×

Hilbert’s fifth problem for local groups. (English) Zbl 1219.22004

The author has started two versions of Hilbert’s fifth problem written below after some definitions:
(a) A local group is a tuple \((G,1,\iota,t, p)\) (or simply, \(G\) in short) where \(G\) is a Hausdorff topological space with a distinguished element \(1\in G\), an \(\iota:\Lambda\to G\) (the inversion map) and \(p:\Omega\to G\) (the product map) are continuous functions with \(\Lambda\subseteq G\) and open \(\Omega\subseteq G\times G\), such that \(1\in\Lambda\), \(\{1\}\times G\subseteq\Omega\), \(G\times \{1\}\subseteq\Omega\), and for all \(x,y,z\in G\):
(1) \(p(1, x)= p(x, 1)= x\); (2) if \(x\in\Lambda\), then \((x, \iota(x))\in\Omega\), \((\iota(x), x)\in\Omega\) and \(p(x,\iota(x))= p(\iota(x), x)= 1\); (3) if \((x, y), (y, z)\in\Omega\) and \((p(x, y),z)\), \((x, p(y,z))\in\Omega\), then (A) \(p(p(x,y),z)= p(x,p(y,z))\);
(b) \(G\) is locally Euclidean if there is an open neighbourhood of 1 homeomorphic to an open subset of \(\mathbb{R}^n\) for some \(n\);
(c) \(G\) is a local Lie group if \(G\) admits a \(C^\infty\) structure such that the maps \(\iota\) and \(p\) are \(C^w\);
(d) with \(U\) an open neighbourhood of 1 in \(G\), the restriction of \(G\) to \(U\) is the local group \(G/U:= (U,\iota/\Lambda_U, p/\Omega_U)\), where \(\Lambda_U:= \Lambda\cap U\cap \iota^{-1}(U)\) and \(\Omega_U:= (U\times U)\cap p^{-1}(U)\);
(e) \(G\) is globalizable if there is a topological group \(H\) and an open neighbourhood \(U\) of \(1_H\) in \(H\) such that \(G= H/U\).
Local \(H5\) first form: If \(G\) is a locally Euclidean local group, then some restriction of \(G\) is a local Lie group. Local \(H5\) second form: If \(G\) is a locally Euclidean local group, then some restriction of \(G\) is globalizable.
The equivalence of the two forms is established. The author follows the non-standard treatment of Hilbert’s fifth problem given by Hirschfeld; in dealing with local \(H5\), the author has a simplified way in the non-standard method.

MSC:

22A30 Other topological algebraic systems and their representations
22E99 Lie groups
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] D. R. Brown and R. S. Houston, ”Cancellative semigroups on manifolds,” Semigroup Forum, vol. 35, iss. 3, pp. 279-302, 1987. · Zbl 0626.22001 · doi:10.1007/BF02573112
[2] M. Davis, Applied Nonstandard Snalysis, New York: Wiley-Interscience [John Wiley & Sons], 1977. · Zbl 0359.02060
[3] L. van den Dries, Unpublished notes, 1981.
[4] A. M. Gleason, ”Groups without small subgroups,” Ann. of Math., vol. 56, pp. 193-212, 1952. · Zbl 0049.30105 · doi:10.2307/1969795
[5] Foundations of Nonstandard Analysis: A Gentle Introduction to Nonstandard Extensions; Nonstandard Analysis: Theory and ApplicationsDordrecht: Kluwer Academic Publishers Group, 1997. · Zbl 0910.03040
[6] J. Hirschfeld, ”The nonstandard treatment of Hilbert’s fifth problem,” Trans. Amer. Math. Soc., vol. 321, iss. 1, pp. 379-400, 1990. · Zbl 0705.03033 · doi:10.2307/2001608
[7] K. H. Hofmann and W. Weiss, ”More on cancellative semigroups on manifolds,” Semigroup Forum, vol. 37, iss. 1, pp. 93-111, 1988. · Zbl 0635.22003 · doi:10.1007/BF02573126
[8] R. Jacoby, ”Some theorems on the structure of locally compact local groups,” Ann. of Math., vol. 66, pp. 36-69, 1957. · Zbl 0084.03202 · doi:10.2307/1970116
[9] I. Kaplansky, Lie Algebras and Locally Compact Groups, Chicago, IL: The University of Chicago Press, 1971. · Zbl 0223.17001
[10] M. Kuranishi, ”On Euclidean local groups satisfying certain conditions,” Proc. Amer. Math. Soc., vol. 1, pp. 372-380, 1950. · Zbl 0038.01701 · doi:10.2307/2032387
[11] T. McGaffey, A partial solution to a generalization of Hilbert’s local fifth problem: the standard part of a locally euclidean local nonstandard Lie group is an analytic Lie group.
[12] D. Montgomery and L. Zippin, Topological Transformation Groups, New York: Interscience Publishers, 1955. · Zbl 0068.01904
[13] D. Montgomery and L. Zippin, ”Small subgroups of finite-dimensional groups,” Ann. of Math., vol. 56, pp. 213-241, 1952. · Zbl 0049.30106 · doi:10.2307/1969796
[14] P. J. Olver, ”Non-associative local Lie groups,” J. Lie Theory, vol. 6, iss. 1, pp. 23-51, 1996. · Zbl 0862.22005
[15] C. Plaut, ”Associativity and the local version of Hilbert’s fifth problem,” University of Tenessee, notes , 1993.
[16] L. Pontrjagin, Topological Group, Princeton, NJ: Princeton Univ. Press, 1939, vol. 2. · JFM 65.0872.02
[17] M. Singer, ”One parameter subgroups and nonstandard analysis,” Manuscripta Math., vol. 18, iss. 1, pp. 1-13, 1976. · Zbl 0328.22011 · doi:10.1007/BF01170531
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.