×

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
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Albeverio, S.; Høegh-Krohn, R. J.; Marion, J. A.; Testard, D. H.; Torrésani, B. S., Noncommutative Distributions (1993), Marcel Dekker: 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: Teubner Leipzig · Zbl 0558.22012
[6] Bourbaki, N., Topological Vector Spaces (1987), Springer-Verlag: Springer-Verlag Berlin · Zbl 0622.46001
[7] Bourbaki, N., Lie Groups and Lie Algebras (1989), Springer-Verlag: Springer-Verlag Berlin · Zbl 0672.22001
[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 · Zbl 0092.11302
[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.; 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. · Zbl 0942.22008
[13] Glöckner, H., Infinite-dimensional Lie groups without completeness restrictions, (Strasburger, A., Geometry and Analysis on Finite- and Infinite-Dimensional Lie Groups, 55 (2002), Banach Center Publications: Banach Center Publications Warsaw), 43-59 · Zbl 1020.58009
[14] H. Glöckner, Differentiable mappings between spaces of sections, submitted.; H. Glöckner, Differentiable mappings between spaces of sections, submitted.
[15] H. Glöckner, Direct limit Lie groups and manifolds, J. Math. Kyoto Unit, to appear.; H. Glöckner, Direct limit Lie groups and manifolds, J. Math. Kyoto Unit, to appear. · Zbl 1056.22013
[16] H. Glöckner, Lie groups of measurable mappings, submitted.; 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.; 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: 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: The Univ. of Chicago Press Chicago · Zbl 0223.17001
[24] Keller, H. H., Differential Calculus in Locally Convex Spaces (1974), Springer-Verlag: Springer-Verlag Berlin · Zbl 0293.58001
[25] Knapp, A. W., Lie Groups Beyond an Introduction (1996), Birkhäuser: 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: 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, (Cordaro, P. D., Multidimensional Complex Analysis and Partial Differential Equations, 205 (1997), AMS: AMS Providence), 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, (DeWitt, B.; Stora, R., Relativity, Groups and Topology II (1983), North-Holland: North-Holland Amsterdam), 1008-1057 · Zbl 0594.22009
[32] Natarajan, L.; Rodrı́guez-Carrington, E.; Wolf, J. A., Differentiable structure for direct limit groups, Lett. Math. Phys., 23, 99-109 (1991) · Zbl 0762.22017
[33] Natarajan, L.; Rodrı́guez-Carrington, E.; Wolf, J. A., Locally convex Lie groups, Nova J. Algebra Geom., 2, 59-87 (1993) · Zbl 0872.22012
[34] L. Natarajan, E. Rodrı́guez-Carrington, and J. A. Wolf, New classes of infinite-dimensional Lie groups, in; L. Natarajan, E. Rodrı́guez-Carrington, and J. A. Wolf, New classes of infinite-dimensional Lie groups, in
[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, inet al.; K. H. Neeb, Some open problems in representation theory related to complex geometry, inet al. · Zbl 0908.22008
[37] K. H. Neeb, Central extensions of infinite-dimensional Lie groups, Ann. Inst. Fourier (Grenoble), to appear.; 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; K. H. Neeb, Infinite-dimensional groups and their representations, in
[39] K. H. Neeb, Borel-Weil theory for loop groups, in; K. H. Neeb, Borel-Weil theory for loop groups, in
[40] K. H. Neeb, A Cartan-Hadamard Theorem for Banach-Finsler manifolds, Geom. Dedicata, to appear.; 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: Addison-Wesley Reading · Zbl 0164.11102
[42] Pressley, A.; Segal, G. B., Loop Groups (1986), Clarendon Press: 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.; 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: Springer-Verlag Berlin · Zbl 0217.16002
[46] Schwartz, L., Théorie des Distributions, Tome I (1957), Hermann: Hermann Paris · Zbl 0078.11003
[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.; 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.; 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: Springer Berlin · Zbl 0506.22011
[51] Upmeier, H., Symmetric Banach Manifolds and Jordan \(C^*\)-Algebras (1985), North-Holland: North-Holland Amsterdam · Zbl 0561.46032
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.