Continuous transformation groups on spaces
A differentiable group is a group in the category of (reduced and nonreduced) differentiable spaces. Special cases are the rationals ℚ, Lie groups, formal groups over ℝ or ℂ; in general there is some mixture of those types, the general structure, however, is not yet completely determined. The following gives as a corollary a first essential answer. It is shown, more generally,that a locally compact topological transformation group, operating effectively on a differentiable space X (which satisfies some mild geometric property) is in fact a Lie group and operates differentiably on X. Special cases have already been known: X a manifold (Montgomery-Zippin), X a reduced (Kerner) or nonreduced (W. Kaup) complex space. The proof requires some analysis on arbitrary differentiable spaces. There one has for example in general no finitely generated ideals as in the case of complex spaces. As a corollary one obtains: The reduction of a locally compact differentiable group is a Lie group (by different methods also proved by Pasternak-Winiarski). It was already proved before that any differentiable group can be uniquely extended to a smallest locally compact differentiable group (as a dense subgroup). The study of the nonreduced parts of differentiable groups remains to be completed.