Casimir Functions of Free Nilpotent Lie Groups of Steps 3 and 4
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Casimir Functions of Free Nilpotent Lie Groups of Steps 3 and 4 A. V. Podobryaev1 Revised: 1 September 2020 / Accepted: 7 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Any free nilpotent Lie algebra is determined by its rank and step. We consider free nilpotent Lie algebras of steps 3 and 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 3-step 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 time-optimal problems on 3-step free Carnot groups, we conclude that extremal controls corresponding to two-dimensional coadjoint orbits have the same behavior as in time-optimal problems on the Heisenberg group or on the Engel group. Keywords Free Carnot group · Coadjoint orbits · Casimir functions · Integration · Geometric control theory · Sub-Riemannian geometry · Sub-Finsler geometry Mathematics Subject Classification (2010) 22E25 · 17B08 · 53C17 · 35R03
1 Introduction The goal of this paper is a description of Casimir functions and coadjoint orbits for some class of nilpotent Lie groups. This knowledge is important for the theory of left-invariant optimal control problems on Lie groups [1]. Casimir functions are integrals of the Hamiltonian system of Pontryagin maximum principle. Moreover, coadjoint orbits are invariant under the flow of the vertical part of this Hamiltonian system. Note that left-invariant sub-Riemannian problems [2], sub-Finsler problems [3] on Lie groups, and in general problem of finding the shortest arcs for intrinsic left-invariant metrics on Lie groups [4] can be also viewed as optimal control problems.
A. V. Podobryaev
[email protected] 1
A. K. Ailamazyan Program Systems Institute of RAS, Pereslavl-Zalessky, Russia
A. V. Podobryaev
Nilpotent Lie groups play a key role in the theory of left-invariant optimal control problems in view of existence of nilpotent approximation [5]. Any nilpotent Lie group is a quotient of a free nilpotent Lie group. A connected and simply connected free nilpotent Lie group is determined by its rank and step. Sub-Riemannian and sub-Finsler problems on such groups were investigated only for steps 2 and 3 and ranks 2 and 3. For step 2, it is known that rank 2 corresponds to the Heisenberg group [6] and rank 3 sub-Riemannian structure was completely investigated by O. Myasnichenko [7]; for bigger ranks, there are some partial results [8–10]. For step 3, sub-Riemannian and sub-Finsler geodesics were investigated in [11, 12], and for sub-Finsler structures, some results were obtained by Yu. L. Sachkov [13]. It turns out that the Hamilto
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