Infinite-dimensional Lie groups and algebras in mathematical physics

Advances in Mathematical Physics

Volumn , Issue , 2010, Pages

  Schmid, Rudolf ^a  

^a EMORY UNIVERSITY   (United States)

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EID: 82155199315     PISSN: 16879120     EISSN: 16879139     Source Type: Journal    
DOI: 10.1155/2010/280362     Document Type: Review

References (63)

1. N. Bourbaki, Lie Groups and Lie Algebras, Herman, Paris, France, 1975.
2. H. H. Keller, Differential Calculus in Locally Convex Spaces, Lecture Notes in Mathematics, Vol. 417, Springer, Berlin, Germany, 1974.
3. A. Pressley and G. Segal, Loop Groups, Oxford Mathematical Monographs, Oxford University Press, New York, NY, USA, 1986.
4. J. Eichhorn and R. Schmid, "Form preserving diffeomorphisms on open manifolds," Annals of Global Analysis and Geometry, vol. 14, no. 2, pp. 147-176, 1996.
5. H. Omori, Infinite-Dimensional Lie Groups, vol. 158 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, USA, 1997.
6. R. Schmid, "The geometry of BRS transformations," Illinois Journal of Mathematics, vol. 34, no. 1, pp. 87-97, 1990.
7. A. Banyaga, The Structure of Classical Diffeomorphism Groups, vol. 400 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1997.
8. D. Bleecker, Gauge Theory and Variational Principles, vol. 1 of Global Analysis Pure and Applied Series A, Addison-Wesley, Reading, Mass, USA, 1981.
9. N. Bourbaki, Variétés Différentielles et Analytiques. Fascicule de Résultats, Actualités Scientifiques et Industrielles, No. 1347, Hermann, Paris, France, 1971.
10. D. G. Ebin and J. Marsden, "Groups of diffeomorphisms and the motion of an incompressible fluid," Annals of Mathematics, vol. 92, pp. 102-163, 1970.
11. G. A. Goldin, "Lectures on diffeomorphism groups in quantum physics," in Contemporary Problems in Mathematical Physics, pp. 3-93, World Scientific, Hackensack, NJ, USA, 2004.
12. R. S. Hamilton, "The inverse function theorem of Nash and Moser," Bulletin of the American Mathematical Society, vol. 7, no. 1, pp. 65-222, 1982.
13. V. G. Kac, Infinite-Dimensional Lie Algebras, Cambridge University Press, Cambridge, Mass, USA, 2nd edition, 1985.
14. S. Lang, Differential Manifolds, Addison-Wesley, Reading, Mass, USA, 1972.
15. J. E. Marsden, D. G. Ebin, and A. E. Fischer, "Diffeomorphism groups, hydrodynamics and relativity," in Proceedings of the 13th Biennial Seminar of the Canadian Mathematical Congress Differential Geometry and Applications, J. R. Vanstone, Ed., vol. 1, pp. 135-279, 1972.
16. P. W. Michor, Manifolds of Differentiable Mappings, vol. 3 of Shiva Mathematics Series, Shiva, Nantwich, UK, 1980.
17. P. J. Olver, Applications of Lie Groups to Differential Equations, vol. 107 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 2nd edition, 1993.
18. R. Palais, Foundations of Global Nonlinear Analysis, Addison-Wesley, Reading, Mass, USA, 1968.
19. R. Schmid, Infinite-Dimensional Hamiltonian Systems, vol. 3 of Monographs and Textbooks in Physical Science. Lecture Notes, Bibliopolis, Naples, Italy, 1987.
20. R. Abraham and J. E. Marsden, Foundations of Mechanics, Benjamin/Cummings Publishing, Reading, Mass, USA, 2nd edition, 1978.
21. J. E. Marsden, Lectures on Mechanics, vol. 174 of London Mathematical Society Lecture Note Series, Cambridge University Press, Cambridge, UK, 1992.
22. J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical System, vol. 17 of Texts in Applied Mathematics, Springer, New York, NY, USA, 1994.
23. J. Marsden and A. Weinstein, "Reduction of symplectic manifolds with symmetry," Reports on Mathematical Physics, vol. 5, no. 1, pp. 121-130, 1974.
24. R. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, Tensor Analysis, and Applications, vol. 2 of Global Analysis Pure and Applied: Series B, Addison-Wesley, Reading, Mass, USA, 1983.
25. P. R. Chernoff and J. E. Marsden, Properties of Infinite Dimensional Hamiltonian Systems, Lecture Notes in Mathematics, Vol. 425, Springer, Berlin, Germany, 1974.
26. Y. Choquet-Bruhat, C. DeWitt-Morette, and M. Dillard-Bleick, Analysis, Manifolds and Physics, North-Holland, Amsterdam, The Netherlands, 2nd edition, 1982.
27. F. Gay-Balmaz and T. S. Ratiu, "Reduced Lagrangian and Hamiltonian formulations of Euler-Yang- Mills fluids," The Journal of Symplectic Geometry, vol. 6, no. 2, pp. 189-237, 2008.
28. F. Gay-Balmaz and T. S. Ratiu, "A new Lagrangian dynamic reduction in field theory," to appear in Annales de l'Institut Fourier.
29. V. Guillemin and S. Sternberg, Symplectic Techniques in Physics, Cambridge University Press, Cambridge, Mass, USA, 1984.
30. R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, vol. 68 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1988.
31. S. Vasylkevych and J. E. Marsden, "The Lie-Poisson structure of the Euler equations of an ideal fluid," Dynamics of Partial Differential Equations, vol. 2, no. 4, pp. 281-300, 2005.
32. J. E. Marsden, A. Weinstein, T. Ratiu, R. Schmid, and R. G. Spencer, "Hamiltonian systems with symmetry, coadjoint orbits and plasma physics," in Proceedings of the IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics, vol. 1, pp. 289-340, Atti della Accademia delle Scienze di Torino. Classe di Scienze Fisiche, Matematiche e Naturali, Torino, Italy, 1982.
33. M. Adams, T. Ratiu, and R. Schmid, "The Lie group structure of diffeomorphism groups and invertible Fourier integral operators, with applications," in Infinite-Dimensional Groups with Applications, vol. 4 of Math. Sci. Res. Inst. Publ., pp. 1-69, Springer, New York, NY, USA, 1985.
34. M. Adams, T. Ratiu, and R. Schmid, "A Lie group structure for pseudodifferential operators," Mathematische Annalen, vol. 273, no. 4, pp. 529-551, 1986.
35. M. Adams, T. Ratiu, and R. Schmid, "A Lie group structure for Fourier integral operators," Mathematische Annalen, vol. 276, no. 1, pp. 19-41, 1986.
36. A. E. Fischer and J. E. Marsden, "The Einstein equations of evolution - a geometric approach," Journal of Mathematical Physics, vol. 13, pp. 546-568, 1972.
37. F. Gay-Balmaz and T. S. Ratiu, "The Lie-Poisson structure of the LAE-α equation," Dynamics of Partial Differential Equations, vol. 2, no. 1, pp. 25-57, 2005.
38. J. E. Marsden and A. Weinstein, "The Hamiltonian structure of the Maxwell-Vlasov equations," Physica D, vol. 4, no. 3, pp. 394-406, 1982.
39. F. Trèves, Introduction to Pseudodifferential and Fourier Integral Operators: Pseudodifferential Operators, vol. 1 of The University Series in Mathematic, Plenum Press, New York, NY, USA, 1980.
40. F. Trèves, Introduction to Pseudodifferential and Fourier Integral Operators: Fourier Integral Operators, vol. 2 of The University Series in Mathematic, Plenum Press, New York, NY, USA, 1980.
41. P. Deligne et al., Ed., Quantum Fields and Strings: A Course for Mathematicians, vol. 1, American Mathematical Society, Providence, RI, USA, 1999.
42. P. Deligne et al., Ed., Quantum Fields and Strings: A Course for Mathematicians, vol. 2, American Mathematical Society, Providence, RI, USA, 1999.
43. C. Becchi, A. Rouet, and R. Stora, "Renormalization of gauge theories," Annals of Physics, vol. 98, no. 2, pp. 287-321, 1976.
44. R. Schmid, "Local cohomology in gauge theories, BRST transformations and anomalies," Differential Geometry and Its Applications, vol. 4, no. 2, pp. 107-116, 1994.
45. C. Chevalley and S. Eilenberg, "Cohomology theory of Lie groups and Lie algebras," Transactions of the American Mathematical Society, vol. 63, pp. 85-124, 1948.
46. R. Schmid, "BRST cohomology and anomalies," in Geometrical and Algebraic Aspects of Nonlinear Field Theory, North-Holland Delta Ser., pp. 159-172, North-Holland, Amsterdam, The Netherlands, 1989.
47. A. Einstein, Mein Weltbild, Ullstein, Frankfurt, Germany, 1934.
48. E. Verlinde, "On the origin of gravity and the laws of Newton," http://arxiv.org/abs/1001.0785.
49. B. Kostant, "Graded manifolds, graded Lie theory, and prequantization," in Differential Geometrical Methods in Mathematical Physics, vol. 570 of Lecture Notes in Mathematics, pp. 177-306, Springer, Berlin, Germany, 1977.
50. E. Binz, J. Śniatycki, and H. Fischer, Geometry of Classical Fields, vol. 154 of North-Holland Mathematics Studies, North-Holland, Amsterdam, The Netherlands, 1988.
51. L. Bonora and P. Cotta-Ramusino, "Some remarks on BRS transformations, anomalies and the cohomology of the Lie algebra of the group of gauge transformations," Communications in Mathematical Physics, vol. 87, no. 4, pp. 589-603, 1982/83.
52. M. Dubois-Violette, "The Weil-BRS algebra of a Lie algebra and the anomalous terms in gauge theory," Journal of Geometry and Physics, vol. 3, no. 4, pp. 525-565, 1986.
53. T. Eguchi, P. B. Gilkey, and A. J. Hanson, "Gravitation, gauge theories and differential geometry," Physics Reports, vol. 66, no. 6, pp. 213-393, 1980.
54. L. D. Faddeev and A. A. Slavnov, Gauge Fields: Introduction to Quantum Theory, vol. 50 of Frontiers in Physics, Benjamin/Cummings Publishing, Reading, Mass, USA, 1980.
55. G. A. Goldin, R. Menikoff, and D. H. Sharp, "Diffeomorphism groups, gauge groups, and quantum theory," Physical Review Letters, vol. 51, no. 25, pp. 2246-2249, 1983.
56. G. A. Goldin, "The diffeomorphism group approach to nonlinear quantum systems," International Journal of Modern Physics B, vol. 6, no. 11-12, pp. 1905-1916, 1992.
57. C. Itzykson and J. B. Zuber, Quantum Field Theory, International Series in Pure and Applied Physic, McGraw-Hill, New York, NY, USA, 1980.
58. D. Kastler and R. Stora, "A differential geometric setting for BRS transformations and anomalies. I, II," Journal of Geometry and Physics, vol. 3, no. 4, pp. 437-505, 1986.
59. B. Kostant and S. Sternberg, "Symplectic reduction, BRS cohomology, and infinite-dimensional Clifford algebras," Annals of Physics, vol. 176, no. 1, pp. 49-113, 1987.
60. T. Glimm, Supersymplectic reduction, Ph.D. thesis, Emory University, 2003, http://www.mathcs.emory.edu/~rudolf/Glimm.pdf.
61. G. M. Tuynman, Supermanifolds and Supergroups: Basic Theory, vol. 570 of Mathematics and Its Applications, Kluwer Academic Publishers, Dordrecht, The Netherlands, 2004.
62. B. DeWitt, Supermanifolds, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, Mass, USA, 2nd edition, 1992.
63. T. Glimm and R. Schmid, "On the supersymmetry group of the classical Bose-Fermi oscillator," Journal of Geometry and Symmetry in Physics, vol. 4, pp. 45-58, 2005.