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Lie group structures on quotient groups and universal complexifications for infinite-dimensional Lie groups. (English) Zbl 1022.22021
The author studies the question whether a given infinite-dimensional Lie group has a complexification. He answers the question completely by giving necessary and sufficient conditions for the so-called Baker-Campbell-Hausdorff (BCH) Lie groups. This class of groups is characterized by the existence of neighborhoods in the Lie algebra on which the exponential function is a diffeomorphism and the multiplication (transported back via the exponential function) is given by the Baker-Campbell-Hausdorff formula. This class of groups includes Banach Lie groups, mapping groups, and direct limit groups but not diffeomorphism groups. In the process the author develops the basic Lie theory for BCH Lie groups as well as the basic properties for various types of mapping groups.

MSC:
22E65 Infinite-dimensional Lie groups and their Lie algebras: general properties
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[1] Albeverio, S.; Høegh-Krohn, R.J.; Marion, J.A.; Testard, D.H.; Torrésani, B.S., Noncommutative distributions, (1993), Marcel Dekker New York · Zbl 0791.22010
[2] Bastiani, A., Applications différentiables et variétés différentiables de dimension infinie, J. analyse math., 13, 1-114, (1964) · Zbl 0196.44103
[3] Bochnak, J.; Siciak, J., Polynomials and multilinear mappings in topological vector spaces, Stud. math., 39, 59-76, (1971) · Zbl 0214.37702
[4] Bochnak, J.; Siciak, J., Analytic functions in topological vector spaces, Stud. math., 39, 77-112, (1971) · Zbl 0214.37703
[5] Boseck, H.; Czichowski, G.; Rudolph, K.-P., Analysis on topological groups—general Lie theory, (1981), Teubner Leipzig · Zbl 0558.22012
[6] Bourbaki, N., Topological vector spaces, (1987), Springer-Verlag Berlin
[7] Bourbaki, N., Lie groups and Lie algebras, (1989), Springer-Verlag Berlin
[8] Douady, A.; Lazard, M., Espaces fibrés en algèbres de Lie et en groupes, Invent. math., 1, 133-151, (1966) · Zbl 0144.01804
[9] Dudley, R.M., On sequential convergence, Trans. AMS, 112, 483-507, (1964) · Zbl 0138.17401
[10] Dupre, M.J.; Glazebrook, J.F., Infinite-dimensional manifold structures on principal bundles, J. Lie theory, 10, 359-373, (2000) · Zbl 0964.58004
[11] Eells, J., On the geometry of function spaces, Symposium internacional de topologı́a algebraica, (1958), Universidad Nacional Autónoma de México and UNESCO, p. 303-308
[12] H. Glöckner, Infinite-Dimensional Complex Groups and Semigroups: Representations of Cones, Tubes, and Conelike Semigroups, Doctoral dissertation, Department of Mathematics, Darmstadt University of Technology, 2000. Advisor: K.-H. Neeb.
[13] Glöckner, H., Infinite-dimensional Lie groups without completeness restrictions, (), 43-59 · Zbl 1020.58009
[14] H. Glöckner, Differentiable mappings between spaces of sections, submitted.
[15] H. Glöckner, Direct limit Lie groups and manifolds, J. Math. Kyoto Unit, to appear.
[16] H. Glöckner, Lie groups of measurable mappings, submitted.
[17] H. Glöckner, and, K.-H. Neeb, Banach-Lie quotients, enlargibility, and universal complexifications, J. Reine Angew. Math, to appear. · Zbl 1029.22029
[18] Goodman, R.; Wallach, N., Structure and unitary cocycle representations of loop groups and the group of diffeomorphisms of the circle, J. reine angew. math., 347, 69-133, (1984) · Zbl 0514.22012
[19] Hamilton, R., The inverse function theorem of Nash and Moser, Bull. amer. math. soc., 7, 65-222, (1982) · Zbl 0499.58003
[20] Hansen, V.L., Some theorems on direct limits of expanding systems of manifolds, Math. scand., 29, 5-36, (1971) · Zbl 0229.58005
[21] Hochschild, G., The structure of Lie groups, (1965), Holden-Day San Franciso · Zbl 0131.02702
[22] Hochschild, G., Complexification of real analytic groups, Trans. amer. math. soc., 125, 406-413, (1966) · Zbl 0149.27603
[23] Kaplansky, I., Lie algebras and locally compact groups, (1971), The Univ. of Chicago Press Chicago
[24] Keller, H.H., Differential calculus in locally convex spaces, (1974), Springer-Verlag Berlin · Zbl 0293.58001
[25] Knapp, A.W., Lie groups beyond an introduction, (1996), Birkhäuser Basel · Zbl 0865.17002
[26] Kriegl, A.; Michor, P.W., The convenient setting of global analysis, (1997), American Mathematical Society Providence · Zbl 0889.58001
[27] Lang, S., Fundamentals of differential geometry, (1999), Springer-Verlag Berlin · Zbl 0932.53001
[28] Lawson, J.D., Polar and ol’shanskĭ decompositions, J. reine angew. math., 448, 191-219, (1994) · Zbl 0786.22012
[29] Lempert, L., The problem of complexifying a Lie group, (), 169-176 · Zbl 0887.22008
[30] Maissen, B., Lie-gruppen mit banachräumen als parameterraäume, Acta math., 108, 229-270, (1962) · Zbl 0207.33701
[31] Milnor, J., Remarks on infinite dimensional Lie groups, (), 1008-1057
[32] Natarajan, L.; Rodrı́guez-Carrington, E.; Wolf, J.A., Differentiable structure for direct limit groups, Lett. math. phys., 23, 99-109, (1991)
[33] Natarajan, L.; Rodrı́guez-Carrington, E.; Wolf, J.A., Locally convex Lie groups, Nova J. algebra geom., 2, 59-87, (1993)
[34] L. Natarajan, E. Rodrı́guez-Carrington, and J. A. Wolf, New classes of infinite-dimensional Lie groups, in: Algebraic Groups and Their GeneralizationsW. J. Habous Ed., Proceedings of Symposia in Pure Mathematics, Vol. 56, Part , 1994, pp. 377-392.
[35] Natarajan, L.; Rodrı́guez-Carrington, E.; Wolf, J.A., The bott – borel – weil theorem for direct limit groups, Trans. amer. math. soc., 353, 4583-4622, (2001) · Zbl 0994.22013
[36] K. H. Neeb, Some open problems in representation theory related to complex geometry, in: Positivity in Lie Theory: Open ProblemsJ. Hilgert, et al. Eds., de Gruyter, Berlin, 1998. · Zbl 0908.22008
[37] K. H. Neeb, Central extensions of infinite-dimensional Lie groups, Ann. Inst. Fourier (Grenoble), to appear. · Zbl 1019.22012
[38] K. H. Neeb, Infinite-dimensional groups and their representations, in: DMV-Seminar Infinite-Dimensional Kähler ManifoldsA. T. Huckleberry and T. Wurzbacher, Eds., pp. 131-178, Oberwolfach, 1995; Birkhäuser Verlag, Basel, 2001.
[39] K. H. Neeb, Borel-Weil theory for loop groups, in DMV-Seminar Infinite-Dimensional Kähler ManifoldsA. T. Huckleberry and T. Wurzbacher Eds., pp. 179-229, Oberwolfach, 1995; Birkhäuser Verlag, Basel, 2001.
[40] K. H. Neeb, A Cartan-Hadamard Theorem for Banach-Finsler manifolds, Geom. Dedicata, to appear. · Zbl 1027.58003
[41] Palais, R., Foundations of global non-linear analysis, (1968), Addison-Wesley Reading · Zbl 0164.11102
[42] Pressley, A.; Segal, G.B., Loop groups, (1986), Clarendon Press Oxford · Zbl 0618.22011
[43] Robart, T., Sur l’intégrabilité des sous-algèbres de Lie en dimension infinie, Can. J. math., 49, 820-839, (1997) · Zbl 0888.22016
[44] E. Rodriguez-Carrington, Kac-Moody Groups—An Analytic Approach, Ph.D. thesis, Rutgers University, 1989. · Zbl 0744.22023
[45] Schaefer, H.H., Topological vector spaces, (1971), Springer-Verlag Berlin · Zbl 0217.16002
[46] Schwartz, L., Théorie des distributions, tome I, (1957), Hermann Paris
[47] Tatsuuma, N.; Shimomura, H.; Hirai, T., On group topologies and unitary representations of inductive limits of topological groups and the case of the group of diffeomorphisms, J. math. Kyoto univ., 38, 551-578, (1998) · Zbl 0930.22002
[48] J. Teichmann, Infinite Dimensional Lie Groups with a View Towards Functional Analysis, Ph.D. thesis, University of Vienna, 1999.
[49] E. G. F. Thomas, Calculus on locally convex spaces, Preprint W-9604, Univ. of Groningen, 1996.
[50] Tits, J., Liesche gruppen und algebren, (1983), Springer Berlin · Zbl 0506.22011
[51] Upmeier, H., Symmetric Banach manifolds and Jordan C*-algebras, (1985), North-Holland Amsterdam · Zbl 0561.46032
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