Rational forms of nilpotent Lie algebras and Anosov diffeomorphisms
type: Item
instance of: scholarly article
title: Rational forms of nilpotent Lie algebras and Anosov diffeomorphisms
zbMATH Open document ID: 1153.22008
author: Jorge Lauret
published in: Monatshefte für Mathematik
publication date: 2008-11-25T00:00:00Z
review text: Let \(N\) be a simply connected nilpotent Lie group and assume that \(N\) admits a lattice \(\Gamma\) (i.e. a cocompact discrete subgroup). Then the compact quotient \(N/\Gamma\) is called a nilmanifold. Then it is interesting to study for such a nilmanifold the dynamics of the automorphisms of \(N\) stabilizing \(\Gamma\) or the geometry, if one equips \(N\) with a left invariant Riemannian metric, the complex structure, the symplectic structure, etc. The dynamical and geometric properties of the nilmanifold often only depend on the commensurability class of the lattice \(\Gamma\). In this connection the following problem appears: (\(\ast\)) Find all rational forms up to isomorphism of a given real nilpotent Lie algebra \(\mathfrak{n}\). Not much is known on this question, and a complete answer seems quite difficult to obtain in explicit examples, even in the low-dimensional or 2-step nilpotent cases. In a paper of \textit{F. Grunewald} and \textit{D. Segal} [``Reflections on the classification of torsion-free nilpotent groups'', Group theory, Essays for Philip Hall, 121--158 (1984; Zbl 0563.20033)], the set of isomorphism classes of rational forms on \(\mathfrak{n}\) is described by using Galois cohomology of the group \(Gal(\overline{\mathbb{Q}}/\mathbb{Q})\) with values in \(Aut(\mathfrak{n)}\). The problem can also be described in terms of rational points in the orbit space of an algebraic variety [\textit{P. Eberlein}, ``Geometry of 2-step nilpotent Lie groups'', in: Modern dynamical systems and applications. Dedicated to Anatole Katok on his 60th birthday. Cambridge: Cambridge University Press. 67--101 (2004; Zbl 1154.22009); \textit{F. J. Grunewald, D. Segal} and \textit{L. S. Sterling}, Math. Z. 179, 219--235 (1982; Zbl 0497.20026)]. A diffeomorphism \(f\) of a compact differentiable manifold \(M\) is called \textit{Anosov} if the most perfect kind of global hyperbolic behavior for a dynamical system holds: the tangent bundle \(TM\) admits a continuous inva…
reviewed by: Ioan Pop
Mathematics Subject Classification ID: 22E25
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 37D20
zbMATH DE Number: 5373499
zbMATH Keywords: Anosov diffeomorphism
zbMATH Keywords: nilmanifold
zbMATH Keywords: nilpotent Lie algebra
zbMATH Keywords: rational form
MaRDI profile type: Publication
full work available at URL: s00605-008-0562-0
OpenAlex ID: W2024719980
cites work: Q4780265
cites work: Nilpotent groups of Hirsch length six
cites work: Examples of Anosov diffeomorphisms
cites work: On Anosov automorphisms of nilmanifolds
cites work: Nilmanifolds of dimension ≤8 admitting Anosov diffeomorphisms
cites work: Q5669004
cites work: Two-step nilpotent Lie algebras
cites work: Q4501779
cites work: Differentiable dynamical systems
DOI: 10.1007/S00605-008-0562-0
description: scientific article; zbMATH DE number 5373499