotent groups of class 2, J. Algebra 126 (1989) 348-399 [32] --, Minimal free multi models for chain algebras, Preprint 1990.
  • [33] J. Huebschmann and T. Kadeishvili, Small models for chain algebras, Math. Z. 207 (1991) 245-280.
  • [34] J. Herz, Pseudo-algebres de Lie, C R. Acad. Sci. Paris 236 (1953) 1935-1937.
  • [35] T. Kimura, Generalized classical BRST and reduction of Poisson manifolds, Comm. Math. Phys.151 (1993) 155-182.
  • [36] B. Kostant and S. Sternberg, Symplectic reduction, BRS cohomology and infinitedimensional Clifford algebras, Ann. Physics 176 (1987) 49-113.
  • [37] J.-L. Koszul, Sur un type d'algebres differentielles en rapport avec la transgression, Colloque de Topologie, Bruxelles, 1950, CBRM, Liege. R. Sjamaar and E. Lerman, Stratified symplectic spaces and reduction, Ann. of Math. 134 (1991) 375-422. J.E. Marsden & A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974) 121-130. R. Palais, The cohomology of Lie rings, Proc. Sympos. Pure Math. Vol. III, Amer. Math. Soc, 1961, 130-137. D. Quillen, Rational homotopy theory, Ann. of Math. 90 (1969) 205-295. G. Rinehart, Differential forms for general commutative algebras, Trans. Amer. Math. Soc. 108 (1963) 195-222. J. Sniatycki & A. Weinstein, Reduction and quantization for singular momentum mappings, Lett. Math. Phys. 7 (1983) 155-161. J.D. Stasheff, Constrained Hamiltonians: A homological approach, Proc. Winter School on Geom. Phys., Suppl. Rend. Circ. Mat. Palermo, II, 16 (1987) 239-252., Constrained Poisson algebras and strong homotopy representations, Bull. Amer. Math. Soc. (1988) 287-290. D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Etudes Sci. Publ. Math. 47 (1977) 269-331. J. Tate, Homology of Noetherian rings and local rings, Illinois J. Math. 1 (1957) 14-27. I. V. Tyutin, Gauge invariance in field theory and statistical physics in operator formulation (in Russian), Lebedev Phys. Inst., Preprint, 39 (1975). A. Weinstein, Coisotropic calculus and Poisson groupoids, J. Math. Soc. Japan 40 (1988) 705-727., Symplectic V-manifolds, periodic orbits of Hamiltonian systems, and the volume of certain Riemannian manifolds, Comm. Pure Appl. Math. 30 (1977) 265-271.