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Towards a Lie theory of locally convex groups. (English) Zbl 1161.22012
The article under review reports on the state of the art in the theory of Lie groups modelled on locally convex spaces. Of particular interest in this theory is the interplay between Lie groups (the global objects of the theory) and their Lie algebras (the infinitesimal objects). In contrast to the finite-dimensional case, considering local Lie groups (the local objects) is also of importance for a general understanding. The thoroughly written article aims at explaining these three concepts, along with its fundamental examples, and at describing the results that allow to translate between them. In addition, a long list of open problems is given, indicating many directions for further research.
In general, the passage from the global to the local level is given by restriction and the passage from the local to the infinitesimal level is given by differentiation at the identity element. This comprises the Lie functor and the core of the theory consists in determining how much this functor forgets and how much can be reconstructed from the infinitesimal and the local level. After a long introduction, referring also to the historical development, the subsequent sections of the article are devoted to different classes of locally convex Lie groups and Lie algebras and describe how much of the theory is known for each of them.
Section 1 digresses from the above described aims into a review of the theory of differential calculus on (not necessarily complete) locally convex spaces and the corresponding concept of locally convex manifolds. All important (and sometimes quite subtle) concepts for a good understanding of the remaining text, such as the Fundamental Theorem of Calculus, weak integrals, vector fields and differential forms, are introduced.
Section 2 deals with the general setup of locally convex Lie groups and their associated Lie algebras. The approach taken here is to define a locally convex Lie group to be a group, endowed with a locally convex manifold structure, such that the group operations are smooth. The corresponding Lie algebra is defined to be the Lie subalgebra of left invariant vector fields on this manifold. In addition, the concept of local Lie group and its Lie algebra are defined. Moreover, the first big class of examples, mapping groups, are introduced.
Section 3 is devoted to the most fundamental concept of interplay between Lie groups and Lie algebras, namely regularity. Regular Lie groups are required to possess solutions to particular initial value problems of ODE’s, determined by Lie algebra-valued curves. These Lie groups share many intuitive phenomena with finite-dimensional Lie groups. For instance, they allow to integrate Lie algebra homomorphisms to Lie group homomorphisms (to the corresponding simply connected cover), they possess exponential functions and allow for a Lie group-valued version of the Fundamental Theorem of Calculus. So far, all known Lie groups, modelled on Mackey-complete spaces, are regular and it is one of the fundamental problems of the theory to determine whether this is a theorem or if there exist counterexamples.
Section 4 deals with locally exponential Lie groups. These are, by definition, Lie groups that have an exponential function, restricting to a diffeomorphism on some zero neighbourhood of the Lie algebra. One big class of examples are Banach-Lie algebras and a typical example of a non-locally exponential but regular Lie group is the diffeomorphism group of a compact manifold. Since the exponential function allows to express the group multiplication in terms of the Lie bracket in the BCH series, there is a close relation to BCH-Lie algebras. As regular ones, locally exponential Lie groups allow a similar integration mechanism of Lie algebra homomorphisms, which is slightly weaker since the target group has to be regular. Moreover, locally exponential Lie groups and BCH-Lie groups allow for more structure on closed subgroups and quotient groups than infinite-dimensional Lie groups do in general, which is exposed in some detail. The remainder of the section concerns integral subgroups, i.e., subgroups of (locally exponential or BCH-) Lie groups induced from inclusions of subalgebras.
Section 5 treats the extensions theory of Lie groups and Lie algebras. After exposing the general ideas and some of the fundamental examples, the cohomology theory of Lie group and Lie algebra extensions are developed. This is done in a rather explicit fashion (largely without using tools from homological algebra), which allows to incorporate smoothness and continuity assumptions on the corresponding cocycles and coboundaries directly. Seemingly, this is done since the corresponding categories, in which the homological algebra would take place, fail to be abelian in this smooth setting (for the same reasons as in the topological setting). One of the main advantages that this direct approach to cohomology allows is to set up a Lie functor between Lie group cohomology and Lie algebra cohomology and one may ask similar questions as for the Lie group-Lie algebra relation now also on the level of cohomology. For instance, it is a reasonable and interesting question to determine which Lie algebra cocycles come from Lie group cocycles (or ”integrate” to Lie group cocycles), and which do not. This can be detected with the aid of vanishing theorems for the so called ”period homomorphisms”, which are associated to Lie algebra cocycles and, moreover, to arbitrary differential forms on Lie groups. Typically, the period homomorphism maps some homotopy group of the Lie group in question to some abelian group and a typical theorem asserts that a Lie algebra cocycle integrates to a Lie group cocycle if the associated period homomorphisms vanish. It thus may be understood as a coupling of algebraic (differential) information, coming from the Lie algebra, to topological (global) information from the Lie group.
Section 6 explains the problem of integrating a given Lie algebra to some Lie group. That this works in finite dimension is ensured by Lie’s Third Theorem, which fails in infinite dimensions. This, probably most fundamental difference to finite-dimensional Lie theory, was first observed by W. T. van Est and T. J. Korthagen in [Nederl. Akad. Wet., Proc., Ser. A 67, 15-31 (1964; Zbl 0121.27503)] and may nicely be shown with techniques from the previous section. In general, a Lie algebra is called integrable if it is (isomorphic to) the Lie algebra of a local Lie group and it is called enlargeable if this local group, in turn, comes from a global Lie group. The phenomena that may occur when integrating Lie algebras are delicate and versatile, for some Lie algebras fail to be integrable (like complexifications of Lie algebras of vector fields, cf. L. Lempert [Contemp. Math. 205, 169–176 (1997; Zbl 0887.22008)]) and others are integrable but not enlargeable (as the above counterexample of van Est and Korthagen). The section explains these phenomena, calculates explicitly some images of period maps and also describes some ad-hoc criteria for (non-) enlargeability and -integrability. Most of the time, the integration problem for locally exponential Lie algebras is considered, which allows for a systematic treatment by the period methods, mentioned above. The last part of the section treats non-locally exponential Lie algebras, for which no such general theory exists.
Section 7 is devoted to direct limits of Lie groups, one of the big classes of examples of infinite-dimensional Lie groups (the other classes, which have already appeared so far, roughly divide into unit groups of Banach algebras, mapping groups and groups of diffeomorphisms). This subject divides itself roughly into direct limits of finite-dimensional groups, for which a good amount of theory has been established by H. Glöckner [Compos. Math. 141, 1551–1577 (2005; Zbl 1082.22012)], and more generally direct limits of infinite-dimensional Lie groups, for which only case-by-case treatments exist.
Section 8 treats (locally exponential Lie subgroups of) unit groups of continuous inverse algebras (CIAs), which are natural generalisations of (subgroups of) unit groups of Banach algebras. These Lie groups are the proper replacement for linear groups in finite dimensions. Some examples and properties are discussed.
Section 9 considers actions of infinite-dimensional Lie groups. Some surprising phenomena occur here, for instance a relatively simple argument implies that there exist no natural Banach-Lie group structures on diffeomorphism groups of compact manifolds. One of the most prominent examples is the action of the diffeomorphism group of some compact manifold $$M$$ on the mapping group $$C^\infty(M,G)$$ by pre-composition, which is considered in some detail.
Section 10, eventually, is on projective limits of Lie groups. It mostly deals with projective limits of finite-dimensional Lie groups (pro-Lie groups), whose structure theory has recently been developed by K. H. Hofmann and S. A. Morris [The Lie theory of connected pro-Lie groups. EMS Tracts in Mathematics 2. (Zürich): European Mathematical Society (EMS). (2007; Zbl 1153.22006)]. In particular, a version of Lie’s Third Theorem and of the Levi-decomposition for pro-Lie groups are given and local exponentiality and regularity are discussed. A short remark also treats projective limits of infinite-dimensional Lie groups.
The article under review ends in an enormous list of references. It also contains several lists of open problems, so that in conclusion it is the perfect introduction to the research going on in the field.

##### MSC:
 2.2e+66 Infinite-dimensional Lie groups and their Lie algebras: general properties 2.2e+16 General properties and structure of real Lie groups
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##### References:
 [1] A. Abouqateb and K.-H. Neeb, Integration of locally exponential Lie algebras of vector fields, submitted. · Zbl 1135.22021 [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. [3] M. Adams, T. Ratiu and R. Schmid, A Lie group structure for pseudodifferential operators, Math. Ann., 273 (1986), 529–551. · Zbl 0587.58047 · doi:10.1007/BF01472130 [4] M. Adams, T. Ratiu and R. Schmid, A Lie group structure for Fourier integral operators, Math. Ann., 276 (1986), 19–41. · Zbl 0619.58010 · doi:10.1007/BF01450921 [5] I. Ado, Über die Darstellung von Lieschen Gruppen durch lineare Substitutionen, Bull. Soc. Phys. Math. Kazan (3), 7 (1936), 3–43. · Zbl 0014.34702 [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. · Zbl 0791.22010 [7] G. R. Allan, A spectral theory for locally convex algebras, Proc. London Math. Soc. (3), 15 (1965), 399–421. · Zbl 0138.38202 · doi:10.1112/plms/s3-15.1.399 [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. · Zbl 0879.17012 [9] B. Allison, G. Benkart and Y. Gao, Central extensions of Lie algebras graded by finite-root systems, Math. Ann., 316 (2000), 499–527. · Zbl 0989.17004 · doi:10.1007/s002080050341 [10] I. Amemiya, Lie algebra of vector fields and complex structure, J. Math. Soc. Japan, 27 (1975), 545–549. · Zbl 0311.57012 · doi:10.2969/jmsj/02740545 [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. [12] V. I. Arnold and B. A. Khesin, Topological Methods in Hydrodynamics, Springer-Verlag, 1998. · Zbl 0902.76001 [13] J. A. de Azcarraga and J. M. Izquierdo, Lie Groups, Lie Algebras, Cohomology and some Applications in Physics, Cambridge Monogr. Math. Phys., 1995. · Zbl 0836.22027 [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. · JFM 32.0159.01 · doi:10.1112/plms/s1-34.1.91 [15] H. F. Baker, On the calculation of the finite equations of a continuous group, Lond. M. S. Proc., 35 (1903), 332–333. · JFM 34.0181.03 · doi:10.1112/plms/s1-35.1.332 [16] A. Banyaga, The Structure of Classical Diffeomorphism Groups, Kluwer Academic Publishers, 1997. · Zbl 0874.58005 [17] A. Bastiani, Applications différentiables et variétés différentiables de dimension infinie, J. Anal. Math., 13 (1964), 1–114. · Zbl 0196.44103 · doi:10.1007/BF02786619 [18] E. J. Beggs, The de Rham complex on infinite dimensional manifolds, Quart. J. Math. Oxford (2), 38 (1987), 131–154. · Zbl 0636.58004 · doi:10.1093/qmath/38.2.131 [19] D. Beltiţă, Asymptotic products and enlargibility of Banach–Lie algebras, J. Lie Theory, 14 (2004), 215–226. · Zbl 1072.22010 [20] D. Beltiţă, Smooth Homogeneous Structures in Operator Theory, Chapman & Hall/CRC Monogr. Surv. Pure Appl. Math., 2006. [21] D. Beltiţă and K.-H. Neeb, Finite-dimensional Lie subalgebras of algebras with continuous inversion, preprint, 2006. [22] D. Beltiţă and T. S. Ratiu, Geometric representation theory for unitary groups of operator algebras, Adv. Math., to appear. [23] D. Beltiţă and T. S. Ratiu, Symplectic leaves in real Banach Lie–Poisson spaces, Geom. Funct. Anal., 15 (2005), 753–779. · Zbl 1083.58011 · doi:10.1007/s00039-005-0524-9 [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. · Zbl 1100.17012 · doi:10.1016/j.jalgebra.2003.10.034 [25] W. Bertram and K.-H. Neeb, Projective completions of Jordan pairs, Part II, Geom. Dedicata, 112 (2005), 75–115. · Zbl 1101.17019 · doi:10.1007/s10711-004-4197-6 [26] W. Bertram, H. Glöckner and K.-H. Neeb, Differential Calculus over General Base Fields and Rings, Expo. Math., 22 (2004), 213–282. · Zbl 1099.58006 [27] Y. Billig, Abelian extensions of the group of diffeomorphisms of a torus, Lett. Math. Phys., 64 (2003), 155–169. · Zbl 1079.58004 · doi:10.1023/A:1025750704319 [28] Y. Billig and A. Pianzola, Free Kac-Moody groups and their Lie algebras, Algebr. Represent. Theory, 5 (2002), 115–136. · Zbl 1007.17003 · doi:10.1023/A:1015691401406 [29] G. Birkhoff, Continuous groups and linear spaces, Mat. Sb., 1 (1936), 635–642. · JFM 62.1231.03 [30] G. Birkhoff, Analytic groups, Trans. Amer. Math. Soc., 43 (1938), 61–101. · JFM 64.1092.02 · doi:10.1090/S0002-9947-1938-1501934-4 [31] B. Blackadar, K-theory for Operator Algebras, 2nd edition, Cambridge Univ. Press, 1998. · Zbl 0913.46054 [32] J. Bochnak and J. Siciak, Analytic functions in topological vector spaces, Studia Math., 39 (1971), 77–112. · Zbl 0214.37703 [33] S. Bochner and D. Montgomery, Groups of differentiable and real or complex analytic transformations, Ann. of Math. (2), 46 (1945), 685–694. · Zbl 0061.04406 · doi:10.2307/1969204 [34] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Ergeb. Math. Grenzgeb., 80, Springer-Verlag, 1973. · Zbl 0271.46039 [35] H. Boseck, G. Czichowski and K.-P. Rudolph, Analysis on Topological Groups – General Lie Theory, Teubner, Leipzig, 1981. · Zbl 0558.22012 [36] J.-B. Bost, Principe d’Oka, K-theorie et systèmes dynamiques non-commutatifs, Invent. Math., 101 (1990), 261–333. · Zbl 0719.46038 · doi:10.1007/BF01231504 [37] R. Bott, On the characteristic classes of groups of diffeomorphisms, Enseign. Math. (2), 23 (1977), 209–220. · Zbl 0367.57004 [38] N. Bourbaki, Topological Vector Spaces, Chaps. 1–5, Springer-Verlag, 1987. · Zbl 0622.46001 [39] N. Bourbaki, Lie Groups and Lie Algebras, Chapter 1–3, Springer-Verlag, 1989. · Zbl 0672.22001 [40] G. E. Bredon, Topology and Geometry, Grad. Texts in Math., 139, Springer-Verlag, 1993. [41] J.-L. Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, Progr. Math., 107, Birkhäuser, 1993. · Zbl 0823.55002 [42] J. I. Burgos Gil, The Regulators of Beilinson and Borel, CRM Monogr., 15, Amer. Math. Soc., 2002. · Zbl 0994.19003 [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. [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. · JFM 28.0321.01 · doi:10.1112/plms/s1-28.1.381 [45] J. E. Campbell, On a law of combination of operators. (second paper), Proc. London Math. Soc., 28 (1897), 381–390. · JFM 28.0321.01 · doi:10.1112/plms/s1-28.1.381 [46] E. Cartan, Les groupes bilinéaires et les systèmes de nombres complexes, Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 12 (1898), B1–B64. · JFM 29.0097.03 [47] E. Cartan, L’intégration des systèmes d’équations aux diffrentielles totales, Ann. Sci. École Norm. Sup. (3), 18 (1901), 241–311. · JFM 32.0351.04 [48] E. Cartan, Sur la structure des groups infinies des transformations, Ann. Sci. École. Norm. Sup., 21 (1904), 153–206; 22 (1905), 219–308. · JFM 35.0176.04 [49] E. Cartan, Le troisième théorème fondamental de Lie, C. R. Math. Acad. Sci. Paris, 190 (1930), 914–916, 1005–1007. · JFM 56.0373.01 [50] E. Cartan, La topologie des groupes de Lie. (Exposés de géométrie Nr. 8.), Actualités. Sci. Indust., 358 (1936), p. 28. · JFM 62.0441.03 [51] E. Cartan, La topologie des espaces représentifs de groupes de Lie, Oeuvres I, Gauthier–Villars, Paris, 2 (1952), 1307–1330. [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. · Zbl 0915.22010 · doi:10.1142/S0129055X9800029X [53] P. Chernoff and J. Marsden, On continuity and smoothness of group actions, Bull. Amer. Math. Soc., 76 (1970), 1044–1049. · Zbl 0202.23202 · doi:10.1090/S0002-9904-1970-12552-6 [54] C. Chevalley, Theory of Lie Groups I, Princeton Univ. Press, 1946. · Zbl 0063.00842 [55] C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc., 63 (1948), 85–124. · Zbl 0031.24803 · doi:10.1090/S0002-9947-1948-0024908-8 [56] A. Connes, Non-commutative Geometry, Academic Press, 1994. [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. · Zbl 0763.46052 · doi:10.1017/S0305004100068626 [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. · Zbl 1028.58011 · doi:10.1016/S0393-0440(02)00117-1 [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. · Zbl 0834.22018 [60] J. Delsartes, Les groups de transformations linéaires dans l’espace de Hilbert, Mém. Sci. Math., 57, Paris. [61] I. Dimitrov and I. Penkov, Weight modules of direct limit Lie algebras, Internat. Math. Res. Notices, 5 (1999), 223–249. · Zbl 0917.17002 · doi:10.1155/S1073792899000124 [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. · Zbl 0596.58010 [63] A. Douady and M. Lazard, Espaces fibrés en algèbres de Lie et en groupes, Invent. Math., 1 (1966), 133–151. · Zbl 0144.01804 · doi:10.1007/BF01389725 [64] A. Dress, Newman’s Theorem on transformation groups, Topology, 8 (1969), 203–207. · Zbl 0176.53201 · doi:10.1016/0040-9383(69)90010-X [65] E. B. Dynkin, Calculation of the coefficients in the Campbell–Hausdorff formula (Russian), Dokl. Akad. Nauk. SSSR (N.S.), 57 (1947), 323–326. [66] E. B. Dynkin, Normed Lie Algebras and Analytic Groups, Amer. Math. Soc. Transl., 97 (1953), p. 66. · Zbl 0052.26202 [67] D. G. Ebin, The manifold of Riemannian metrics, In: Global Analysis, Berkeley, Calif., 1968, Proc. Sympos. Pure Math., 15 (1970), pp. 11–40. · Zbl 0172.22905 [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. · Zbl 0183.54502 · doi:10.1090/S0002-9904-1969-12315-3 [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. · Zbl 0211.57401 · doi:10.2307/1970699 [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. [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. [72] J. Eells, Jr., A setting for global analysis, Bull. Amer. Math. Soc., 72 (1966), 751–807. · doi:10.1090/S0002-9904-1966-11558-6 [73] J. Eichhorn and R. Schmid, Form preserving diffeomorphisms on open manifolds, Ann. Global Anal. Geom., 14 (1996), 147–176. · Zbl 0862.58007 · doi:10.1007/BF00127971 [74] J. Eichhorn and R. Schmid, Lie groups of Fourier integral operators on open manifolds, Comm. Anal. Geom., 9 (2001), 983–1040. · Zbl 1038.58032 [75] M. Eichler, A new proof of the Baker–Campbell–Hausdorff formula, J. Math. Soc. Japan, 20 (1968), 23–25. · Zbl 0157.07601 · doi:10.2969/jmsj/02010023 [76] W. T. van Est, Local and global groups, Proc. Konink. Nederl. Akad. Wetensch. Ser. A, 65; Indag. Math., 24 (1962), 391–425. · Zbl 0109.02003 [77] W. T. van Est, On Ado’s theorem, Proc. Konink. Nederl. Akad. Wetensch. Ser. A, 69; Indag. Math., 28 (1966), 176–191. · Zbl 0156.03901 [78] W. T. van Est, Rapport sur les S-atlas, Astérisque, 116 (1984), 235–292. · Zbl 0543.58003 [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. [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. · Zbl 0121.27503 [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. · Zbl 0268.43011 [82] P. I. Etinghof and I. B. Frenkel, Central extensions of current groups in two dimensions, Comm. Math. Phys., 165 (1994), 429–444. · Zbl 0822.22014 · doi:10.1007/BF02099419 [83] R. P. Filipkiewicz, Isomorphisms between diffeomorphism groups, Ergodic Theory Dynam. Systems, 2 (1983), 159–171. · Zbl 0521.58016 [84] K. Floret, Lokalkonvexe Sequenzen mit kompakten Abbildungen, J. Reine Angew. Math., 247 (1971), 155–195. · Zbl 0209.43001 · doi:10.1515/crll.1971.247.155 [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. [86] A. Frölicher and W. Bucher, Calculus in Vector Spaces without Norm, Lecture Notes in Math., 30, Springer-Verlag, 1966. · Zbl 0156.38303 [87] A. Frölicher and A. Kriegl, Linear Spaces and Differentiation Theory, J. Wiley, Interscience, 1988. [88] D. B. Fuks, Cohomology of Infinite-Dimensional Lie Algebras, Consultants Bureau, New York, London, 1986. · Zbl 0667.17005 [89] G. Galanis, Projective limits of Banach–Lie groups, Period. Math. Hungar., 32 (1996), 179–191. · Zbl 0866.58009 · doi:10.1007/BF02109787 [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. · Zbl 0902.34052 [91] S. L. Glashow, and M. Gell-Mann, Gauge theories of vector particles, Ann. Physics, 15 (1961), 437–460. · Zbl 0099.43601 · doi:10.1016/0003-4916(61)90193-2 [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. [93] H. Glöckner, Algebras whose groups of units are Lie groups, Studia Math., 153 (2002), 147–177. · Zbl 1009.22021 · doi:10.4064/sm153-2-4 [94] H. Glöckner, Lie group structures on quotient groups and universal complexifications for infinite-dimensional Lie groups, J. Funct. Anal., 194 (2002), 347–409. · Zbl 1022.22021 · doi:10.1006/jfan.2002.3942 [95] H. Glöckner, Patched locally convex spaces, almost local mappings, and diffeomorphism groups of non-compact manifolds, TU Darmstadt, manuscript, 26.6.02. [96] H. Glöckner, Implicit functions from topological vector spaces to Banach spaces, Israel J. Math., to appear, math.GM/0303320. [97] H. Glöckner, Direct limit Lie groups and manifolds, J. Math. Kyoto Univ., 43 (2003), 1–26. [98] H. Glöckner, Lie groups of measurable mappings, Canad. J. Math., 55 (2003), 969–999. · Zbl 1053.22013 · doi:10.4153/CJM-2003-039-9 [99] H. Glöckner, Tensor products in the category of topological vector spaces are not associative, Comment. Math. Univ. Carolin., 45 (2004), 607–614. · Zbl 1103.46001 [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. [101] H. Glöckner, Fundamentals of direct limit Lie theory, Compositio Math., 141 (2005), 1551–1577. · Zbl 1082.22012 · doi:10.1112/S0010437X05001491 [102] H. Glöckner, Discontinuous non-linear mappings on locally convex direct limits, Publ. Math. Debrecen, 68 (2006) 1–13. [103] H. Glöckner, Fundamental problems in the theory of infinite-dimensional Lie groups, J. Geom. Symmetry Phys., 5 (2006), 24–35. · Zbl 1109.22014 [104] H. Glöckner, Direct limits of infinite-dimensional Lie groups compared to direct limits in related categories, in preparation. · Zbl 1119.22012 [105] H. Glöckner, Direct limit groups do not have small subgroups, preprint, math.GR/0602407. [106] H. Glöckner and K.-H. Neeb, Banach–Lie quotients, enlargibility, and universal complexifications, J. Reine Angew. Math., 560 (2003), 1–28. · Zbl 1029.22029 · doi:10.1515/crll.2003.056 [107] H. Glöckner and K.-H. Neeb, Infinite-dimensional Lie groups, Vol. I, Basic Theory and Main Examples, book in preparation. · Zbl 1167.22013 [108] H. Glöckner and K.-H. Neeb, Infinite-dimensional Lie groups, Vol. II, Geometry and Topology, book in preparation. · Zbl 1167.22013 [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. [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. · Zbl 0514.22012 · doi:10.1515/crll.1984.347.69 [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. · Zbl 0636.22013 · doi:10.1016/0022-1236(85)90090-4 [112] M. Goto, On an arcwise connected subgroup of a Lie group, Proc. Amer. Math. Soc., 20 (1969), 157–162. · Zbl 0182.04602 · doi:10.1090/S0002-9939-1969-0233923-X [113] J. Grabowski Free subgroups of diffeomorphism groups, Fund. Math., 131 (1988), 103–121. · Zbl 0666.58011 [114] J. Grabowski, Derivative of the exponential mapping for infinite-dimensional Lie groups, Ann. Global Anal. Geom., 11 (1993), 213–220. · Zbl 0836.22028 [115] J. M. Gracia-Bondia, J. C. Vasilly and H. Figueroa, Elements of Non-commutative Geometry, Birkhäuser Advanced Texts, Birkhäuser, Basel, 2001. [116] B. Gramsch, Relative Inversion in der Störungstheorie von Operatoren und $$\Psi$$-Algebren, Math. Ann., 269 (1984), 22–71. · Zbl 0661.47037 · doi:10.1007/BF01455995 [117] H. Grundling and K.- H. Neeb, Lie group extensions associated to modules of continuous inverse algebras, in preparation. [118] J. Gutknecht, Die C $$\Gamma$$-Struktur auf der Diffeomorphismengruppe einer kompakten Mannigfaltigkeit, Ph. D. thesis, Eidgenössische Technische Hochschule Zürich, Diss. No. 5879, Juris Druck + Verlag, Zurich, 1977. [119] R. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc., 7 (1982), 65–222. · Zbl 0499.58003 · doi:10.1090/S0273-0979-1982-15004-2 [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. · Zbl 0256.22015 [121] L. A. Harris and W. Kaup, Linear algebraic groups in infinite dimensions, Illinois J. Math., 21 (1977), 666–674. · Zbl 0385.22011 [122] F. Hausdorff, Die symbolische Exponentialformel in der Gruppentheorie, Leipziger Berichte, 58 (1906), 19–48. · JFM 37.0176.02 [123] M. Hausner and J. T. Schwartz, Lie Groups; Lie Algebras, Gordon and Breach, New York, London, Paris, 1968. · Zbl 0192.35902 [124] G. Hector and E. Macías-Virgós, Diffeological groups, Res. Exp. Math., 25 (2002), 247–260. · Zbl 1018.58001 [125] A. Ya. Helemskii, Banach and Locally Convex Algebras, Oxford Sci. Publications, Oxford University Press, New York, 1993. [126] S. Hiltunen, Implicit functions from locally convex spaces to Banach spaces, Studia Math., 134 (1999), 235–250. · Zbl 0934.58008 [127] G. Hochschild, Group extensions of Lie groups I, II, Ann. of Math., 54 (1951), 96–109; 54 (1951), 537–551. · Zbl 0045.30802 [128] G. Hochschild, The Structure of Lie Groups, Holden Day, San Francisco, 1965. · Zbl 0131.02702 [129] K. H. Hofmann, Introduction to the Theory of Compact Groups. Part I, Dept. Math. Tulane Univ., New Orleans, LA, 1968. · Zbl 0229.22006 [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. [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. [132] K. H. Hofmann and S. A. Morris, The Structure of Compact Groups, de Gruyter Stud. Math., de Gruyter, Berlin, 1998. [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. · Zbl 1061.22001 · doi:10.1515/jgth.2005.8.1.115 [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). · Zbl 1153.22006 [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. · Zbl 1041.22005 · doi:10.1515/form.2004.004 [136] K. H. Hofmann and K.-H. Neeb, Pro-Lie groups which are infinite-dimensional Lie groups, submitted. · Zbl 1165.22017 [137] H. Hopf, Fundamentalgruppe und zweite Bettische Gruppe, Comment. Math. Helv., 14 (1942), 257–309. · Zbl 0027.09503 · doi:10.1007/BF02565622 [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. · Zbl 0049.33903 [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). · Zbl 0052.08701 [140] R. S. Ismagilov, Representations of Infinite-Dimensional Groups, Transl. Math. Monogr., 152 (1996). · Zbl 0856.22001 [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. · Zbl 0614.22006 [142] V. G. Kac, Infinite-dimensional Lie Algebras, Cambridge University Press, 1990. · Zbl 0716.17022 [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. · Zbl 0578.17014 [144] N. Kamran and T. Robart, A manifold structure for analytic Lie pseudogroups of infinite type, J. Lie Theory, 11 (2001), 57–80. · Zbl 0977.58021 [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. · Zbl 1086.58010 · doi:10.1155/S1073792804132388 [146] W. Kaup, Über die Klassifikation der symmetrischen hermiteschen Mannigfaltigkeiten unendlicher Dimension I, Math. Ann., 257 (1981), 463–486. · Zbl 0482.32010 · doi:10.1007/BF01465868 [147] W. Kaup, A Riemann mapping theorem for bounded symmetric domains in complex Banach spaces, Math. Z., 183 (1983), 503–529. · Zbl 0519.32024 · doi:10.1007/BF01173928 [148] W. Kaup, Über die Klassifikation der symmetrischen hermiteschen Mannigfaltigkeiten unendlicher Dimension II, Math. Ann., 262 (1983), 57–75. · Zbl 0504.32026 · doi:10.1007/BF01474170 [149] J. Kedra, D. Kotchick and S. Morita, Crossed flux homomorphisms and vanishing theorems for flux groups, preprint, Aug. 2005, math.AT/0503230. [150] H. H. Keller, Differential Calculus in Locally Convex Spaces, Springer-Verlag, 1974. · Zbl 0293.58001 [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. · Zbl 1147.22301 [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. · Zbl 0671.58007 · doi:10.1007/BF01077802 [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. · Zbl 0546.58005 · doi:10.1007/BF01438267 [154] N. Kopell, Commuting diffeomorphisms, Proc. Sympos. Pure Math., 14 (1970), 165–184. [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. [156] G. Köthe, Topological Vector Spaces I, Grundlehren der Math. Wissenschaften, 159, Springer-Verlag, Berlin etc., 1969. [157] A. Kriegl and P. Michor, The Convenient Setting of Global Analysis, Math. Surveys Monogr., 53 (1997). · Zbl 0889.58001 [158] A. Kriegl and P. Michor, Regular infinite-dimensional Lie groups, J. Lie Theory, 7 (1997), 61–99. · Zbl 0893.22012 [159] N. H. Kuiper, The homotopy type of the unitary group of Hilbert space, Topology, 3 (1965), 19–30. · Zbl 0129.38901 · doi:10.1016/0040-9383(65)90067-4 [160] S. Kumar, Kac-Moody Groups, their Flag Varieties and Representation Theory, Progr. Math., 204, Birkhäuser, Boston, MA, 2002. · Zbl 1026.17030 [161] M. Kuranishi, On the local theory of continuous infinite pseudo groups I, Nagoya Math. J., 15 (1959), 225–260. · Zbl 0212.56501 [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. · Zbl 0974.53062 [163] S. Lang, Fundamentals of Differential Geometry, Grad. Texts in Math., 191, Springer-Verlag, 1999. · Zbl 0932.53001 [164] V. T. Laredo, Integration of unitary representations of infinite dimensional Lie groups, J. Funct. Anal., 161 (1999), 478–508. · Zbl 0919.22007 · doi:10.1006/jfan.1998.3359 [165] R. K. Lashof, Lie algebras of locally compact groups, Pacific J. Math., 7 (1957), 1145–1162. · Zbl 0081.02204 [166] D. Laugwitz, Über unendliche kontinuierliche Gruppen. I. Grundlagen der Theorie; Untergruppen, Math. Ann., 130 (1955), 337–350. · Zbl 0065.26203 · doi:10.1007/BF01343900 [167] D. Laugwitz, Über unendliche kontinuierliche Gruppen. II. Strukturtheorie lokal Banachscher Gruppen, Bayer. Akad. Wiss. Math. Natur. Kl. Sitzungsber., 1956, 261–286 (1957). · Zbl 0077.03303 [168] M. Lazard and J. Tits, Domaines d’injectivité de l’application exponentielle, Topology, 4 (1966), 315–322. · Zbl 0156.03203 · doi:10.1016/0040-9383(66)90030-9 [169] P. Lecomte, Sur l’algèbre de Lie des sections d’un fibré en algèbre de Lie, Ann. Inst. Fourier, 30 (1980), 35–50. · Zbl 0433.58002 [170] P. Lecomte, Sur la suite exacte canonique associée à un fibré principal, Bull. Soc. Math. France, 13 (1985), 259–271. · Zbl 0592.55010 [171] L. Lempert, The Virasoro group as a complex manifold, Math. Res. Lett., 2 (1995), 479–495. · Zbl 0847.22006 [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. · Zbl 0887.22008 [173] J. A. Leslie, On a theorem of E. Cartan, Ann. Mat. Pura Appl. (4), 74 (1966), 173–177. · Zbl 0144.26605 · doi:10.1007/BF02416455 [174] J. A. Leslie, On a differential structure for the group of diffeomorphisms, Topology, 6 (1967), 263–271. · Zbl 0147.23601 · doi:10.1016/0040-9383(67)90038-9 [175] J. A. Leslie, Some Frobenius theorems in global analysis, J. Differential Geom., 2 (1968), 279–297. · Zbl 0169.53201 [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. · Zbl 0526.58011 · doi:10.1090/S0002-9947-1982-0675073-5 [177] J. A. Leslie, A Lie group structure for the group of analytic diffeomorphisms, Boll. Un. Mat. Ital. A (6), 2 (1983), 29–37. · Zbl 0525.58005 [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. · Zbl 0810.17015 [179] J. A. Leslie, Some integrable subalgebras of infinite-dimensional Lie groups, Trans. Amer. Math. Soc., 333 (1992), 423–443. · Zbl 0781.22015 · doi:10.2307/2154117 [180] J. A. Leslie, On the integrability of some infinite dimensional Lie algebras, Howard University, preprint, 1993. [181] J. A. Leslie, On a diffeological group realization of certain generalized symmetrizable Kac-Moody Lie algebras, J. Lie Theory, 13 (2003), 427–442. · Zbl 1119.17303 [182] D. Lewis, Formal power series transformations, Duke Math. J., 5 (1939), 794–805. · Zbl 0022.32703 · doi:10.1215/S0012-7094-39-00565-X [183] S. Lie, Theorie der Transformationsgruppen I, Math. Ann., 16 (1880), 441–528. · JFM 12.0292.01 · doi:10.1007/BF01446218 [184] S. Lie, Unendliche kontinuierliche Gruppen, Abh. Sächs. Ges. Wiss., 21 (1895), 43–150. [185] J.-L. Loday, Cyclic Homology, Grundlehren Math. Wiss., 301, Springer-Verlag, Berlin, 1998. [186] O. Loos, Symmetric Spaces I: General Theory, Benjamin, New York, Amsterdam, 1969. · Zbl 0175.48601 [187] M. V. Losik, Fréchet manifolds as diffeologic spaces, Russian Math., 36 (1992), 31–37. · Zbl 0774.58002 [188] D. Luminet and A. Valette, Faithful uniformly continuous representations of Lie groups, J. London Math. Soc. (2), 49 (1994), 100–108. · Zbl 0789.22010 [189] S. MacLane, Homology, Grundlehren Math. Wiss., 114, Springer-Verlag, 1963. [190] S. MacLane, Origins of the cohomology of groups, Enseig. Math., 24 (1978), 1–29. · Zbl 0379.18012 [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. · Zbl 0582.58034 · doi:10.3836/tjm/1270151569 [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. [193] P. Maier and K.-H. Neeb, Central extensions of current groups, Math. Ann., 326 (2003), 367–415. · Zbl 1029.22025 · doi:10.1007/s00208-003-0425-x [194] B. Maissen, Lie-Gruppen mit Banachräumen als Parameterräume, Acta Math., 108 (1962), 229–269. · Zbl 0207.33701 · doi:10.1007/BF02545768 [195] B. Maissen, Über Topologien im Endomorphismenraum eines topologischen Vektorraums, Math. Ann., 151 (1963), 283–285. · Zbl 0119.10002 · doi:10.1007/BF01470820 [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. · Zbl 0989.46039 · doi:10.1007/BF02255990 [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. · Zbl 0159.54801 [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. · Zbl 0211.57402 [199] L. Maurer, Über allgemeinere Invarianten-Systeme, Münchner Berichte, 43 (1888), 103–150. [200] W. Mayer and T. Y. Thomas, Foundations of the theory of Lie groups, Ann. of Math., 36 (1935), 770–822. · Zbl 0012.05502 · doi:10.2307/1968658 [201] D. McDuff, Enlarging the Hamiltonian group, preprint, May 2005, math.SG/0503268. · Zbl 1109.53078 [202] D. McDuff and D. Salamon, Introduction to Symplectic Topology, Oxford Math. Monogr., 1998. · Zbl 0844.58029 [203] E. Michael, Convex structures and continuous selections, Canad. J. Math., 11 (1959), 556–575. · Zbl 0093.36603 · doi:10.4153/CJM-1959-051-9 [204] A. D. Michal, Differential calculus in linear topological spaces, Proc. Nat. Acad. Sci. U. S. A., 24 (1938), 340–342. · JFM 64.0366.02 · doi:10.1073/pnas.24.8.340 [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. · JFM 66.0544.01 · doi:10.1073/pnas.26.5.356 [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. · Zbl 0060.27206 · doi:10.1073/pnas.31.9.315 [207] A. D. Michal, Differentiable infinite continuous groups in abstract spaces, Rev. Ci., Lima 50 (1948), 131–140. · Zbl 0031.31202 [208] A. D. Michal and V. Elconin, Differential properties of abstract transformation groups with abstract parameters, Amer. J. Math., 59 (1937), 129–143. · Zbl 0015.39401 · doi:10.2307/2371567 [209] P. W. Michor, Manifolds of Differentiable Mappings, Shiva Publishing, Orpington, Kent (U.K.), 1980. · Zbl 0433.58001 [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. · Zbl 0548.58001 [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. [212] P. W. Michor, Gauge Theory for Fiber Bundles, Bibliopolis, ed. di fil. sci., Napoli, 1991. [213] P. Michor and J. Teichmann, Description of infinite dimensional abelian regular Lie groups, J. Lie Theory, 9 (1999), 487–489. · Zbl 1012.22036 [214] J. Mickelsson, Kac-Moody groups, topology of the Dirac determinant bundle, and fermionization, Comm. Math. Phys., 110 (1987), 173–183. · Zbl 0625.58043 · doi:10.1007/BF01207361 [215] J. Mickelsson, Current algebras and groups, Plenum Press, New York, 1989. · Zbl 0726.22015 [216] J. Milnor, On infinite-dimensional Lie groups, Institute of Adv. Stud. Princeton, preprint, 1982. [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. [218] D. Montgomery and L. Zippin, Topological Transformation Groups, Interscience, New York, 1955. · Zbl 0068.01904 [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. · Zbl 0104.30503 · doi:10.1073/pnas.47.11.1824 [220] S. B. Myers, Algebras of differentiable functions, Proc. Amer. Math. Soc., 5 (1954), 917–922. · Zbl 0057.09504 · doi:10.1090/S0002-9939-1954-0065823-1 [221] T. Nagano, Linear differential systems with singularities and an application to transitive Lie algebras, J. Math. Soc. Japan, 18 (1966), 398–404. · Zbl 0147.23502 · doi:10.2969/jmsj/01840398 [222] M. Nagumo, Einige analytische Untersuchungen in linearen metrischen Ringen, Japan. J. Math., 13 (1936), 61–80. · Zbl 0015.24401 [223] L. Natarajan, E. Rodriguez-Carrington and J. A. Wolf, Differentiable structure for direct limit groups, Lett. Math. Phys., 23 (1991), 99–109. · Zbl 0762.22017 · doi:10.1007/BF00703721 [224] L. Natarajan, E. Rodriguez-Carrington and J. A. Wolf, Locally convex Lie groups, Nova J. Algebra Geom., 2 (1993), 59–87. · Zbl 0872.22012 [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. · Zbl 0826.22022 [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. · Zbl 0994.22013 · doi:10.1090/S0002-9947-01-02452-7 [227] D. S. Nathan, One-parameter groups of transformations in abstract vector spaces, Duke Math. J., 1 (1935), 518–526. · Zbl 0013.20904 · doi:10.1215/S0012-7094-35-00139-9 [228] K.-H. Neeb, Holomorphic highest weight representations of infinite dimensional complex classical groups, J. Reine Angew. Math., 497 (1998), 171–222. · Zbl 0894.22007 · doi:10.1515/crll.1998.038 [229] K.-H. Neeb, Holomorphy and Convexity in Lie Theory, Expositions in Mathematics, 28, de Gruyter Verlag, Berlin, 1999. · Zbl 0936.22001 [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. · Zbl 0995.22007 [231] K.-H. Neeb, Locally finite Lie algebras with unitary highest weight representations, Manuscripta Math., 104 (2001), 343–358. · Zbl 1033.17009 · doi:10.1007/s002290170033 [232] K.-H. Neeb, Central extensions of infinite-dimensional Lie groups, Ann. Inst. Fourier, 52 (2002), 1365–1442. · Zbl 1019.22012 [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. · Zbl 1010.22024 [234] K.-H. Neeb, A Cartan–Hadamard Theorem for Banach–Finsler manifolds, Geom. Dedicata, 95 (2002), 115–156. · Zbl 1027.58003 · doi:10.1023/A:1021221029301 [235] K.-H. Neeb, Universal central extensions of Lie groups, Acta Appl. Math., 73 (2002), 175–219. · Zbl 1019.22011 · doi:10.1023/A:1019743224737 [236] K.-H. Neeb, Locally convex root graded Lie algebras, Trav. Math., 14 (2003), 25–120. · Zbl 1062.22038 [237] K.-H. Neeb, Abelian extensions of infinite-dimensional Lie groups, Trav. Math., 15 (2004), 69–194. · Zbl 1079.22018 [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. [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. [240] K.-H. Neeb, Non-abelian extensions of infinite-dimensional Lie groups, Ann. Inst. Fourier, to appear. · Zbl 1167.22013 [241] K.-H. Neeb, Lie algebra extensions and higher order cocycles, J. Geom. Symmetry Phys., 5 (2006), 48–74. · Zbl 1105.53064 [242] K.-H. Neeb, Non-abelian extensions of topological Lie algebras, Comm. Algebra, 34 (2006), 991–1041. · Zbl 1158.17308 · doi:10.1080/00927870500441973 [243] K.-H. Neeb, On the period group of a continuous inverse algebra, in preparation. [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. · Zbl 0993.17011 · doi:10.1515/crll.2001.025 [245] K.-H. Neeb and C. Vizman, Flux homomorphisms and principal bundles over infinite-dimensional manifolds, Monatsh. Math., 139 (2003), 309–333. · Zbl 1029.22027 · doi:10.1007/s00605-002-0001-6 [246] K.-H. Neeb and F. Wagemann, The second cohomology of current algebras of general Lie algebras, Canad. J. Math., to appear. · Zbl 1162.17019 [247] K.-H. Neeb and F. Wagemann, Lie group structures on groups of maps on non-compact manifolds, in preparation. · Zbl 1143.22016 [248] E. Neher, Generators and relations for 3-graded Lie algebras, J. Algebra, 155 (1993), 1–35. · Zbl 0769.17019 · doi:10.1006/jabr.1993.1029 [249] E. Neher, Lie algebras graded by 3-graded root systems and Jordan pairs covered by grids, Amer. J. Math., 118 (1996), 439–491. · Zbl 0857.17019 · doi:10.1353/ajm.1996.0018 [250] J. von Neumann, Über die analytischen Eigenschaften von Gruppen linearer Transformationen, Math. Z., 30 (1929), 3–42. · JFM 55.0245.05 · doi:10.1007/BF01187749 [251] P. J. Olver, Applications of Lie Groups to Differential Equations, second edition, Grad. Texts in Math., 107, Springer-Verlag, New York, 1993. · Zbl 0785.58003 [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. [253] H. Omori, Groups of iffeomorphisms and their subgroups, Trans. Amer. Math. Soc., 179 (1973), 85–122. · Zbl 0269.58005 · doi:10.1090/S0002-9947-1973-0377975-0 [254] H. Omori, Infinite Dimensional Lie Transformation Groups, Lecture Notes Math., 427, Springer-Verlag, Berlin-New York, 1974. · Zbl 0328.58005 [255] H. Omori, On Banach–Lie groups acting on finite-dimensional manifolds, Tôhoku Math. J., 30 (1978), 223–250. · Zbl 0409.58009 · doi:10.2748/tmj/1178230027 [256] H. Omori, A method of classifying expansive singularities, J. Differential. Geom., 15 (1980), 493–512. · Zbl 0476.32010 [257] H. Omori, A remark on non-enlargible Lie algebras, J. Math. Soc. Japan, 33 (1981), 707–710. · Zbl 0463.22015 · doi:10.2969/jmsj/03340707 [258] H. Omori, Infinite-Dimensional Lie Groups, Transl. Math. Monogr., 158, Amer. Math. Soc., 1997. · Zbl 0871.58007 [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. · Zbl 0218.58003 [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. · Zbl 0271.58006 [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. · Zbl 0461.58003 · doi:10.3836/tjm/1270473002 [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. · Zbl 0486.58002 · doi:10.3836/tjm/1270215153 [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. · Zbl 0486.58003 · doi:10.3836/tjm/1270215154 [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. · Zbl 0515.58004 · doi:10.3836/tjm/1270214899 [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. · Zbl 0526.58003 · doi:10.3836/tjm/1270214325 [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. · Zbl 0537.58005 · doi:10.3836/tjm/1270213867 [267] K. Ono, Floer-Novikov cohomology and the flux conjecture, preprint, 2004. [268] J. T. Ottesen, Infinite Dimensional Groups and Algebras in Quantum Physics, Springer-Verlag, Lecture Notes in Phys., m 27, 1995. · Zbl 0848.22026 [269] R. S. Palais, A Global Formulation of the Lie Theory of Transformation Groups, Mem. Amer. Math. Soc., 22, Amer. Math. Soc., 1957. · Zbl 0178.26502 [270] V. P. Palamodov, Homological methods in the theory of locally convex spaces, Russian Math. Surveys, 26 (1971), 1–64. · Zbl 0247.46070 · doi:10.1070/RM1971v026n01ABEH003815 [271] J. Palis, On Morse-Smale dynamical systems, Topology, 8 (1968), 385–404. · Zbl 0189.23902 · doi:10.1016/0040-9383(69)90024-X [272] J. Palis, Vector fields generate few diffeomorphisms, Bull. Amer. Math. Soc., 80 (1974), 503–505. · Zbl 0296.57008 · doi:10.1090/S0002-9904-1974-13470-1 [273] V. G. Pestov, Nonstandard hulls of Banach–Lie groups and algebras, Nova J. Algebra Geom., 1 (1992), 371–381. · Zbl 0876.46037 [274] V. G. Pestov, Free Banach–Lie algebras, couniversal Banach–Lie groups, and more, Pacific J. Math., 157 (1993), 137–144. · Zbl 0785.22025 [275] V. G. Pestov, Enlargible Banach–Lie algebras and free topological groups, Bull. Austral. Math. Soc., 48 (1993), 13–22. · Zbl 0785.22024 · doi:10.1017/S0004972700015409 [276] V. G. Pestov, Correction to ”Free Banach–Lie algebras, couniversal Banach–Lie groups, and more”, Pacific J. Math., 171 (1995), 585–588. · Zbl 0853.22017 [277] V. G. Pestov, Regular Lie groups and a theorem of Lie-Palais, J. Lie Theory, 5 (1995), 173–178. · Zbl 0848.22025 [278] D. Pickrell, Invariant Measures for Unitary Groups Associated to Kac-Moody Lie Algebras, Mem. Amer. Math. Soc., 693, 2000. · Zbl 0960.43001 [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. · Zbl 1025.58003 · doi:10.2140/pjm.2000.193.177 [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. · Zbl 0346.32034 [281] D. Pisanelli, An example of an infinite Lie group, Proc. Amer. Math. Soc., 62 (1977), 156–160. · Zbl 0353.22013 · doi:10.1090/S0002-9939-1977-0436234-7 [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. · Zbl 0424.22012 [283] L. Polterovich, The geometry of the group of symplectic diffeomorphisms, Lectures Math., ETH Zürich, Birkhäuser, 2001. · Zbl 1197.53003 [284] L. Pontrjagin, Topological Groups, Princeton Math. Ser., 2, Princeton University Press, Princeton, 1939. · JFM 65.0872.02 [285] A. Pressley and G. Segal, Loop Groups, Oxford University Press, Oxford, 1986. · Zbl 0618.22011 [286] M. E. Pursell, Algebraic structures associated with smooth manifolds, Thesis, Purdue Univ., 1952. [287] M. E. Pursell and M. E. Shanks, The Lie algebra of a smooth manifold, Proc. Amer. Math. Soc., 5 (1954), 468–472. · Zbl 0055.42105 · doi:10.1090/S0002-9939-1954-0064764-3 [288] C. R. Putnam and A. Winter, The orthogonal group in Hilbert space, Amer. J. Math., 74 (1952), 52–78. · Zbl 0049.35502 · doi:10.2307/2372068 [289] T. Ratiu and A. Odzijewicz, Banach Lie–Poisson spaces and reduction, Comm. Math. Phys., 243 (2003), 1–54. · Zbl 1044.53057 · doi:10.1007/s00220-003-0948-8 [290] T. Ratiu and A. Odzijewicz, Extensions of Banach Lie–Poisson spaces, J. Funct. Anal., 217 (2004), 103–125. · Zbl 1067.53065 · doi:10.1016/j.jfa.2004.02.012 [291] T. Ratiu and R. Schmid, The differentiable structure of three remarkable diffeomorphism groups, Math. Z., 177 (1981), 81–100. · Zbl 0451.58011 · doi:10.1007/BF01214340 [292] J. F. Ritt, Differential groups and formal Lie theory for an infinite number of parameters, Ann. of Math. (2), 52 (1950), 708–726. · Zbl 0038.16801 · doi:10.2307/1969444 [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. · Zbl 0870.57050 [294] T. Robart, Sur l’intégrabilité des sous-algèbres de Lie en dimension infinie, Canad. J. Math., 49 (1997), 820–839. · Zbl 0888.22016 · doi:10.4153/CJM-1997-042-7 [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. · Zbl 1039.22014 [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. · Zbl 1130.17306 [297] T. Robart and N. Kamran, Sur la théorie locale des pseudogroupes de transformations continus infinis. I, Math. Ann., 308 (1997), 593–613. · Zbl 0874.22019 · doi:10.1007/s002080050092 [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. [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. · Zbl 0892.17018 · doi:10.1016/0034-4877(96)89288-3 [300] J. Rosenberg, Algebraic K-theory and its Applications, Grad. Texts in Math., 147, Springer-Verlag, 1994. · Zbl 0801.19001 [301] W. Rudin, Functional Analysis, McGraw Hill, 1973. [302] R. Schmid, Infinite-dimensional Hamiltonian Systems, Monographs and Textbooks in Physical Science, Lecture Notes, 3, Bibliopolis, Naples, 1987. [303] R. Schmid, Infinite dimensional Lie groups with applications to mathematical physics, J. Geom. Symmetry Phys., 1 (2004), 54–120. · Zbl 1063.22020 [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. [305] J. R. Schue, Hilbert space methods in the theory of Lie algebras, Trans. Amer. Math. Soc., 95 (1960), 69–80. · Zbl 0093.30601 · doi:10.1090/S0002-9947-1960-0117575-1 [306] J. R. Schue, Cartan decompositions for L *-algebras, Trans. Amer. Math. Soc., 98 (1961), 334–349. [307] F. Schur, Neue Begründung der Theorie der endlichen Transformationsgruppen, Math. Ann., 35 (1890), 161–197. · JFM 21.0371.01 · doi:10.1007/BF01443876 [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. · JFM 22.0375.01 [309] G. Segal, Unitary representations of some infinite-dimensional groups, Comm. Math. Phys., 80 (1981), 301–342. · Zbl 0495.22017 · doi:10.1007/BF01208274 [310] J.-P. Serre, Lie Algebras and Lie Groups, Lecture Notes in Math., 1500, Springer-Verlag, 1965 (1st ed.). · Zbl 0132.27803 [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. · Zbl 0277.58008 · doi:10.1007/BF02787690 [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. [313] J.-M. Souriau, Un algorithme générateur de structures quantiques, Soc. Math. Fr., Astérisque, hors série, 1985, 341–399. [314] S. Sternberg, Infinite Lie groups and the formal aspects of dynamical systems, J. Math. Mech., 10 (1961), 451–474. · Zbl 0131.26802 [315] N. Stumme, The Structure of Locally Finite Split Lie algebras, Ph. D. thesis, Darmstadt University of Technology, 1999. · Zbl 1027.17019 [316] H. J. Sussmann, Orbits of families of vector fields and integrability of distributions, Trans. Amer. Math. Soc., 180 (1973), 171–188. · Zbl 0274.58002 · doi:10.1090/S0002-9947-1973-0321133-2 [317] K. Suto, Groups associated with unitary forms of Kac–Moody algebras, J. Math. Soc. Japan, 40 (1988), 85–104. · Zbl 0651.17010 · doi:10.2969/jmsj/04010085 [318] K. Suto, Borel–Weil type theorem for the flag manifold of a generalized Kac–Moody algebra, J. Algebra, 193 (1997), 529–551. · Zbl 0879.17014 · doi:10.1006/jabr.1996.6989 [319] R. G. Swan, Vector bundles and projective modules, Trans. Amer. Math. Soc., 105 (1962), 264–277. · Zbl 0109.41601 · doi:10.1090/S0002-9947-1962-0143225-6 [320] S. Swierczkowski, Embedding theorems for local analytic groups, Acta Math., 114 (1965), 207–235. · Zbl 0135.07002 · doi:10.1007/BF02391822 [321] S. Swierczkowski, Cohomology of local group extensions, Trans. Amer. Math. Soc., 128 (1967), 291–320. · Zbl 0149.27604 [322] S. Swierczkowski, The path-functor on Banach Lie algebras, Nederl. Akad. Wet., Proc. Ser. A, 74; Indag. Math., 33 (1971), 235–239. · Zbl 0213.13702 [323] A. Tagnoli, La varietà analitiche reali come spazi omogenei, Boll. Un. Mat. Ital. (s4), 1 (1968), 422–426. · Zbl 0159.25101 [324] J. Tits, Liesche Gruppen und Algebren, Springer-Verlag, 1983. [325] F. Treves, Topological Vector Spaces, Distributions, and Kernels, Academic Press, New York, 1967. [326] H. Upmeier, Symmetric Banach Manifolds and Jordan C *-algebras, North Holland Mathematics Studies, 1985. [327] V. S. Varadarajan, Lie Groups, Lie Algebras, and Their Representations, Grad. Texts in Math., 102, Springer-Verlag, 1984. · Zbl 0955.22500 [328] D. Vogt, On the functors Ext1 (E,F) for Fréchet spaces, Studia Math., 85 (1987), 163–197. [329] L. Waelbroeck, Les algèbres à inverse continu, C. R. Acad. Sci. Paris, 238 (1954), 640–641. · Zbl 0056.33701 [330] L. Waelbroeck, Le calcul symbolique dans les algèbres commutatives, J. Math. Pures Appl., 33 (1954), 147–186. · Zbl 0056.33601 [331] L. Waelbroeck, Structure des algèbres à inverse continu, C. R. Acad. Sci. Paris, 238 (1954), 762–764. · Zbl 0055.33901 [332] L. Waelbroeck, Topological Vector Spaces and Algebras, Springer-Verlag, 1971. · Zbl 0225.46001 [333] G. Warner, Harmonic Analysis on Semisimple Lie Groups I, Springer-Verlag, 1972. · Zbl 0265.22020 [334] A. Weinstein, Symplectic structures on Banach manifolds, Bull. Amer. Math. Soc., 75 (1969), 1040–1041. · Zbl 0179.50104 · doi:10.1090/S0002-9904-1969-12353-0 [335] D. Werner, Funktionalanalysis, Springer-Verlag, 1995. [336] H. Wielandt, Über die Unbeschränktheit der Operatoren der Quantenmechanik, Math. Ann., 121 (1949), p. 21. · Zbl 0035.19903 · doi:10.1007/BF01329611 [337] Chr. Wockel, The Topology of Gauge Groups, submitted, math-ph/0504076. [338] Chr. Wockel, Smooth Extensions and Spaces of Smooth and Holomorphic Mappings, J. Geom. Symmetry Phys., 5 (2006), 118–126, math.DG/0511064. · Zbl 1108.58006 [339] W. Wojtyński, Effective integration of Lie algebras, J. Lie Theory, 16 (2006), 601–620. · Zbl 1113.22015 [340] J. A. Wolf, Principal series representations of direct limit groups, Compositio Math., 141 (2005), 1504–1530. · Zbl 1086.22009 · doi:10.1112/S0010437X05001430 [341] M. Wüstner, Supplements on the theory of exponential Lie groups, J. Algebra, 265 (2003), 148–170. · Zbl 1024.22004 · doi:10.1016/S0021-8693(03)00165-0 [342] M. Wüstner, The classification of all simple Lie groups with surjective exponential map, J. Lie Theory, 15 (2005), 269–278. · Zbl 1072.22003 [343] K. Yosida, On the groups embedded in the metrical complete ring, Japan. J. Math., 13 (1936), 7–26. · JFM 62.0441.04
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