Lie Groups of Measurable Mappings

Glöckner, Helge

Canadian journal of mathematics, Tome 55 (2003) no. 5, pp. 969-999

Résumé

We describe new construction principles for infinite-dimensional Lie groups. In particular, given any measure space $\left( X,\,\sum ,\,\mu\right)$ and (possibly infinite-dimensional) Lie group $G$ , we construct a Lie group ${{L}^{\infty }}\left( X,G \right)$ , which is a Fréchet-Lie group if $G$ is so. We also show that the weak direct product $\prod{_{i\in I}^{*}{{G}_{i}}}$ of an arbitrary family ${{\left( {{G}_{i}} \right)}_{i\in I}}$ of Lie groups can be made a Lie group, modelled on the locally convex direct sum ${{\oplus }_{i\in I}}L\left( {{G}_{i}} \right)$ .

Détail

DOI : 10.4153/CJM-2003-039-9

Mots-clés : Primary:, 22E65, secondary:, 46E40, 46E30, 22E67, 46T20, 46T25

Bibliographie

[1] [1] Albeverio, S., Høegh-Krohn, R. J., Marion, J. A., Testard, D. H. and Torrésani, B. S., Noncommutative Distributions. Marcel Dekker, 1993. Google Scholar

[2] [2] Bastiani, A., Applications différentiables et variétés différentiables de dimension infinie. J. Analyse Math. 13(1964), 1–114. Google Scholar

[3] [3] Bauer, H., Maß- und Integrationstheorie. 2nd edition, de Gruyter, 1992. Google Scholar

[4] [4] Bloch, A., El Hadrami, M. O., Flaschka, H. and Ratiu, T. S., Maximal tori of some symplectomorphism groups and applications to convexity. In: Deformation Theory and Symplectic Geometry (Ascona, 1996), Math. Phys. Stud. 20, Kluwer, Dordrecht, 1997, 201–222. Google Scholar

[5] [5] Bochnak, J. and Siciak, J., Analytic functions in topological vector spaces. Studia Math. 39(1971), 77–112. Google Scholar

[6] [6] Boseck, H., Czichowski, G. and Rudolph, K.-P., Analysis on Topological Groups . General Lie Theory. Teubner, Leipzig, 1981. Google Scholar

[7] [7] Bourbaki, N., Topological Vector Spaces, Chapters 1.5. Springer-Verlag, 1987. Google Scholar

[8] [8] Bourbaki, N., Lie Groups and Lie Algebras, Chapters 1–3. Springer-Verlag, 1989. Google Scholar

[9] [9] Engelking, R., General Topology. Heldermann-Verlag, Berlin, 1989. Google Scholar

[10] [10] Glöckner, H., Infinite-dimensional Lie groups without completeness restrictions. In: Geometry and Analysis on Finite- and Infinite-Dimensional Lie Groups (eds. Strasburger, A. et al.), Banach Center Publications, Vol. 55, Warsaw, 2002, 43–59. Google Scholar

[11] [11] Glöckner, H., Algebras whose groups of units are Lie groups. Studia Math. 153(2002), 147–177. Google Scholar

[12] [12] Glöckner, H., Lie group structures on quotient groups and universal complexi_cations for infinite-dimensional Lie groups. J. Funct. Anal. 194(2002), 347–409. Google Scholar

[13] [13] Glöckner, H., Direct limit Lie groups and manifolds. Math. J. Kyoto Univ. (2003), in print. Google Scholar

[14] [14] Glöckner, H., Discontinuous non-linear mappings on locally convex direct limits. Submitted. Google Scholar

[15] [15] Glöckner, H., Patched locally convex spaces, almost local mappings, and diffeomorphism groups of non-compact manifolds. In preparation. Google Scholar

[16] [16] Glöckner, H., Lie groups over non-discrete topological fields. In preparation. Google Scholar

[17] [17] Hamilton, R., The inverse function theorem of Nash and Moser. Bull. Amer.Math. Soc. 7(1982), 65–222. Google Scholar

[18] [18] Hewitt, E. and Ross, K. A., Abstract Harmonic Analysis I. Springer-Verlag, 1979. Google Scholar

[19] [19] Hille, E. and Phillips, R. S., Functional Analysis and Semi-Groups. Amer.Math. Soc., Providence, 1957. Google Scholar

[20] [20] Kamran, N. and Robart, T., On the parametrization problem of Lie pseudogroups of infinite type. C. R. Acad. Sci. Paris Sér. I Math. 331(2000), 899–903. Google Scholar

[21] [21] Kamran, N. and Robart, T., A manifold structure for analytic isotropy Lie pseudogroups of infinite type. J. Lie Theory 11(2001), 57–80. Google Scholar

[22] [22] Keller, H. H., Differential Calculus in Locally Convex Spaces. Springer-Verlag, Berlin, 1974. Google Scholar

[23] [23] Kriegl, A. and Michor, P. W., The Convenient Setting of Global Analysis. Amer. Math. Soc., Providence R. I., 1997. Google Scholar

[24] [24] Milnor, J., Remarks on infinite dimensional Lie groups. In: Relativity, Groups and Topology II (eds. DeWitt, B. and Stora, R.), North-Holland, 1983, 1008–1057. Google Scholar

[25] [25] Natarajan, L., Rodríguez-Carrington, E. and Wolf, J. A., Differentiable structure for direct limit groups. Lett. Math. Phys. 23(1991), 99–109. Google Scholar

[26] [26] Natarajan, L., Rodríguez-Carrington, E. and Wolf, J. A., Locally convex Lie groups. Nova J. Algebra Geom. (1) 2(1993), 59–87. Google Scholar

[27] [27] Natarajan, L., Rodríguez-Carrington, E. and Wolf, J. A., New classes of infinite-dimensional Lie groups. In: Algebraic Groups and their Generalizations, Proc. Sympos. Pure Math. 56, Part II, 1994, 377–392. Google Scholar

[28] [28] Natarajan, L., Rodríguez-Carrington, E. and Wolf, J. A., The Bott-Borel-Weil Theorem for direct limit groups. Trans. Amer.Math. Soc. 353(2001), 4583–4622. Google Scholar

[29] [29] Neeb, K.-H., Infinite-dimensional groups and their representations. In: Infinite-dimensional Kähler manifolds (eds. Huckleberry, A. T. and Wurzbacher, T.), Birkhäuser Verlag, 2001, 131–178. Google Scholar

[30] [30] Neeb, K.-H., Central extensions of infinite-dimensional Lie groups. Ann. Inst. Fourier (Grenoble) 52(2002), 1365–1442. Google Scholar

[31] [31] Pressley, A. and Segal, G. B., Loop Groups. Clarendon Press, Oxford, 1986. Google Scholar

[32] [32] Robart, T., Sur l'intégrabilité des sous-algèbres de Lie en dimension infinie. Canad. J. Math. 49(1997), 820–839. Google Scholar

[33] [33] Rudin, W., Real and Complex Analysis. McGraw-Hill, 1987. Google Scholar

[34] [34] Schaefer, H. H., Topological Vector Spaces. Springer-Verlag, 1971. Google Scholar

[35] [35] Schubert, H., Topologie. Teubner-Verlag, 1964. Google Scholar

[36] [36] Tatsuuma, N., Shimomura, H. and Hirai, T., On group topologies and unitary representations on inductive limits of topological groups and the case of the group of diffeomorphisms. Math. J. Kyoto Univ. 38(1998), 551–578. Google Scholar

[37] [37] Thomas, E. G. F., Calculus on locally convex spaces. Preprint W-9604, Groningen Univ., 1996. Google Scholar