Casimir functions of free nilpotent Lie groups of steps three and four
Abstract
Any free nilpotent Lie algebra is determined by its rank and step. We consider free nilpotent Lie algebras of steps 3, 4 and corresponding connected and simply connected Lie groups. We construct Casimir functions of such groups, i.e., invariants of the coadjoint representation. For free 3step nilpotent Lie groups we get a full description of coadjoint orbits. It turns out that general coadjoint orbits are affine subspaces, and special coadjoint orbits are affine subspaces or direct products of nonsingular quadrics. The knowledge of Casimir functions is useful for investigation of integration properties of dynamical systems and optimal control problems on Carnot groups. In particular, for some wide class of timeoptimal problems on 3step free Carnot groups we conclude that extremal controls corresponding to twodimensional coadjoint orbits have the same behavior as in timeoptimal problems on the Heisenberg group or on the Engel group.
 Publication:

arXiv eprints
 Pub Date:
 May 2020
 DOI:
 10.48550/arXiv.2006.00224
 arXiv:
 arXiv:2006.00224
 Bibcode:
 2020arXiv200600224P
 Keywords:

 Mathematics  Differential Geometry;
 Mathematics  Optimization and Control;
 22E25;
 17B08;
 53C17;
 35R03
 EPrint:
 19 pages