Carnot algebras and sub-Riemannian structures with growth vector \((2,\,3,\,5,\,6)\)
Given a Carnot algebra \(\mathfrak{g}\) (also known as a stratified Lie algebra) of step \(s\) with stratification \(\mathfrak{g} = \bigoplus_{i=1}^s \mathfrak{g}^{(i)}\), its growth vector is the sequence \((n_1, \dots, n_s)\) where \(n_j = \sum_{i=1}^j \dim \mathfrak{g}^{(i)}\). The paper under review considers Carnot algebras \(\mathfrak{g}\) with growth vector \((2,3,5,6)\), which is to say those of step \(s=4\) and dimension \(\dim \mathfrak{g} = 6\) with \(\dim \mathfrak{g}^{(1)} = 2\), \(\dim \mathfrak{g}^{(2)} = 1\), \(\dim \mathfrak{g}^{(3)} = 2\), \(\dim \mathfrak{g}^{(4)} = 1\). It is known that, up to Lie algebra isomorphism, there are exactly three Carnot algebras of growth vector \((2,3,5,6)\) [\textit{V. V. Morozov}, Izv. Vyssh. Uchebn. Zaved., Mat. 1958, No. 4(5), 161--171 (1958; Zbl 0198.05501); Izv. Vyssh. Uchebn. Zaved., Mat. 1958, No. 4(5), 161--171 (1958); \textit{M.-P. Gong}, Classification of nilpotent Lie algebras of dimension 7 (over algebraically closed fields and R). Waterloo: University of Waterloo (Ph.D. Thesis) (1998); \textit{E. Le Donne} and \textit{F. Tripaldi}, ``A cornucopia of Carnot groups in low dimensions'', Preprint, \url{arXiv:2008.12356}]. In this paper, the authors begin by identifying the three types via a signature invariant \(\mathbf{s} \in \{-1, 0, 1\}\), and discuss how they arise as quotients of the free nilpotent Lie algebra \(\mathfrak{g}^8\) of step \(4\) with \(2\) generators, whose growth vector is \((2,3,5,8)\). Next, the authors consider left-invariant sub-Riemannian structures on such Carnot algebras \(\mathfrak{g}\), corresponding to a choice of inner product \(g\) on the horizontal distribution \(\Delta = \mathfrak{g}^{(1)}\). It is shown that the family of such structures is classified up to isometric isomorphism by a real parameter \(\rho \in [-1,1]\), whose sign corresponds to the signature \(\mathbf{s}\) giving the isomorphism type of the underlying Lie algebra \(\mathfrak{g}\). Moreover, it is shown that this also classifies them up to isometry of their sub-Riemannian (or Carnot-Carathéodory) distances. Finally, the authors compute the Casimir functions and symplectic foliations of each of the three Carnot algebras under study.
left-invariant sub-Riemannian structures
scientific article; zbMATH DE number 7465825
scientific article