Completeness of Infinite-dimensional Lie Groups in Their Left Uniformity

Glöckner, Helge

Canadian journal of mathematics, Tome 71 (2019) no. 1, pp. 131-152

Résumé

We prove completeness for the main examples of infinite-dimensional Lie groups and some related topological groups. Consider a sequence $G_{1}\subseteq G_{2}\subseteq \cdots \,$ of topological groups $G_{n}$ n such that $G_{n}$ is a subgroup of $G_{n+1}$ and the latter induces the given topology on $G_{n}$, for each $n\in \mathbb{N}$. Let $G$ be the direct limit of the sequence in the category of topological groups. We show that $G$ induces the given topology on each $G_{n}$ whenever $\cup _{n\in \mathbb{N}}V_{1}V_{2}\cdots V_{n}$ is an identity neighbourhood in $G$ for all identity neighbourhoods $V_{n}\subseteq G_{n}$. If, moreover, each $G_{n}$ is complete, then $G$ is complete. We also show that the weak direct product $\oplus _{j\in J}G_{j}$ is complete for each family $(G_{j})_{j\in J}$ of complete Lie groups $G_{j}$. As a consequence, every strict direct limit $G=\cup _{n\in \mathbb{N}}G_{n}$ of finite-dimensional Lie groups is complete, as well as the diffeomorphism group $\text{Diff}_{c}(M)$ of a paracompact finite-dimensional smooth manifold $M$ and the test function group $C_{c}^{k}(M,H)$, for each $k\in \mathbb{N}_{0}\cup \{\infty \}$ and complete Lie group $H$ modelled on a complete locally convex space.

Détail

DOI : 10.4153/CJM-2017-048-5

Mots-clés : infinite-dimensional Lie group, direct sum, box product, weak direct product, (LB)-space, inductive limit, direct limit, ascending sequence, product set, bamboo shoot topology, compact support, test function group, diffeomorphism group, Banach–Lie group, left uniform structure, Cauchy net, Cauchy filter, strong (ILB)-Lie group, projective limit, inverse limit

Bibliographie

[1] Albeverio, S., Høegh-Krohn, R., Marion, J. A., Testard, D. H., and Torrésani, B. S., Noncommutative distributions: unitary representation of gauge groups and algebras. Marcel Dekker, New York, 1993. Google Scholar | DOI

[2] Arhangel’Skii, A. and Tkachenko, M., Topological groups and related structures. World Scientific, Paris, 2008. Google Scholar | DOI

[3] Banakh, T., Mine, K., Repovš, D., Sakai, K., and Yagasaki, T., Detecting topological groups which are (locally) homeomorphic to LF-spaces . Topology Appl. 160(2013), 2272–2284. . Google Scholar | DOI

[4] Banakh, T. and Repovš, D., Topological structure of direct limits in the category of uniform spaces . Topology Appl. 157(2010), 1091–1100. . Google Scholar | DOI

[5] Banakh, T. and Yagasaki, Y., Diffeomorphism groups of non-compact manifolds endowed with the Whitney C ∞ -topology . Topology Appl. 179(2015), 51–61. . Google Scholar | DOI

[6] Banyaga, A., The structure of classical diffeomorphism groups. Kluwer Academic Publishers, Dordrecht, 1997. Google Scholar | DOI

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

[8] Bertram, W., Glöckner, H., and Neeb, K.-H., Differential calculus over general base fields and rings . Expo. Math. 22(2004), 213–282. . Google Scholar | DOI

[9] Bourbaki, N., Topological vector spaces. Chapters 1–5, Springer, Berlin, 1987. Google Scholar | DOI

[10] Bourbaki, N., Lie Groups and Lie algebras. Chapters 1–3, Springer, Berlin, 1989. Google Scholar

[11] Cabau, P. and Pelletier, F., Integrability on direct limit Banach manifolds. arxiv:1408.3715v2. Google Scholar

[12] Dahmen, R., Analytic mappings between LB-spaces and applications in infinite-dimensional Lie theory . Math. Z. 266(2010), 115–140. . Google Scholar | DOI

[13] Dahmen, R., Regularity in Milnor’s sense for ascending unions of Banach–Lie groups . J. Lie Theory 24(2014), 545–560. Google Scholar

[14] Dierolf, S. and Wengenroth, J., Inductive limits of topological algebras . In: Linear Topological Spaces and Complex Analysis III. Scientific and Technical Research Council of Turkey (TÜBİTAK), Ankata, 1997, pp. 45–49. Google Scholar

[15] Dieudonné, J., Sur la completion des groupes topologiques . C. R. Acad. Sci. Paris 218(1944), 774–776. Google Scholar

[16] Glöckner, H., Infinite-dimensional Lie groups without completeness restrictions . In: Geometry and analysis on finite- and infinite-dimensional Lie groups, Banach Center Publications 55. Warsaw, 2002, pp. 43–59. Google Scholar

[17] Glöckner, H., Lie group structures on quotient groups and universal complexifications for infinite-dimensional Lie groups . J. Funct. Anal. 94(2002), 347–409. Google Scholar | DOI

[18] Glöckner, H., Lie groups of measurable mappings . Canadian J. Math. 55(2003), 969–999. . Google Scholar | DOI

[19] Glöckner, H., Fundamentals of direct limit Lie theory . Compositio Math. 141(2005), 1551–1577. . Google Scholar | DOI

[20] Glöckner, H., Implicit functions from topological vector spaces to Banach spaces . Israel J. Math. 155(2006), 205–252. . Google Scholar | DOI

[21] Glöckner, H., Direct limits of infinite-dimensional Lie groups compared to direct limits in related categories . J. Funct. Anal. 245(2007), 19–61. . Google Scholar | DOI

[22] Glöckner, H., Completeness of locally k -groups and related infinite-dimensional Lie groups . Topol. Appl. 228(2017), 277–284. . Google Scholar | DOI

[23] Glöckner, H., Lie groups over non-discrete topological fields. arxiv:math/0408008. Google Scholar

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

[25] Hewitt, E. and Ross, K. A., Abstract harmonic analysis, Vol. 1. Grundlehren der Mathematischen Wissenschaften, 115. Springer-Verlag, Berlin, 1979. Google Scholar | DOI

[26] Hirai, T., Shimomura, H., Tatsuuma, N., and Hirai, E., Inductive limits of topologies, their direct products, and problems related to algebraic structures . J. Math. Kyoto Univ. 41(2001), 475–505. . Google Scholar | DOI

[27] Hirsch, M. W., Differential topology. Springer, New York, 1976. Google Scholar | DOI

[28] Hjelle, E. O. and Schmeding, A., Strong topologies for spaces of smooth maps with infinite-dimensional target . Expo. Math. 35(2017), 13–53. . Google Scholar | DOI

[29] Hunt, D. C. and Morris, S. A., Free subgroups of free topological groups, Lect. Notes Math., 372. Springer, Berlin, 1974, pp. 377–387. Google Scholar

[30] Illman, S., The very-strong C ∞ -topology on C ∞ (M, N) and K-equivariant maps . Osaka J. Math. 40(2003), 409–428. Google Scholar

[31] Kelley, J. L., General topology. Springer, New York, 1975. Google Scholar

[32] Kriegl, A. and Michor, P. W., The convenient setting of global analysis, Mathematical Surveys and Monographs, 53, American Mathematical Society, Providence, RI, 1997. Google Scholar | DOI

[33] Michor, P. W., Manifolds of differentiable mappings. Shiva Publications, Nantwich, 1980. Google Scholar

[34] Milnor, J., Remarks on infinite-dimensional Lie groups . In: Relativity, groups and topology, II. North-Holland, Amsterdam, 1984, pp. 1007–1057. Google Scholar

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

[36] Neeb, K.-H., Towards a Lie theory of locally convex groups . Jpn. J. Math. 1(2006), 291–468. . Google Scholar | DOI

[37] Neeb, K.-H. and Wagemann, F., Lie group structures on groups of smooth and holomorphic maps on non-compact manifolds . Geom. Dedicata 134(2008), 17–60. . Google Scholar | DOI

[38] Omori, H., Infinited-dimensional Lie groups, Translations of Mathematical Monographs, 158. American Mathematical Society, Providence, RI, 1997. Google Scholar

[39] Pressley, A. and Segal, G., Loop groups . Clarendon Press, Oxford, 1986. Google Scholar

[40] Roelcke, W. and Dierolf, S., Uniform structures on topological groups and their quotients. McGraw-Hill, New York, 1981. Google Scholar

[41] Schütt, J., Symmetry groups of principal bundles over non-compact bases. arxiv:1310.8538. Google Scholar

[42] Tatsuuma, N., Shimomura, H., and 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(1998), 551–578. . Google Scholar | DOI

[43] Waelbroeck, L., Topological vector spaces and algebras. Springer, Berlin, 1971. Google Scholar

[44] Wells, J. C., Invariant manifolds of nonlinear operators . Pacific J. Math. 62(1976), 285–293. . Google Scholar | DOI

[45] Wockel, C., Lie group structures on symmetry groups of principal bundles . J. Funct. Anal. 251(2007), 254–288. . Google Scholar | DOI

[46] Yamasaki, A., Inductive limit of general linear groups . J. Math. Kyoto Univ. 38(1998), 769–779. . Google Scholar | DOI