The generalized Gluškov-Iwasawa local splitting theorem (Q2902438)

The author obtains a generalized Gluškov-Iwasawa local splitting theorem.NEWLINENEWLINELet \(G\) be an almost connected, projective Lie group such that there exists a family of normal subgroups of \(G\) such that their quotients are Lie groups, which converge to the identity element \(1\) of \(G\), and whose Lie algebras have nil-radicals finite dimensional. Then, for any open neighbourhood \(U\) of \(1 \in G\), there exists an almost connected, normal subgroup \(N\) of \(G\), contained in \(U\), with \(G/ N\) a Lie group, and there is an open neighbourhood \(W\) of the identity element of \(G/N\) and a map \(\varphi\) from \(W\) to \(G\) such that the map from \(N \times W\) to \(G\) induced by \(\varphi\) as \((n, w) \mapsto n \varphi (w)\) is a local homeomorphism and a local isomorphism from \(N \times (G/N) \) to \(G\).NEWLINENEWLINEAs a note, the classical Gluškov-Iwasawa local splitting theorem is given.NEWLINENEWLINELet \(G\) be a locally compact group. Then, for any open neighbourhood \(U\) of \(1 \in G\), there exists a compact normal subgroup \(N\) of \(G\), contained in \(U\), with \(G/N\) a Lie group such that \(G\) is locally isomorphic to \(N \times (G/N)\).