Invariant orderings in solvable Lie groups
type: Item
instance of: scholarly article
title: Invariant orderings in solvable Lie groups
zbMATH Open document ID: 0681.22016
DOI: 10.1007/BF01054214
published in: Siberian Mathematical Journal
publication date: 1989-01-01T00:00:00Z
review text: The author defines the concepts of oscillator Lie algebra and Lie group. An invariant cone in the Lie algebra \({\mathfrak g}\) is a cone C with acute vertex, closed, and invariant with respect to inner automorphisms of \({\mathfrak g}\). Theorem 1.1. Let \({\mathfrak g}\) be a solvable real Lie algebra. If \({\mathfrak g}\) admits an invariant cone C with int \(C\neq \emptyset\), then \({\mathfrak g}\) can be equipped with the structure of an oscillator Lie algebra. Every oscillator Lie algebra has an invariant cone C with int \(C\neq \emptyset.\) A continuous ordering in a Lie group G is a closed subsemigroup P which is invariant with respect to inner automorphisms, \(P\cap P^{-1}=\{e\}\), P (topologically) can be generated by any neighborhood of e, and the cone \(\{\) \(x\in {\mathfrak g}:\) \((\forall t>O)\) \((\exp(tx)\in P)\}\) has non-empty interior. Here \({\mathfrak g}\) is Lie algebra of G. Theorem 1.3-1.4. For every invariant cone C with int \(C\neq \emptyset\) in an oscillator Lie algebra there exists a unique continuous ordering in the corresponding simply connected Lie group. - A connected solvable Lie group admits a continuous ordering if and only if when its maximal compact subgroup is contained in the center of the group, and the Lie algebra admits an invariant cone C with int \(C\neq \emptyset\) having trivial intersection with the subalgebra that corresponds to the maximal compact subgroup. Every such cone corresponds to a unique continuous ordering.
Mathematics Subject Classification ID: 22E60
Mathematics Subject Classification ID: 06F25
Mathematics Subject Classification ID: 17B30
Mathematics Subject Classification ID: 22E25
zbMATH DE Number: 4114917
zbMATH Keywords: oscillator Lie algebra
zbMATH Keywords: inner automorphisms
zbMATH Keywords: solvable real Lie algebra
zbMATH Keywords: invariant cone
zbMATH Keywords: continuous ordering
zbMATH Keywords: simply connected Lie group
zbMATH Keywords: connected solvable Lie group
Wikidata QID: Q115394544
author: Victor Gichev
reviewed by: A. K. Guts
MaRDI profile type: Publication
cites work: Deformations of bouquets of quasihomogeneous one-dimensional singularities
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cites work: Lorentzian cones in real Lie algebras
cites work: Q3954461
full work available at URL: https://doi.org/10.1007/bf01054214
OpenAlex ID: W2022081278
description: scientific article; zbMATH DE number 4114917