Carnot algebras and sub-Riemannian structures with growth vector \((2,\,3,\,5,\,6)\)
type: Item
instance of: scholarly article
title: Carnot algebras and sub-Riemannian structures with growth vector \((2,\,3,\,5,\,6)\)
zbMATH Open document ID: 1495.53053
DOI: 10.1134/S0081543821050175
author: Yu L. Sachkov
author: E. F. Sachkova
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2022-01-28T00:00:00Z
review text: 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 tha…
reviewed by: Nathaniel Eldredge
Mathematics Subject Classification ID: 53C17
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 22E25
zbMATH DE Number: 7465825
zbMATH Keywords: sub-Riemannian geometry
zbMATH Keywords: Carnot algebras
zbMATH Keywords: Carnot groups
zbMATH Keywords: left-invariant sub-Riemannian structures
zbMATH Keywords: stratified Lie algebras
Wikidata QID: Q115247967
MaRDI profile type: Publication
cites work: A Comprehensive Introduction to Sub-Riemannian Geometry
cites work: Maxwell strata and cut locus in the sub-Riemannian problem on the Engel group
cites work: Q4530758
cites work: On the motion planning of the ball with a trailer
cites work: Invariant Carnot–Caratheodory Metrics on $S^3$, $SO(3)$, $SL(2)$, and Lens Spaces
cites work: Cut locus and optimal synthesis in sub-Riemannian problem on the Lie group SH(2)
cites work: Q4830318
cites work: Isometries of nilpotent metric groups
cites work: Liouville integrability of sub-Riemannian problems on Carnot groups of step 4 or greater
cites work: Complete description of the Maxwell strata in the generalized Dido problem
cites work: Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane
cites work: Degenerate abnormal trajectories in a sub-Riemannian problem with growth vector \((2, 3, 5, 8)\)
cites work: The structure of abnormal extremals in a sub-Riemannian problem with growth vector
cites work: Q4711784
full work available at URL: s0081543821050175
OpenAlex ID: W4205400913
description: scientific article; zbMATH DE number 7465825