Japanese Journal of Mathematics

, Volume 1, Issue 2, pp 291–468 | Cite as

Towards a Lie theory of locally convex groups

Article

Abstract.

In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie subgroups, and integrability of Lie algebra extensions to Lie group extensions. We further describe how regularity or local exponentiality of a Lie group can be used to obtain quite satisfactory answers to some of the fundamental problems. These results are illustrated by specialization to some specific classes of Lie groups, such as direct limit groups, linear Lie groups, groups of smooth maps and groups of diffeomorphisms.

Keywords and Phrases.

infinite-dimensional Lie group infinite-dimensional Lie algebra continuous inverse algebra diffeomorphism group gauge group pro-Lie group BCH–Lie group exponential function Maurer–Cartan equation Lie functor integrable Lie algebra 

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References

  1. A. Abouqateb and K.-H. Neeb, Integration of locally exponential Lie algebras of vector fields, submitted.Google Scholar
  2. 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, Berkeley, Calif., 1984, Math. Sci. Res. Inst. Publ., 4, Springer-Verlag, 1985, pp. 1–69.Google Scholar
  3. M. Adams, T. Ratiu and R. Schmid, A Lie group structure for pseudodifferential operators, Math. Ann., 273 (1986), 529–551.MATHMathSciNetCrossRefGoogle Scholar
  4. M. Adams, T. Ratiu and R. Schmid, A Lie group structure for Fourier integral operators, Math. Ann., 276 (1986), 19–41.MATHMathSciNetCrossRefGoogle Scholar
  5. I. Ado, Über die Darstellung von Lieschen Gruppen durch lineare Substitutionen, Bull. Soc. Phys. Math. Kazan (3), 7 (1936), 3–43.MATHGoogle Scholar
  6. S. A. Albeverio, R. J. Høegh-Krohn, J. A. Marion, D. H. Testard and B. S. Torrésani, Noncommutative distributions. Unitary representation of Gauge Groups and Algebras, Monogr. Textbooks Pure Appl. Math., 175, Marcel Dekker, Inc., New York, 1993.Google Scholar
  7. G. R. Allan, A spectral theory for locally convex algebras, Proc. London Math. Soc. (3), 15 (1965), 399–421.MATHMathSciNetGoogle Scholar
  8. B. N. Allison, S. Azam, S. Berman, Y. Gao and A. Pianzola, Extended Affine Lie Algebras and Their Root Systems, Mem. Amer. Math. Soc., 603, Providence, R.I., 1997.Google Scholar
  9. B. Allison, G. Benkart and Y. Gao, Central extensions of Lie algebras graded by finite-root systems, Math. Ann., 316 (2000), 499–527.MATHMathSciNetCrossRefGoogle Scholar
  10. I. Amemiya, Lie algebra of vector fields and complex structure, J. Math. Soc. Japan, 27 (1975), 545–549.MATHMathSciNetCrossRefGoogle Scholar
  11. V. I. Arnold, Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier, 16 (1966), 319–361.MATHGoogle Scholar
  12. V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Springer-Verlag, 1998.Google Scholar
  13. J. A. de Azcarraga and J. M. Izquierdo, Lie Groups, Lie Algebras, Cohomology and some Applications in Physics, Cambridge Monogr. Math. Phys., 1995.Google Scholar
  14. H. F. Baker, On the exponential theorem for a simply transitive continuous group, and the calculation of the finite equations from the constants of structure, J. London Math. Soc., 34 (1901), 91–127.MATHGoogle Scholar
  15. H. F. Baker, On the calculation of the finite equations of a continuous group, Lond. M. S. Proc., 35 (1903), 332–333.MATHGoogle Scholar
  16. A. Banyaga, The Structure of Classical Diffeomorphism Groups, Kluwer Academic Publishers, 1997.Google Scholar
  17. A. Bastiani, Applications différentiables et variétés différentiables de dimension infinie, J. Anal. Math., 13 (1964), 1–114.MATHMathSciNetGoogle Scholar
  18. E. J. Beggs, The de Rham complex on infinite dimensional manifolds, Quart. J. Math. Oxford (2), 38 (1987), 131–154.MATHMathSciNetGoogle Scholar
  19. D. Beltiţă, Asymptotic products and enlargibility of Banach–Lie algebras, J. Lie Theory, 14 (2004), 215–226.MathSciNetMATHGoogle Scholar
  20. D. Beltiţă, Smooth Homogeneous Structures in Operator Theory, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., 2006.Google Scholar
  21. D. Beltiţă and K.-H. Neeb, Finite-dimensional Lie subalgebras of algebras with continuous inversion, preprint, 2006.Google Scholar
  22. D. Beltiţă and T. S. Ratiu, Geometric representation theory for unitary groups of operator algebras, Adv. Math., to appear.Google Scholar
  23. D. Beltiţă and T. S. Ratiu, Symplectic leaves in real Banach Lie–Poisson spaces, Geom. Funct. Anal., 15 (2005), 753–779.MathSciNetCrossRefMATHGoogle Scholar
  24. W. Bertram and K.-H. Neeb, Projective completions of Jordan pairs, Part I. The generalized projective geometry of a Lie algebra, J. Algebra, 277 (2004), 474–519.MATHMathSciNetCrossRefGoogle Scholar
  25. W. Bertram and K.-H. Neeb, Projective completions of Jordan pairs, Part II, Geom. Dedicata, 112 (2005), 75–115.MathSciNetCrossRefGoogle Scholar
  26. W. Bertram, H. Glöckner and K.-H. Neeb, Differential Calculus over General Base Fields and Rings, Expo. Math., 22 (2004), 213–282.MATHMathSciNetGoogle Scholar
  27. Y. Billig, Abelian extensions of the group of diffeomorphisms of a torus, Lett. Math. Phys., 64 (2003), 155–169.MATHMathSciNetCrossRefGoogle Scholar
  28. Y. Billig and A. Pianzola, Free Kac-Moody groups and their Lie algebras, Algebr. Represent. Theory, 5 (2002), 115–136.MATHMathSciNetCrossRefGoogle Scholar
  29. G. Birkhoff, Continuous groups and linear spaces, Mat. Sb., 1 (1936), 635–642.MATHGoogle Scholar
  30. G. Birkhoff, Analytic groups, Trans. Amer. Math. Soc., 43 (1938), 61–101.MATHMathSciNetCrossRefGoogle Scholar
  31. B. Blackadar, K-theory for Operator Algebras, 2nd edition, Cambridge Univ. Press, 1998.Google Scholar
  32. J. Bochnak and J. Siciak, Analytic functions in topological vector spaces, Studia Math., 39 (1971), 77–112.MATHMathSciNetGoogle Scholar
  33. S. Bochner and D. Montgomery, Groups of differentiable and real or complex analytic transformations, Ann. of Math. (2), 46 (1945), 685–694.MATHMathSciNetCrossRefGoogle Scholar
  34. F. F. Bonsall and J. Duncan, Complete Normed Algebras, Ergeb. Math. Grenzgeb., 80, Springer-Verlag, 1973.Google Scholar
  35. H. Boseck, G. Czichowski and K.-P. Rudolph, Analysis on Topological Groups – General Lie Theory, Teubner, Leipzig, 1981.Google Scholar
  36. J.-B. Bost, Principe d’Oka, K-theorie et systèmes dynamiques non-commutatifs, Invent. Math., 101 (1990), 261–333.MATHMathSciNetCrossRefGoogle Scholar
  37. R. Bott, On the characteristic classes of groups of diffeomorphisms, Enseign. Math. (2), 23 (1977), 209–220.MATHMathSciNetGoogle Scholar
  38. N. Bourbaki, Topological Vector Spaces, Chaps. 1–5, Springer-Verlag, 1987.Google Scholar
  39. N. Bourbaki, Lie Groups and Lie Algebras, Chapter 1–3, Springer-Verlag, 1989.Google Scholar
  40. G. E. Bredon, Topology and Geometry, Grad. Texts in Math., 139, Springer-Verlag, 1993.Google Scholar
  41. J.-L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, Progr. Math., 107, Birkhäuser, 1993.Google Scholar
  42. J. I. Burgos Gil, The Regulators of Beilinson and Borel, CRM Monogr., 15, Amer. Math. Soc., 2002.Google Scholar
  43. E. Calabi, On the group of automorphisms of a symplectic manifold, In: Probl. Analysis. Sympos. in Honor of Salomon Bochner, Princeton Univ. Press, Princeton, N.J., 1970, pp. 1–26.Google Scholar
  44. J. E. Campbell, On a law of combination of operators bearing on the theory of continuous transformation groups, Proc. London Math. Soc., 28 (1897), 381–390.MATHGoogle Scholar
  45. J. E. Campbell, On a law of combination of operators. (second paper), Proc. London Math. Soc., 28 (1897), 381–390.MATHGoogle Scholar
  46. E. Cartan, Les groupes bilinéaires et les systèmes de nombres complexes, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 12 (1898), B1–B64.MathSciNetGoogle Scholar
  47. E. Cartan, L’intégration des systèmes d’équations aux diffrentielles totales, Ann. Sci. École Norm. Sup. (3), 18 (1901), 241–311.MATHMathSciNetGoogle Scholar
  48. E. Cartan, Sur la structure des groups infinies des transformations, Ann. Sci. École. Norm. Sup., 21 (1904), 153–206; 22 (1905), 219–308.Google Scholar
  49. E. Cartan, Le troisième théorème fondamental de Lie, C. R. Math. Acad. Sci. Paris, 190 (1930), 914–916, 1005–1007.Google Scholar
  50. E. Cartan, La topologie des groupes de Lie. (Exposés de géométrie Nr. 8.), Actualités. Sci. Indust., 358 (1936), p. 28.MATHGoogle Scholar
  51. E. Cartan, La topologie des espaces représentifs de groupes de Lie, Oeuvres I, Gauthier–Villars, Paris, 2 (1952), 1307–1330.Google Scholar
  52. G. Cassinelli, E. de Vito, P. Lahti and A. Levrero, Symmetries of the quantum state space and group representations, Rev. Math. Phys., 10 (1998), 893–924.MATHMathSciNetCrossRefGoogle Scholar
  53. P. Chernoff and J. Marsden, On continuity and smoothness of group actions, Bull. Amer. Math. Soc., 76 (1970), 1044–1049.MATHMathSciNetGoogle Scholar
  54. C. Chevalley, Theory of Lie Groups I, Princeton Univ. Press, 1946.Google Scholar
  55. C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc., 63 (1948), 85–124.MATHMathSciNetCrossRefGoogle Scholar
  56. A. Connes, Non-commutative Geometry, Academic Press, 1994.Google Scholar
  57. J. A. Cuenca Mira, A. Garcia Martin and C. Martin Gonzalez, Structure theory of L *-algebras, Math. Proc. Cambridge Philos. Soc., 107 (1990), 361–365.MATHMathSciNetCrossRefGoogle Scholar
  58. J. Dai and D. Pickrell, The orbit method and the Virasoro extension of ( \(\hbox{Diff}_+{\user2{\mathbb{S}}}^{1}\)). I. Orbital integrals, J. Geom. Phys., 44 (2003), 623–653.MATHMathSciNetCrossRefGoogle Scholar
  59. P. Dazord, Lie groups and algebras in infinite dimension: a new approach, In: Symplectic Geometry and Quantization, Contemp. Math., 179, Amer. Math. Soc., Providence, RI, 1994, pp. 17–44.Google Scholar
  60. J. Delsartes, Les groups de transformations linéaires dans l’espace de Hilbert, Mém. Sci. Math., 57, Paris.Google Scholar
  61. I. Dimitrov and I. Penkov, Weight modules of direct limit Lie algebras, Internat. Math. Res. Notices, 5 (1999), 223–249.MATHMathSciNetCrossRefGoogle Scholar
  62. P. Donato and P. Iglesias, Examples de groupes difféologiques: flots irrationnels sur le tore, C. R. Acad. Sci. Paris Ser. I Math., 301 (1985), 127–130.MATHMathSciNetGoogle Scholar
  63. A. Douady and M. Lazard, Espaces fibrés en algèbres de Lie et en groupes, Invent. Math., 1 (1966), 133–151.MATHMathSciNetCrossRefGoogle Scholar
  64. A. Dress, Newman’s Theorem on transformation groups, Topology, 8 (1969), 203–207.MATHMathSciNetCrossRefGoogle Scholar
  65. E. B. Dynkin, Calculation of the coefficients in the Campbell–Hausdorff formula (Russian), Dokl. Akad. Nauk. SSSR (N.S.), 57 (1947), 323–326.MATHMathSciNetGoogle Scholar
  66. E. B. Dynkin, Normed Lie Algebras and Analytic Groups, Amer. Math. Soc. Transl., 97 (1953), p. 66.MathSciNetGoogle Scholar
  67. D. G. Ebin, The manifold of Riemannian metrics, In: Global Analysis, Berkeley, Calif., 1968, Proc. Sympos. Pure Math., 15 (1970), pp. 11–40.Google Scholar
  68. D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the solution of the classical Euler equations for a perfect fluid, Bull. Amer. Math. Soc., 75 (1969), 962–967.MATHMathSciNetGoogle Scholar
  69. D. G. Ebin and J. E. Marsden, Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math., 92 (1970), 102–163.MATHMathSciNetCrossRefGoogle Scholar
  70. D. G. Ebin and G. Misiolek, The exponential map on \({\user1{\mathcal{D}}}^{s}_{\mu }\). In: The Arnoldfest, Toronto, ON, 1997, Fields Inst. Commun., 24, Amer. Math. Soc., Providence, RI, 1999, 153–163.Google Scholar
  71. J. Eells, Jr., On the geometry of function spaces, In: International Symposium on Algebraic Topology, Universidad Nacional Autonoma de México and UNESCO, Mexico City, pp. 303–308.Google Scholar
  72. J. Eells, Jr., A setting for global analysis, Bull. Amer. Math. Soc., 72 (1966), 751–807.MathSciNetGoogle Scholar
  73. J. Eichhorn and R. Schmid, Form preserving diffeomorphisms on open manifolds, Ann. Global Anal. Geom., 14 (1996), 147–176.MATHMathSciNetCrossRefGoogle Scholar
  74. J. Eichhorn and R. Schmid, Lie groups of Fourier integral operators on open manifolds, Comm. Anal. Geom., 9 (2001), 983–1040.MATHMathSciNetGoogle Scholar
  75. M. Eichler, A new proof of the Baker–Campbell–Hausdorff formula, J. Math. Soc. Japan, 20 (1968), 23–25.MATHMathSciNetGoogle Scholar
  76. W. T. van Est, Local and global groups, Proc. Konink. Nederl. Akad. Wetensch. Ser. A, 65; Indag. Math., 24 (1962), 391–425.Google Scholar
  77. W. T. van Est, On Ado’s theorem, Proc. Konink. Nederl. Akad. Wetensch. Ser. A, 69; Indag. Math., 28 (1966), 176–191.Google Scholar
  78. W. T. van Est, Rapport sur les S-atlas, Astérisque, 116 (1984), 235–292.MATHGoogle Scholar
  79. W. T. van Est, Une démonstration de É. Cartan du troisième théorème de Lie, In: Seminaire Sud-Rhodanien de Geometrie VIII: Actions Hamiltoniennes de Groupes; Troisième Théorème de Lie, (eds. P. Dazord et al.), Hermann, Paris, 1988.Google Scholar
  80. W. T. van Est and Th. J. Korthagen, Non enlargible Lie algebras, Proc. Konink. Nederl. Akad. Wetensch. Ser. A; Indag. Math., 26 (1964), 15–31.Google Scholar
  81. W. T. van Est and S. Świerczkowski, The path functor and faithful representability of Banach Lie algebras, In: Collection of articles dedicated to the memory of Hannare Neumann, I., J. Austral. Math. Soc., 16 (1973), 54–69.Google Scholar
  82. P. I. Etinghof and I. B. Frenkel, Central extensions of current groups in two dimensions, Comm. Math. Phys., 165 (1994), 429–444.MathSciNetCrossRefGoogle Scholar
  83. R. P. Filipkiewicz, Isomorphisms between diffeomorphism groups, Ergodic Theory Dynam. Systems, 2 (1983), 159–171.MathSciNetGoogle Scholar
  84. K. Floret, Lokalkonvexe Sequenzen mit kompakten Abbildungen, J. Reine Angew. Math., 247 (1971), 155–195.MATHMathSciNetGoogle Scholar
  85. Ch. Freifeld, One-parameter subgroups do not fill a neighborhood of the identity in an infinite-dimensional Lie (pseudo-) group, Battelle Rencontres, 1967, Lectures Math. Phys., Benjamin, New York, 1968, 538–543.Google Scholar
  86. A. Frölicher and W. Bucher, Calculus in Vector Spaces without Norm, Lecture Notes in Math., 30, Springer-Verlag, 1966.Google Scholar
  87. A. Frölicher and A. Kriegl, Linear Spaces and Differentiation Theory, J. Wiley, Interscience, 1988.Google Scholar
  88. D. B. Fuks, Cohomology of Infinite-Dimensional Lie Algebras, Consultants Bureau, New York, London, 1986.Google Scholar
  89. G. Galanis, Projective limits of Banach–Lie groups, Period. Math. Hungar., 32 (1996), 179–191.MATHMathSciNetCrossRefGoogle Scholar
  90. G. Galanis, On a type of linear differential equations in Fréchet spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 24 (1997), 501–510.MATHMathSciNetGoogle Scholar
  91. S. L. Glashow, and M. Gell-Mann, Gauge theories of vector particles, Ann. Physics, 15 (1961), 437–460.MATHMathSciNetCrossRefGoogle Scholar
  92. H. Glöckner, Infinite-dimensional Lie groups without completeness restrictions, In: Geometry and Analysis on Finite and Infinite-dimensional Lie Groups, (eds. A. Strasburger, W. Wojtynski, J. Hilgert and K.-H. Neeb), Banach Center Publ., 55 (2002), 43–59.Google Scholar
  93. H. Glöckner, Algebras whose groups of units are Lie groups, Studia Math., 153 (2002), 147–177.MATHMathSciNetGoogle Scholar
  94. H. Glöckner, Lie group structures on quotient groups and universal complexifications for infinite-dimensional Lie groups, J. Funct. Anal., 194 (2002), 347–409.MATHMathSciNetCrossRefGoogle Scholar
  95. H. Glöckner, Patched locally convex spaces, almost local mappings, and diffeomorphism groups of non-compact manifolds, TU Darmstadt, manuscript, 26.6.02.Google Scholar
  96. H. Glöckner, Implicit functions from topological vector spaces to Banach spaces, Israel J. Math., to appear, math.GM/0303320.Google Scholar
  97. H. Glöckner, Direct limit Lie groups and manifolds, J. Math. Kyoto Univ., 43 (2003), 1–26.MATHGoogle Scholar
  98. H. Glöckner, Lie groups of measurable mappings, Canad. J. Math., 55 (2003), 969–999.MATHMathSciNetGoogle Scholar
  99. H. Glöckner, Tensor products in the category of topological vector spaces are not associative, Comment. Math. Univ. Carolin., 45 (2004), 607–614.MathSciNetMATHGoogle Scholar
  100. H. Glöckner, Lie groups of germs of analytic mappings, In: Infinite Dimensional Groups and Manifolds, (eds. V. Turaev and T. Wurzbacher), IRMA Lect. Math. Theor. Phys., de Gruyter, 2004, pp. 1–16.Google Scholar
  101. H. Glöckner, Fundamentals of direct limit Lie theory, Compositio Math., 141 (2005), 1551–1577.MATHCrossRefGoogle Scholar
  102. H. Glöckner, Discontinuous non-linear mappings on locally convex direct limits, Publ. Math. Debrecen, 68 (2006) 1–13.MATHMathSciNetGoogle Scholar
  103. H. Glöckner, Fundamental problems in the theory of infinite-dimensional Lie groups, J. Geom. Symmetry Phys., 5 (2006), 24–35.MATHMathSciNetGoogle Scholar
  104. H. Glöckner, Direct limits of infinite-dimensional Lie groups compared to direct limits in related categories, in preparation.Google Scholar
  105. H. Glöckner, Direct limit groups do not have small subgroups, preprint, math.GR/0602407.Google Scholar
  106. H. Glöckner and K.-H. Neeb, Banach–Lie quotients, enlargibility, and universal complexifications, J. Reine Angew. Math., 560 (2003), 1–28.MATHMathSciNetGoogle Scholar
  107. H. Glöckner and K.-H. Neeb, Infinite-dimensional Lie groups, Vol. I, Basic Theory and Main Examples, book in preparation.Google Scholar
  108. H. Glöckner and K.-H. Neeb, Infinite-dimensional Lie groups, Vol. II, Geometry and Topology, book in preparation.Google Scholar
  109. G. A. Goldin, Lectures on diffeomorphism groups in quantum physics, In: Contemporary Problems in Mathematical Physics, Cotonue, 2003, Proc. of the third internat. workshop, 2004, pp. 3–93.Google Scholar
  110. R. Goodman and N. R. Wallach, Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle, J. Reine Angew. Math., 347 (1984), 69–133.MATHMathSciNetGoogle Scholar
  111. R. Goodman and N. R. Wallach, Projective unitary positive energy representations of Diff \({\user2{\mathbb{S}}}^{1}\), J. Funct. Anal., 63 (1985), 299–312.MATHMathSciNetCrossRefGoogle Scholar
  112. M. Goto, On an arcwise connected subgroup of a Lie group, Proc. Amer. Math. Soc., 20 (1969), 157–162.MATHMathSciNetCrossRefGoogle Scholar
  113. J. Grabowski Free subgroups of diffeomorphism groups, Fund. Math., 131 (1988), 103–121.MATHMathSciNetGoogle Scholar
  114. J. Grabowski, Derivative of the exponential mapping for infinite-dimensional Lie groups, Ann. Global Anal. Geom., 11 (1993), 213–220.MATHMathSciNetGoogle Scholar
  115. J. M. Gracia-Bondia, J. C. Vasilly and H. Figueroa, Elements of Non-commutative Geometry, Birkhäuser Advanced Texts, Birkhäuser, Basel, 2001.Google Scholar
  116. B. Gramsch, Relative Inversion in der Störungstheorie von Operatoren und Ψ-Algebren, Math. Ann., 269 (1984), 22–71.MathSciNetCrossRefGoogle Scholar
  117. H. Grundling and K.- H. Neeb, Lie group extensions associated to modules of continuous inverse algebras, in preparation.Google Scholar
  118. J. Gutknecht, Die C Γ-Struktur auf der Diffeomorphismengruppe einer kompakten Mannigfaltigkeit, Ph. D. thesis, Eidgenössische Technische Hochschule Zürich, Diss. No. 5879, Juris Druck + Verlag, Zurich, 1977.Google Scholar
  119. R. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc., 7 (1982), 65–222.MATHMathSciNetGoogle Scholar
  120. P. de la Harpe, Classical Banach–Lie Algebras and Banach–Lie Groups of Operators in Hilbert Space, Lecture Notes in Math., 285, Springer-Verlag, 1972.Google Scholar
  121. L. A. Harris and W. Kaup, Linear algebraic groups in infinite dimensions, Illinois J. Math., 21 (1977), 666–674.MATHMathSciNetGoogle Scholar
  122. F. Hausdorff, Die symbolische Exponentialformel in der Gruppentheorie, Leipziger Berichte, 58 (1906), 19–48.MATHGoogle Scholar
  123. M. Hausner and J. T. Schwartz, Lie Groups; Lie Algebras, Gordon and Breach, New York, London, Paris, 1968.Google Scholar
  124. G. Hector and E. Macías-Virgós, Diffeological groups, Res. Exp. Math., 25 (2002), 247–260.MATHGoogle Scholar
  125. A. Ya. Helemskii, Banach and Locally Convex Algebras, Oxford Sci. Publications, Oxford University Press, New York, 1993.Google Scholar
  126. S. Hiltunen, Implicit functions from locally convex spaces to Banach spaces, Studia Math., 134 (1999), 235–250.MATHMathSciNetGoogle Scholar
  127. G. Hochschild, Group extensions of Lie groups I, II, Ann. of Math., 54 (1951), 96–109; 54 (1951), 537–551.Google Scholar
  128. G. Hochschild, The Structure of Lie Groups, Holden Day, San Francisco, 1965.MATHGoogle Scholar
  129. K. H. Hofmann, Introduction to the Theory of Compact Groups. Part I, Dept. Math. Tulane Univ., New Orleans, LA, 1968.MATHGoogle Scholar
  130. K. H. Hofmann, Die Formel von Campbell, Hausdorff und Dynkin und die Definition Liescher Gruppen, In: Theory Sets Topology in Honour of Felix Hausdorff, 1868–1942, VEB Deutsch, Verlag Wissensch., Berlin, 1972, pp. 251–264.Google Scholar
  131. K. H. Hofmann, Analytic groups without analysis, Sympos. Math., 16, Convegno sui Gruppi Topologici e Gruppi di Lie, INDAM, Rome, 1974, Academic Press, London, 1975, pp. 357–374.Google Scholar
  132. K. H. Hofmann and S. A. Morris, The Structure of Compact Groups, de Gruyter Stud. Math., de Gruyter, Berlin, 1998.MATHGoogle Scholar
  133. K. H. Hofmann and S. A. Morris, Sophus Lie’s third fundamental theorem and the adjoint functor theorem, J. Group Theory, 8 (2005), 115–123.MATHMathSciNetCrossRefGoogle Scholar
  134. K. H. Hofmann and S. A. Morris, The Lie Theory of Connected Pro-Lie Groups–A Structure Theory for Pro-Lie Algebras, Pro-Lie Groups and Connected Locally Compact Groups, EMS Publishing House, Zürich, to appear (2006).Google Scholar
  135. K. H. Hofmann, S. A. Morris and D. Poguntke, The exponential function of locally connected compact abelian groups, Forum Math., 16 (2004), 1–16.MATHMathSciNetCrossRefGoogle Scholar
  136. K. H. Hofmann and K.-H. Neeb, Pro-Lie groups which are infinite-dimensional Lie groups, submitted.Google Scholar
  137. H. Hopf, Fundamentalgruppe und zweite Bettische Gruppe, Comment. Math. Helv., 14 (1942), 257–309.MATHMathSciNetGoogle Scholar
  138. L. van Hove, Topologie des espaces fonctionnels analytiques, et des groups infinis des transformations, Acad. Roy. Belgique, Bull. Cl. Sci. (5), 38 (1952), 333–351.MATHMathSciNetGoogle Scholar
  139. L. van Hove, L’ensemble des fonctions analytiques sur un compact en tant qu’algèbre topologique, Bull. Soc. Math. Belg., 1952, 8–17 (1953).Google Scholar
  140. R. S. Ismagilov, Representations of Infinite-Dimensional Groups, Transl. Math. Monogr., 152 (1996).Google Scholar
  141. V. G. Kac, Constructing groups associated to infinite-dimensional Lie algebras, In: Infinite-Dimensional Groups with Applications, (ed. V. Kac), MSRI Publications, 4, Springer-Verlag, 1985.Google Scholar
  142. V. G. Kac, Infinite-dimensional Lie Algebras, Cambridge University Press, 1990.Google Scholar
  143. V. G. Kac and D. H. Peterson, Regular functions on certain infinite-dimensional groups, In: Arithmetic and Geometry, (eds. M. Artin and J. Tate), 2, Birkhäuser, Boston, 1983.Google Scholar
  144. N. Kamran and T. Robart, A manifold structure for analytic Lie pseudogroups of infinite type, J. Lie Theory, 11 (2001), 57–80.MATHMathSciNetGoogle Scholar
  145. N. Kamran and T. Robart, An infinite-dimensional manifold structure for analytic Lie pseudogroups of infinite type, Internat. Math. Res. Notices, 34 (2004), 1761–1783.MATHMathSciNetCrossRefGoogle Scholar
  146. W. Kaup, Über die Klassifikation der symmetrischen hermiteschen Mannigfaltigkeiten unendlicher Dimension I, Math. Ann., 257 (1981), 463–486.MATHMathSciNetCrossRefGoogle Scholar
  147. W. Kaup, A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Z., 183 (1983), 503–529.MATHMathSciNetCrossRefGoogle Scholar
  148. W. Kaup, Über die Klassifikation der symmetrischen hermiteschen Mannigfaltigkeiten unendlicher Dimension II, Math. Ann., 262 (1983), 57–75.MATHMathSciNetCrossRefGoogle Scholar
  149. J. Kedra, D. Kotchick and S. Morita, Crossed flux homomorphisms and vanishing theorems for flux groups, preprint, Aug. 2005, math.AT/0503230.Google Scholar
  150. H. H. Keller, Differential Calculus in Locally Convex Spaces, Springer-Verlag, 1974.Google Scholar
  151. A. Kirillov, The orbit method beyond Lie groups. Infinite-dimensional groups, Surveys in modern mathematics, 292–304; London Math. Soc. Lecture Note Ser., 321, Cambridge Univ. Press, Cambridge, 2005.Google Scholar
  152. A. A. Kirillov and D. V. Yuriev, Kähler geometry of the infinite-dimensional homogeneous space \(M = \hbox{Diff}_+({\user2{\mathbb{S}}}^{1})/\hbox{Rot}({\user2{\mathbb{S}}}^{1})\), Funct. Anal. Appl., 21 (1987), 284–294.MATHCrossRefGoogle Scholar
  153. O. Kobayashi, A. Yoshioka, Y. Maeda and H. Omori, The theory of infinite-dimensional Lie groups and its applications, Acta Appl. Math., 3 (1985), 71–106.MATHMathSciNetCrossRefGoogle Scholar
  154. N. Kopell, Commuting diffeomorphisms, Proc. Sympos. Pure Math., 14 (1970), 165–184.MATHMathSciNetGoogle Scholar
  155. B. Kostant, Quantization and unitary representations, In: Lectures in Modern Analysis and Applications III, Lecture Notes in Math., 170, Springer-Verlag, 1970, pp. 87–208.Google Scholar
  156. G. Köthe, Topological Vector Spaces I, Grundlehren der Math. Wissenschaften, 159, Springer-Verlag, Berlin etc., 1969.Google Scholar
  157. A. Kriegl and P. Michor, The Convenient Setting of Global Analysis, Math. Surveys Monogr., 53 (1997).Google Scholar
  158. A. Kriegl and P. Michor, Regular infinite-dimensional Lie groups, J. Lie Theory, 7 (1997), 61–99.MATHMathSciNetGoogle Scholar
  159. N. H. Kuiper, The homotopy type of the unitary group of Hilbert space, Topology, 3 (1965), 19–30.MATHMathSciNetCrossRefGoogle Scholar
  160. S. Kumar, Kac-Moody Groups, their Flag Varieties and Representation Theory, Progr. Math., 204, Birkhäuser, Boston, MA, 2002.Google Scholar
  161. M. Kuranishi, On the local theory of continuous infinite pseudo groups I, Nagoya Math. J., 15 (1959), 225–260.MathSciNetGoogle Scholar
  162. F. Lalonde, D. McDuff and L. Polterovich, On the flux conjectures, In: Geometry, topology, and dynamics, Montreal, PQ, 1995, CRM Proc. Lecture Notes, 15, Amer. Math. Soc., Providence, RI, 1998, pp. 69–85.Google Scholar
  163. S. Lang, Fundamentals of Differential Geometry, Grad. Texts in Math., 191, Springer-Verlag, 1999.Google Scholar
  164. V. T. Laredo, Integration of unitary representations of infinite dimensional Lie groups, J. Funct. Anal., 161 (1999), 478–508.MATHMathSciNetCrossRefGoogle Scholar
  165. R. K. Lashof, Lie algebras of locally compact groups, Pacific J. Math., 7 (1957), 1145–1162.MATHMathSciNetGoogle Scholar
  166. D. Laugwitz, Über unendliche kontinuierliche Gruppen. I. Grundlagen der Theorie; Untergruppen, Math. Ann., 130 (1955), 337–350.MATHMathSciNetCrossRefGoogle Scholar
  167. D. Laugwitz, Über unendliche kontinuierliche Gruppen. II. Strukturtheorie lokal Banachscher Gruppen, Bayer. Akad. Wiss. Math. Natur. Kl. Sitzungsber., 1956, 261–286 (1957).Google Scholar
  168. M. Lazard and J. Tits, Domaines d’injectivité de l’application exponentielle, Topology, 4 (1966), 315–322.MATHMathSciNetCrossRefGoogle Scholar
  169. P. Lecomte, Sur l’algèbre de Lie des sections d’un fibré en algèbre de Lie, Ann. Inst. Fourier, 30 (1980), 35–50.MATHMathSciNetGoogle Scholar
  170. P. Lecomte, Sur la suite exacte canonique associée à un fibré principal, Bull. Soc. Math. France, 13 (1985), 259–271.MathSciNetGoogle Scholar
  171. L. Lempert, The Virasoro group as a complex manifold, Math. Res. Lett., 2 (1995), 479–495.MATHMathSciNetGoogle Scholar
  172. L. Lempert, The problem of complexifying a Lie group, In: Multidimensional Complex Analysis and Partial Differential Equations, (eds. P. D. Cordaro et al.), Amer. Math. Soc., Contemp. Math., 205 (1997), 169–176.Google Scholar
  173. J. A. Leslie, On a theorem of E. Cartan, Ann. Mat. Pura Appl. (4), 74 (1966), 173–177.MATHMathSciNetGoogle Scholar
  174. J. A. Leslie, On a differential structure for the group of diffeomorphisms, Topology, 6 (1967), 263–271.MATHMathSciNetCrossRefGoogle Scholar
  175. J. A. Leslie, Some Frobenius theorems in global analysis, J. Differential Geom., 2 (1968), 279–297.MATHMathSciNetGoogle Scholar
  176. J. A. Leslie, On the group of real analytic diffeomorphisms of a compact real analytic manifold, Trans. Amer. Math. Soc., 274 (1982), 651–669.MATHMathSciNetCrossRefGoogle Scholar
  177. J. A. Leslie, A Lie group structure for the group of analytic diffeomorphisms, Boll. Un. Mat. Ital. A (6), 2 (1983), 29–37.MATHMathSciNetGoogle Scholar
  178. J. A. Leslie, A path functor for Kac-Moody Lie algebras, In: Lie Theory, Differential Equations and Representation Theory, Montreal, PQ, 1989, Univ. Montreal, Montreal, QC, 1990, pp. 265–270.Google Scholar
  179. J. A. Leslie, Some integrable subalgebras of infinite-dimensional Lie groups, Trans. Amer. Math. Soc., 333 (1992), 423–443.MATHMathSciNetCrossRefGoogle Scholar
  180. J. A. Leslie, On the integrability of some infinite dimensional Lie algebras, Howard University, preprint, 1993.Google Scholar
  181. J. A. Leslie, On a diffeological group realization of certain generalized symmetrizable Kac-Moody Lie algebras, J. Lie Theory, 13 (2003), 427–442.MATHMathSciNetGoogle Scholar
  182. D. Lewis, Formal power series transformations, Duke Math. J., 5 (1939), 794–805.MATHMathSciNetCrossRefGoogle Scholar
  183. S. Lie, Theorie der Transformationsgruppen I, Math. Ann., 16 (1880), 441–528.MATHMathSciNetCrossRefGoogle Scholar
  184. S. Lie, Unendliche kontinuierliche Gruppen, Abh. Sächs. Ges. Wiss., 21 (1895), 43–150.Google Scholar
  185. J.-L. Loday, Cyclic Homology, Grundlehren Math. Wiss., 301, Springer-Verlag, Berlin, 1998.Google Scholar
  186. O. Loos, Symmetric Spaces I: General Theory, Benjamin, New York, Amsterdam, 1969.MATHGoogle Scholar
  187. M. V. Losik, Fréchet manifolds as diffeologic spaces, Russian Math., 36 (1992), 31–37.MATHGoogle Scholar
  188. D. Luminet and A. Valette, Faithful uniformly continuous representations of Lie groups, J. London Math. Soc. (2), 49 (1994), 100–108.MATHMathSciNetGoogle Scholar
  189. S. MacLane, Homology, Grundlehren Math. Wiss., 114, Springer-Verlag, 1963.Google Scholar
  190. S. MacLane, Origins of the cohomology of groups, Enseig. Math., 24 (1978), 1–29.MathSciNetGoogle Scholar
  191. Y. Maeda, H. Omori, O. Kobayashi and A. Yoshioka, On regular Fréchet-Lie groups. VIII. Primordial operators and Fourier integral operators, Tokyo J. Math., 8 (1985), 1–47.MATHMathSciNetGoogle Scholar
  192. P. Maier, Central extensions of topological current algebras, In: Geometry and Analysis on Finite-and Infinite-Dimensional Lie Groups, (eds. A. Strasburger et al.), Banach Center Publ., 55, Warszawa, 2002.Google Scholar
  193. P. Maier and K.-H. Neeb, Central extensions of current groups, Math. Ann., 326 (2003), 367–415.MATHMathSciNetCrossRefGoogle Scholar
  194. B. Maissen, Lie-Gruppen mit Banachräumen als Parameterräume, Acta Math., 108 (1962), 229–269.MATHMathSciNetGoogle Scholar
  195. B. Maissen, Über Topologien im Endomorphismenraum eines topologischen Vektorraums, Math. Ann., 151 (1963), 283–285.MATHMathSciNetCrossRefGoogle Scholar
  196. J. Marion and T. Robart, Regular Fréchet Lie groups of invertibe elements in some inverse limits of unital involutive Banach algebras, Georgian Math. J., 2 (1995), 425–444.MATHMathSciNetCrossRefGoogle Scholar
  197. J. E. Marsden, Hamiltonian one parameter groups: A mathematical exposition of infinite dimensional Hamiltonian systems with applications in classical and quantum mechanics, Arch. Rational Mech. Anal., 28 (1968), 362–396.MATHMathSciNetGoogle Scholar
  198. J. E. Marsden and R. Abraham, Hamiltonian mechanics on Lie groups and Hydrodynamics, In: Global Analysis, (eds. S. S. Chern and S. Smale), Proc. Sympos. Pure Math., 16, 1970, Amer. Math. Soc., Providence, RI, pp. 237–244.Google Scholar
  199. L. Maurer, Über allgemeinere Invarianten-Systeme, Münchner Berichte, 43 (1888), 103–150.Google Scholar
  200. W. Mayer and T. Y. Thomas, Foundations of the theory of Lie groups, Ann. of Math., 36 (1935), 770–822.MATHMathSciNetCrossRefGoogle Scholar
  201. D. McDuff, Enlarging the Hamiltonian group, preprint, May 2005, math.SG/0503268.Google Scholar
  202. D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford Math. Monogr., 1998.Google Scholar
  203. E. Michael, Convex structures and continuous selections, Canad. J. Math., 11 (1959), 556–575.MATHMathSciNetGoogle Scholar
  204. A. D. Michal, Differential calculus in linear topological spaces, Proc. Nat. Acad. Sci. U. S. A., 24 (1938), 340–342.MATHCrossRefGoogle Scholar
  205. A. D. Michal, Differential of functions with arguments and values in topological abelian groups, Proc. Nat. Acad. Sci. U. S. A., 26 (1940), 356–359.MATHMathSciNetCrossRefGoogle Scholar
  206. A. D. Michal, The total differential equation for the exponential function in non-commutative normed linear rings, Proc. Nat. Acad. Sci. U. S. A., 31 (1945), 315–317.MATHMathSciNetCrossRefGoogle Scholar
  207. A. D. Michal, Differentiable infinite continuous groups in abstract spaces, Rev. Ci., Lima 50 (1948), 131–140.MATHMathSciNetGoogle Scholar
  208. A. D. Michal and V. Elconin, Differential properties of abstract transformation groups with abstract parameters, Amer. J. Math., 59 (1937), 129–143.MATHMathSciNetCrossRefGoogle Scholar
  209. P. W. Michor, Manifolds of Differentiable Mappings, Shiva Publishing, Orpington, Kent (U.K.), 1980.Google Scholar
  210. P. W. Michor, A convenient setting for differential geometry and global analysis I, II, Cahiers. Topologie Géom. Différentielle Catég., 25 (1984), 63–109, 113–178.MATHMathSciNetGoogle Scholar
  211. P. W. Michor, The cohomology of the diffeomorphism group of a manifold is a Gelfand-Fuks cohomology, In: Proc. of the 14th Winter School on Abstr. Analysis, Srni, 1986, Rend. Circ. Mat. Palermo (2) Suppl., 14, 1987, pp. 235–246.Google Scholar
  212. P. W. Michor, Gauge Theory for Fiber Bundles, Bibliopolis, ed. di fil. sci., Napoli, 1991.Google Scholar
  213. P. Michor and J. Teichmann, Description of infinite dimensional abelian regular Lie groups, J. Lie Theory, 9 (1999), 487–489.MATHMathSciNetGoogle Scholar
  214. J. Mickelsson, Kac-Moody groups, topology of the Dirac determinant bundle, and fermionization, Comm. Math. Phys., 110 (1987), 173–183.MATHMathSciNetCrossRefGoogle Scholar
  215. J. Mickelsson, Current algebras and groups, Plenum Press, New York, 1989.MATHGoogle Scholar
  216. J. Milnor, On infinite-dimensional Lie groups, Institute of Adv. Stud. Princeton, preprint, 1982.Google Scholar
  217. J. Milnor, Remarks on infinite-dimensional Lie groups, In: Relativité, groupes et topologie II, (eds. B. DeWitt and R. Stora), Les Houches, 1983, North Holland, Amsterdam, 1984, pp. 1007–1057.Google Scholar
  218. D. Montgomery and L. Zippin, Topological Transformation Groups, Interscience, New York, 1955.MATHGoogle Scholar
  219. J. Moser, A new technique for the construction of solutions of nonlinear differential equations, Proc. Nat. Acad. Sci. U. S. A., 47 (1961), 1824–1831.MATHMathSciNetCrossRefGoogle Scholar
  220. S. B. Myers, Algebras of differentiable functions, Proc. Amer. Math. Soc., 5 (1954), 917–922.MATHMathSciNetCrossRefGoogle Scholar
  221. T. Nagano, Linear differential systems with singularities and an application to transitive Lie algebras, J. Math. Soc. Japan, 18 (1966), 398–404.MATHMathSciNetGoogle Scholar
  222. M. Nagumo, Einige analytische Untersuchungen in linearen metrischen Ringen, Japan. J. Math., 13 (1936), 61–80.MATHGoogle Scholar
  223. L. Natarajan, E. Rodriguez-Carrington and J. A. Wolf, Differentiable structure for direct limit groups, Lett. Math. Phys., 23 (1991), 99–109.MATHMathSciNetCrossRefGoogle Scholar
  224. L. Natarajan, E. Rodriguez-Carrington and J. A. Wolf, Locally convex Lie groups, Nova J. Algebra Geom., 2 (1993), 59–87.MATHMathSciNetGoogle Scholar
  225. L. Natarajan, E. Rodriguez-Carrington and J. A. Wolf, New classes of infinite dimensional Lie groups, Proc. Sympos. Pure Math., 56 (1994), 377–392.MATHMathSciNetGoogle Scholar
  226. L. Natarajan, E. Rodriguez-Carrington and J. A. Wolf, The Bott–Borel–Weil theorem for direct limit groups, Trans. Amer. Math. Soc., 353 (2001), 4583–4622.MATHMathSciNetCrossRefGoogle Scholar
  227. D. S. Nathan, One-parameter groups of transformations in abstract vector spaces, Duke Math. J., 1 (1935), 518–526.MATHMathSciNetCrossRefGoogle Scholar
  228. K.-H. Neeb, Holomorphic highest weight representations of infinite dimensional complex classical groups, J. Reine Angew. Math., 497 (1998), 171–222.MATHMathSciNetGoogle Scholar
  229. K.-H. Neeb, Holomorphy and Convexity in Lie Theory, Expositions in Mathematics, 28, de Gruyter Verlag, Berlin, 1999.Google Scholar
  230. K.-H. Neeb, Representations of infinite dimensional groups, In: Infinite Dimensional Kähler Manifolds, (eds. A. Huckleberry and T. Wurzbacher), DMV Sem., 31, Birkhäuser, 2001, pp. 131–178.Google Scholar
  231. K.-H. Neeb, Locally finite Lie algebras with unitary highest weight representations, Manuscripta Math., 104 (2001), 343–358.MathSciNetCrossRefGoogle Scholar
  232. K.-H. Neeb, Central extensions of infinite-dimensional Lie groups, Ann. Inst. Fourier, 52 (2002), 1365–1442.MATHMathSciNetGoogle Scholar
  233. K.-H. Neeb, Classical Hilbert–Lie groups, their extensions and their homotopy groups, In: Geometry and Analysis on Finite and Infinite-dimensional Lie Groups, (eds. A. Strasburger, W. Wojtynski, J. Hilgert and K.-H. Neeb), Banach Center Publ., 55, Warszawa, 2002, pp. 87–151.Google Scholar
  234. K.-H. Neeb, A Cartan–Hadamard Theorem for Banach–Finsler manifolds, Geom. Dedicata, 95 (2002), 115–156.MATHMathSciNetCrossRefGoogle Scholar
  235. K.-H. Neeb, Universal central extensions of Lie groups, Acta Appl. Math., 73 (2002), 175–219.MATHMathSciNetCrossRefGoogle Scholar
  236. K.-H. Neeb, Locally convex root graded Lie algebras, Trav. Math., 14 (2003), 25–120.MathSciNetGoogle Scholar
  237. K.-H. Neeb, Abelian extensions of infinite-dimensional Lie groups, Trav. Math., 15 (2004), 69–194.MathSciNetGoogle Scholar
  238. K.-H. Neeb, Infinite-dimensional Lie groups and their representations, In: Lie Theory: Lie Algebras and Representations, (eds. J. P. Anker and B. Ørsted), Progr. Math., 228, Birkhäuser, 2004, pp. 213–328.Google Scholar
  239. K.-H. Neeb, Current groups for non-compact manifolds and their central extensions, In: Infinite Dimensional Groups and Manifolds, (ed. T. Wurzbacher), IRMA Lect. Math. Theor. Phys., 5, de Gruyter Verlag, Berlin, 2004, pp. 109–183.Google Scholar
  240. K.-H. Neeb, Non-abelian extensions of infinite-dimensional Lie groups, Ann. Inst. Fourier, to appear.Google Scholar
  241. K.-H. Neeb, Lie algebra extensions and higher order cocycles, J. Geom. Symmetry Phys., 5 (2006), 48–74.MATHMathSciNetGoogle Scholar
  242. K.-H. Neeb, Non-abelian extensions of topological Lie algebras, Comm. Algebra, 34 (2006), 991–1041.MATHMathSciNetCrossRefGoogle Scholar
  243. K.-H. Neeb, On the period group of a continuous inverse algebra, in preparation.Google Scholar
  244. K.-H. Neeb and N. Stumme, On the classification of locally finite split simple Lie algebras, J. Reine Angew. Math., 533 (2001), 25–53.MATHMathSciNetGoogle Scholar
  245. K.-H. Neeb and C. Vizman, Flux homomorphisms and principal bundles over infinite-dimensional manifolds, Monatsh. Math., 139 (2003), 309–333.MATHMathSciNetCrossRefGoogle Scholar
  246. K.-H. Neeb and F. Wagemann, The second cohomology of current algebras of general Lie algebras, Canad. J. Math., to appear.Google Scholar
  247. K.-H. Neeb and F. Wagemann, Lie group structures on groups of maps on non-compact manifolds, in preparation.Google Scholar
  248. E. Neher, Generators and relations for 3-graded Lie algebras, J. Algebra, 155 (1993), 1–35.MATHMathSciNetCrossRefGoogle Scholar
  249. E. Neher, Lie algebras graded by 3-graded root systems and Jordan pairs covered by grids, Amer. J. Math., 118 (1996), 439–491.MATHMathSciNetGoogle Scholar
  250. J. von Neumann, Über die analytischen Eigenschaften von Gruppen linearer Transformationen, Math. Z., 30 (1929), 3–42.MATHMathSciNetCrossRefGoogle Scholar
  251. P. J. Olver, Applications of Lie Groups to Differential Equations, second edition, Grad. Texts in Math., 107, Springer-Verlag, New York, 1993.Google Scholar
  252. H. Omori, On the group of diffeomorphisms on a compact manifold, In: Global Analysis, Proc. Sympos. Pure Math., 15, Berkeley, Calif., 1968, Amer. Math. Soc., Providence, R.I., 1970, pp. 167–183.Google Scholar
  253. H. Omori, Groups of iffeomorphisms and their subgroups, Trans. Amer. Math. Soc., 179 (1973), 85–122.MATHMathSciNetCrossRefGoogle Scholar
  254. H. Omori, Infinite Dimensional Lie Transformation Groups, Lecture Notes Math., 427, Springer-Verlag, Berlin-New York, 1974.MATHGoogle Scholar
  255. H. Omori, On Banach–Lie groups acting on finite-dimensional manifolds, Tôhoku Math. J., 30 (1978), 223–250.MATHMathSciNetGoogle Scholar
  256. H. Omori, A method of classifying expansive singularities, J. Differential. Geom., 15 (1980), 493–512.MATHMathSciNetGoogle Scholar
  257. H. Omori, A remark on non-enlargible Lie algebras, J. Math. Soc. Japan, 33 (1981), 707–710.MATHMathSciNetGoogle Scholar
  258. H. Omori, Infinite-Dimensional Lie Groups, Transl. Math. Monogr., 158, Amer. Math. Soc., 1997.Google Scholar
  259. H. Omori and P. de la Harpe, Opération de groupes de Lie banachiques sur les variétés différentielles de dimension finie, C. R. Acad. Sci. Paris Sér. A-B, 273 (1971), A395–A397.Google Scholar
  260. H. Omori and P. de la Harpe, About interactions between Banach–Lie groups and finite dimensional manifolds, J. Math. Kyoto Univ., 12 (1972), 543–570.MATHMathSciNetGoogle Scholar
  261. H. Omori, Y. Maeda and A. Yoshioka, On regular Fréchet-Lie groups. I. Some differential geometrical expressions of Fourier integral operators on a Riemannian manifold, Tokyo J. Math., 3 (1980), 353–390.MATHMathSciNetGoogle Scholar
  262. H. Omori, Y. Maeda and A. Yoshioka, On regular Fréchet-Lie groups. II. Composition rules of Fourier-integral operators on a Riemannian manifold, Tokyo J. Math., 4 (1981), 221–253.MATHMathSciNetGoogle Scholar
  263. H. Omori, Y. Maeda, A. Yoshioka and O. Kobayashi, On regular Fréchet-Lie groups. III. A second cohomology class related to the Lie algebra of pseudodifferential operators of order one, Tokyo J. Math., 4 (1981), 255–277.MATHMathSciNetGoogle Scholar
  264. H. Omori, Y. Maeda, A. Yoshioka and O. Kobayashi, On regular Fréchet-Lie groups IV. Definition and fundamental theorems, Tokyo J. Math., 5 (1982), 365–398.MATHMathSciNetCrossRefGoogle Scholar
  265. H. Omori, Y. Maeda, A. Yoshioka and O. Kobayashi, On regular Fréchet-Lie groups. V. Several basic properties, Tokyo J. Math., 6 (1983), 39–64.MATHMathSciNetGoogle Scholar
  266. H. Omori, Y. Maeda, A. Yoshioka and O. Kobayashi, On regular Fréchet-Lie groups. VI. Infinite-dimensional Lie groups which appear in general relativity, Tokyo J. Math., 6 (1983), 217–246.MATHMathSciNetGoogle Scholar
  267. K. Ono, Floer-Novikov cohomology and the flux conjecture, preprint, 2004.Google Scholar
  268. J. T. Ottesen, Infinite Dimensional Groups and Algebras in Quantum Physics, Springer-Verlag, Lecture Notes in Phys., m 27, 1995.Google Scholar
  269. R. S. Palais, A Global Formulation of the Lie Theory of Transformation Groups, Mem. Amer. Math. Soc., 22, Amer. Math. Soc., 1957.Google Scholar
  270. V. P. Palamodov, Homological methods in the theory of locally convex spaces, Russian Math. Surveys, 26 (1971), 1–64.MATHMathSciNetCrossRefGoogle Scholar
  271. J. Palis, On Morse-Smale dynamical systems, Topology, 8 (1968), 385–404.MathSciNetCrossRefGoogle Scholar
  272. J. Palis, Vector fields generate few diffeomorphisms, Bull. Amer. Math. Soc., 80 (1974), 503–505.MATHMathSciNetGoogle Scholar
  273. V. G. Pestov, Nonstandard hulls of Banach–Lie groups and algebras, Nova J. Algebra Geom., 1 (1992), 371–381.MATHMathSciNetGoogle Scholar
  274. V. G. Pestov, Free Banach–Lie algebras, couniversal Banach–Lie groups, and more, Pacific J. Math., 157 (1993), 137–144.MATHMathSciNetGoogle Scholar
  275. V. G. Pestov, Enlargible Banach–Lie algebras and free topological groups, Bull. Austral. Math. Soc., 48 (1993), 13–22.MATHMathSciNetGoogle Scholar
  276. V. G. Pestov, Correction to “Free Banach–Lie algebras, couniversal Banach–Lie groups, and more”, Pacific J. Math., 171 (1995), 585–588.MATHMathSciNetGoogle Scholar
  277. V. G. Pestov, Regular Lie groups and a theorem of Lie-Palais, J. Lie Theory, 5 (1995), 173–178.MATHMathSciNetGoogle Scholar
  278. D. Pickrell, Invariant Measures for Unitary Groups Associated to Kac-Moody Lie Algebras, Mem. Amer. Math. Soc., 693, 2000.Google Scholar
  279. D. Pickrell, On the action of the group of diffeomorphisms of a surface on sections of the determinant line bundle, Pacific J. Math., 193 (2000), 177–199.MATHMathSciNetCrossRefGoogle Scholar
  280. D. Pisanelli, An extension of the exponential of a matrix and a counter example to the inversion theorem in a space H(K), Rend. Mat. (6), 9 (1976), 465–475.MATHMathSciNetGoogle Scholar
  281. D. Pisanelli, An example of an infinite Lie group, Proc. Amer. Math. Soc., 62 (1977), 156–160.MATHMathSciNetCrossRefGoogle Scholar
  282. D. Pisanelli, The second Lie theorem in the group Gh(n, \({\user2{\mathbb{C}}}\)), In: Advances in Holomorphy, (ed. J. A. Barroso), North Holland Publ., 1979.Google Scholar
  283. L. Polterovich, The geometry of the group of symplectic diffeomorphisms, Lectures Math., ETH Zürich, Birkhäuser, 2001.Google Scholar
  284. L. Pontrjagin, Topological Groups, Princeton Math. Ser., 2, Princeton University Press, Princeton, 1939.Google Scholar
  285. A. Pressley and G. Segal, Loop Groups, Oxford University Press, Oxford, 1986.MATHGoogle Scholar
  286. M. E. Pursell, Algebraic structures associated with smooth manifolds, Thesis, Purdue Univ., 1952.Google Scholar
  287. M. E. Pursell and M. E. Shanks, The Lie algebra of a smooth manifold, Proc. Amer. Math. Soc., 5 (1954), 468–472.MATHMathSciNetCrossRefGoogle Scholar
  288. C. R. Putnam and A. Winter, The orthogonal group in Hilbert space, Amer. J. Math., 74 (1952), 52–78.MATHMathSciNetCrossRefGoogle Scholar
  289. T. Ratiu and A. Odzijewicz, Banach Lie–Poisson spaces and reduction, Comm. Math. Phys., 243 (2003), 1–54.MATHMathSciNetCrossRefGoogle Scholar
  290. T. Ratiu and A. Odzijewicz, Extensions of Banach Lie–Poisson spaces, J. Funct. Anal., 217 (2004), 103–125.MATHMathSciNetCrossRefGoogle Scholar
  291. T. Ratiu and R. Schmid, The differentiable structure of three remarkable diffeomorphism groups, Math. Z., 177 (1981), 81–100.MATHMathSciNetCrossRefGoogle Scholar
  292. J. F. Ritt, Differential groups and formal Lie theory for an infinite number of parameters, Ann. of Math. (2), 52 (1950), 708–726.MATHMathSciNetCrossRefGoogle Scholar
  293. T. Robart, Groupes de Lie de dimension infinie. Second et troisième théorèmes de Lie. I. Groupes de première espèce, C. R. Acad. Sci. Paris Sér. I Math., 322 (1996), 1071–1074.MATHMathSciNetGoogle Scholar
  294. T. Robart, Sur l’intégrabilité des sous-algèbres de Lie en dimension infinie, Canad. J. Math., 49 (1997), 820–839.MATHMathSciNetGoogle Scholar
  295. T. Robart, Around the exponential mapping, In: Infinite Dimensional Lie Groups in Geometry and Representation Theory, World Sci. Publ., River Edge, NJ, 2002, pp. 11–30.Google Scholar
  296. T. Robart, On Milnor’s regularity and the path-functor for the class of infinite dimensional Lie algebras of CBH type, Algebras Groups Geom., 21 (2004), 367–386.MATHMathSciNetGoogle Scholar
  297. T. Robart and N. Kamran, Sur la théorie locale des pseudogroupes de transformations continus infinis. I, Math. Ann., 308 (1997), 593–613.MATHMathSciNetCrossRefGoogle Scholar
  298. E. Rodriguez-Carrington, Lie groups associated to Kac–Moody Lie algebras: an analytic approach, In: Infinite-dimensional Lie Algebras and Groups, Luminy-Marseille, 1988, Adv. Ser. Math. Phys., 7, World Sci. Publ., Teaneck, NJ, 1989, pp. 57–69.Google Scholar
  299. C. Roger, Extensions centrales d’algèbres et de groupes de Lie de dimension infinie, algèbres de Virasoro et généralisations, Rep. Math. Phys., 35 (1995), 225–266.MATHMathSciNetCrossRefGoogle Scholar
  300. J. Rosenberg, Algebraic K-theory and its Applications, Grad. Texts in Math., 147, Springer-Verlag, 1994.Google Scholar
  301. W. Rudin, Functional Analysis, McGraw Hill, 1973.Google Scholar
  302. R. Schmid, Infinite-dimensional Hamiltonian Systems, Monographs and Textbooks in Physical Science, Lecture Notes, 3, Bibliopolis, Naples, 1987.Google Scholar
  303. R. Schmid, Infinite dimensional Lie groups with applications to mathematical physics, J. Geom. Symmetry Phys., 1 (2004), 54–120.MATHMathSciNetGoogle Scholar
  304. R. Schmid, M. Adams and T. Ratiu, The group of Fourier integral operators as symmetry group, In: XIIIth International Colloquium on Group Theoretical Methods in Physics, College Park, Md., 1984, World Sci. Publ., Singapore, 1984, pp. 246–249.Google Scholar
  305. J. R. Schue, Hilbert space methods in the theory of Lie algebras, Trans. Amer. Math. Soc., 95 (1960), 69–80.MATHMathSciNetCrossRefGoogle Scholar
  306. J. R. Schue, Cartan decompositions for L *-algebras, Trans. Amer. Math. Soc., 98 (1961), 334–349.MATHMathSciNetCrossRefGoogle Scholar
  307. F. Schur, Neue Begründung der Theorie der endlichen Transformationsgruppen, Math. Ann., 35 (1890), 161–197.MATHMathSciNetCrossRefGoogle Scholar
  308. F. Schur, Beweis für die Darstellbarkeit der infinitesimalen Transformationen aller transitiven endlichen Gruppen durch Quotienten beständig convergenter Potenzreihen, Leipz. Ber., 42 (1890), 1–7.MATHGoogle Scholar
  309. G. Segal, Unitary representations of some infinite-dimensional groups, Comm. Math. Phys., 80 (1981), 301–342.MATHMathSciNetCrossRefGoogle Scholar
  310. J.-P. Serre, Lie Algebras and Lie Groups, Lecture Notes in Math., 1500, Springer-Verlag, 1965 (1st ed.).Google Scholar
  311. I. M. Singer and S. Sternberg, The infinite groups of Lie and Cartan. I. The transitive groups, J. Anal. Math., 15 (1965), 1–114.MATHMathSciNetGoogle Scholar
  312. J.-M. Souriau, Groupes différentiels de physique mathématique, In: Feuilletages et Quantification Géometrique, (eds. P. Dazord and N. Desolneux-Moulis), Journ. lyonnaises Soc. math. France, 1983, Sémin. sud-rhodanien de Géom. II, Hermann, Paris, 1984, pp. 73–119.Google Scholar
  313. J.-M. Souriau, Un algorithme générateur de structures quantiques, Soc. Math. Fr., Astérisque, hors série, 1985, 341–399.Google Scholar
  314. S. Sternberg, Infinite Lie groups and the formal aspects of dynamical systems, J. Math. Mech., 10 (1961), 451–474.MATHMathSciNetGoogle Scholar
  315. N. Stumme, The Structure of Locally Finite Split Lie algebras, Ph. D. thesis, Darmstadt University of Technology, 1999.Google Scholar
  316. H. J. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc., 180 (1973), 171–188.MATHMathSciNetCrossRefGoogle Scholar
  317. K. Suto, Groups associated with unitary forms of Kac–Moody algebras, J. Math. Soc. Japan, 40 (1988), 85–104.MATHMathSciNetCrossRefGoogle Scholar
  318. K. Suto, Borel–Weil type theorem for the flag manifold of a generalized Kac–Moody algebra, J. Algebra, 193 (1997), 529–551.MATHMathSciNetCrossRefGoogle Scholar
  319. R. G. Swan, Vector bundles and projective modules, Trans. Amer. Math. Soc., 105 (1962), 264–277.MATHMathSciNetCrossRefGoogle Scholar
  320. S. Swierczkowski, Embedding theorems for local analytic groups, Acta Math., 114 (1965), 207–235.MATHMathSciNetCrossRefGoogle Scholar
  321. S. Swierczkowski, Cohomology of local group extensions, Trans. Amer. Math. Soc., 128 (1967), 291–320.MATHMathSciNetCrossRefGoogle Scholar
  322. S. Swierczkowski, The path-functor on Banach Lie algebras, Nederl. Akad. Wet., Proc. Ser. A, 74; Indag. Math., 33 (1971), 235–239.Google Scholar
  323. A. Tagnoli, La varietà analitiche reali come spazi omogenei, Boll. Un. Mat. Ital. (s4), 1 (1968), 422–426.MathSciNetGoogle Scholar
  324. J. Tits, Liesche Gruppen und Algebren, Springer-Verlag, 1983.Google Scholar
  325. F. Treves, Topological Vector Spaces, Distributions, and Kernels, Academic Press, New York, 1967.MATHGoogle Scholar
  326. H. Upmeier, Symmetric Banach Manifolds and Jordan C *-algebras, North Holland Mathematics Studies, 1985.Google Scholar
  327. V. S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations, Grad. Texts in Math., 102, Springer-Verlag, 1984.Google Scholar
  328. D. Vogt, On the functors Ext1 (E,F) for Fréchet spaces, Studia Math., 85 (1987), 163–197.MathSciNetGoogle Scholar
  329. L. Waelbroeck, Les algèbres à inverse continu, C. R. Acad. Sci. Paris, 238 (1954), 640–641.MATHMathSciNetGoogle Scholar
  330. L. Waelbroeck, Le calcul symbolique dans les algèbres commutatives, J. Math. Pures Appl., 33 (1954), 147–186.MATHMathSciNetGoogle Scholar
  331. L. Waelbroeck, Structure des algèbres à inverse continu, C. R. Acad. Sci. Paris, 238 (1954), 762–764.MATHMathSciNetGoogle Scholar
  332. L. Waelbroeck, Topological Vector Spaces and Algebras, Springer-Verlag, 1971.Google Scholar
  333. G. Warner, Harmonic Analysis on Semisimple Lie Groups I, Springer-Verlag, 1972.Google Scholar
  334. A. Weinstein, Symplectic structures on Banach manifolds, Bull. Amer. Math. Soc., 75 (1969), 1040–1041.MATHMathSciNetCrossRefGoogle Scholar
  335. D. Werner, Funktionalanalysis, Springer-Verlag, 1995.Google Scholar
  336. H. Wielandt, Über die Unbeschränktheit der Operatoren der Quantenmechanik, Math. Ann., 121 (1949), p. 21.MATHMathSciNetCrossRefGoogle Scholar
  337. Chr. Wockel, The Topology of Gauge Groups, submitted, math-ph/0504076.Google Scholar
  338. Chr. Wockel, Smooth Extensions and Spaces of Smooth and Holomorphic Mappings, J. Geom. Symmetry Phys., 5 (2006), 118–126, math.DG/0511064.MATHMathSciNetGoogle Scholar
  339. W. Wojtyński, Effective integration of Lie algebras, J. Lie Theory, 16 (2006), 601–620.MATHMathSciNetGoogle Scholar
  340. J. A. Wolf, Principal series representations of direct limit groups, Compositio Math., 141 (2005), 1504–1530.MATHCrossRefGoogle Scholar
  341. M. Wüstner, Supplements on the theory of exponential Lie groups, J. Algebra, 265 (2003), 148–170.MATHMathSciNetCrossRefGoogle Scholar
  342. M. Wüstner, The classification of all simple Lie groups with surjective exponential map, J. Lie Theory, 15 (2005), 269–278.MATHMathSciNetGoogle Scholar
  343. K. Yosida, On the groups embedded in the metrical complete ring, Japan. J. Math., 13 (1936), 7–26.MATHGoogle Scholar

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© The Mathematical Society of Japan and Springer-Verlag 2006

Authors and Affiliations

  1. 1.Technische Universität DarmstadtDarmstadtDeutschland

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