|
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. ==Classic formulation== A formulation that was accepted for a long period was that the question was to characterize Lie groups as the topological groups that were also topological manifolds. In terms closer to those that Hilbert would have used, near the identity element of the group in question, there is an open set in Euclidean space containing , and on some open subset of there is a continuous mapping : that satisfies the group axioms where those are defined. This much is a fragment of a typical locally Euclidean topological group. The problem is then to show that is a smooth function near (since topological groups are homogeneous spaces, they look the same everywhere as they do near ). Another way to put this is that the possible differentiability class of does not matter: the group axioms collapse the whole gamut. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hilbert's fifth problem」の詳細全文を読む スポンサード リンク
|