A rigidity phenomenon for germs of actions of \(\mathbb{R}^2\). (English) Zbl 1214.22001

The authors deal with the maximality of Lie algebras generated by commuting vector fields without linear part. The work is restricted to those commuting vector fields whose Taylor development starts with a given pair of polynomial homogeneous commuting vector fields. The authors restrict their interest to three special maximal Lie algebras generated by commuting quadratic homogeneous vector fields in \(\mathbb R^2\), \(\mathbb R^3\) and \(\mathbb R^4\) respectively. In the first case it is proved that the quadratic algebra is a smooth normal form. In the second and third ones, the authors prove that the orbit structure is, from a topological viewpoint, the one of the quadratic part.


22E05 Local Lie groups
17B66 Lie algebras of vector fields and related (super) algebras
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