Local splitting of locally compact groups and pro-Lie groups
The note provides a new proof for a local splitting theorem for pro-Lie groups, i.e., (not necessarily locally compact) topological groups which are projective limits of (finite-dimensional, real) Lie groups. A profound theory of pro-Lie groups was developed in a recent monograph by the authors [The Lie theory of connected pro-Lie groups. A structure theory for pro-Lie algebras, pro-Lie groups, and connected locally compact groups. Zürich: European Mathematical Society (2007; Zbl 1153.22006)], (*). The book already contains a local splitting result (op. cit., Theorem 13.19) for connected pro-Lie groups, stimulated by the classical work of \textit{K. Iwasawa} [Ann.\ Math. (2) 50, 507--558 (1949; Zbl 0034.01803)] which applies to locally compact groups. The condition of connectedness was removed afterwards by \textit{A. A. George Michael} (see [J. Math. Sci. Univ. Tokyo 17, 123--133 (2010; Zbl 1254.22001)]). Based on the general theory developed in (*), the authors give a short new proof for George Michael's result.NEWLINENEWLINERecall that a pro-Lie group \(G\) is said to be almost connected if \(G/G_e\) is compact (where \(G_e\) is the connected component of the identity element). Let \({\mathfrak g}\) be the Lie algebra associated to \(G\) (as in (*)) and \({\mathfrak s}\) be its coreductive radical, i.e., the smallest closed ideal of \({\mathfrak g}\) such that \({\mathfrak g}/{\mathfrak s}\) is reductive (viz.\ a direct product of a family of simple finite-dimensional Lie algebras and copies of \({\mathbb R}\)). George Michael's result is established in the following form (Theorem 2): Let \(G\) be an almost connected pro-Lie group such that the coreductive radical of \({\mathfrak g}\) is finite-dimensional. Then there are arbitrarily small normal subgroups \(N\) such that there is a simply connected Lie group \(L\) and a morphism \(\alpha: L\to G\) such that \(N\times L\to G\), \((n,x)\mapsto n\alpha(x)\) is open and has discrete kernel. In particular, \(G\) and \(N\times L\) are locally isomorphic.
scientific article; zbMATH DE number 6031523
scientific article