Английская Википедия:Hilbert's fifth problem
Шаблон:Short description Hilbert's fifth problem is the fifth mathematical problem from the problem list publicized in 1900 by mathematician David Hilbert, and concerns the characterization of Lie groups.
The theory of Lie groups describes continuous symmetry in mathematics; its importance there and in theoretical physics (for example quark theory) grew steadily in the twentieth century. In rough terms, Lie group theory is the common ground of group theory and the theory of topological manifolds. The question Hilbert asked was an acute one of making this precise: is there any difference if a restriction to smooth manifolds is imposed?
The expected answer was in the negative (the classical groups, the most central examples in Lie group theory, are smooth manifolds). This was eventually confirmed in the early 1950s. Since the precise notion of "manifold" was not available to Hilbert, there is room for some debate about the formulation of the problem in contemporary mathematical language.
Formulation of the problem
A modern formulation of the problem (in its simplest interpretation) is as follows:Шаблон:Sfn
An equivalent formulation of this problem closer to that of Hilbert, in terms of composition laws, goes as follows:[1]
In this form the problem was solved by Montgomery–Zippin and Gleason.
A stronger interpretation (viewing Шаблон:Math as a transformation group rather than an abstract group) results in the Hilbert–Smith conjecture about group actions on manifolds, which in full generality is still open. It is known classically for actions on 2-dimensional manifolds and has recently been solved for three dimensions by John Pardon.
Solution
The first major result was that of John von Neumann in 1933,[2] giving an affirmative answer for compact groups. The locally compact abelian group case was solved in 1934 by Lev Pontryagin. The final resolution, at least in the interpretation of what Hilbert meant given above, came with the work of Andrew Gleason, Deane Montgomery and Leo Zippin in the 1950s.
In 1953, Hidehiko Yamabe obtained further results about topological groups that may not be manifolds:Шаблон:Efn
It follows that every locally compact group contains an open subgroup that is a projective limit of Lie groups, by van Dantzig's theorem (this last statement is called the Gleason–Yamabe Theorem in Шаблон:Harvtxt).
No small subgroups
An important condition in the theory is no small subgroups. A topological group Шаблон:Math, or a partial piece of a group like Шаблон:Math above, is said to have no small subgroups if there is a neighbourhood Шаблон:Math of Шаблон:Math containing no subgroup bigger than Шаблон:Math. For example, the circle group satisfies the condition, while the [[p-adic integers|Шаблон:Math-adic integers]] Шаблон:Math as additive group does not, because Шаблон:Math will contain the subgroups: Шаблон:Math, for all large integers Шаблон:Math. This gives an idea of what the difficulty is like in the problem. In the Hilbert–Smith conjecture case it is a matter of a known reduction to whether Шаблон:Math can act faithfully on a closed manifold. Gleason, Montgomery and Zippin characterized Lie groups amongst locally compact groups, as those having no small subgroups.
Infinite dimensions
Researchers have also considered Hilbert's fifth problem without supposing finite dimensionality. This was the subject of Per Enflo's doctoral thesis; his work is discussed in Шаблон:Harvtxt.
See also
Notes
Шаблон:Notelist Шаблон:Reflist
References
- Шаблон:Cite journal
- Шаблон:Cite book
- Шаблон:Cite book
- Шаблон:Cite book
- Yamabe, Hidehiko, On an arcwise connected subgroup of a Lie group, Osaka Mathematical Journal v.2, no. 1 Mar. (1950), 13–14.
- Irving Kaplansky, Lie Algebras and Locally Compact Groups, Chicago Lectures in Mathematics, 1971.
- Шаблон:Cite book
- Enflo, Per. (1970) Investigations on Hilbert’s fifth problem for non locally compact groups. (Ph.D. thesis of five articles of Enflo from 1969 to 1970)
- Enflo, Per; 1969a: Topological groups in which multiplication on one side is differentiable or linear. Math. Scand., 24, 195–197.
- Шаблон:Cite journal
- Enflo, Per; 1969b: On a problem of Smirnov. Ark. Mat. 8, 107–109.
- Шаблон:Cite journal
- Шаблон:Cite journal
Шаблон:Hilbert's problems Шаблон:Authority control
- ↑ Hilbert, David. "5. Lie's concept of a continuous group of transformations without the assumption of the differentiability of the functions defining the group". Mathematical Problems – via Wikisource.
- ↑ Шаблон:Cite journal
