scholarly journals A fixed point theorem for Lie groups acting on buildings and applications to Kac–Moody theory

2015 ◽  
Vol 27 (1) ◽  
Author(s):  
Timothée Marquis
Keyword(s):  
Fixed Point ◽  
Finite Elements ◽  
Fixed Point Theorem ◽  
Lie Groups ◽  
Finite Rank ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Point Property ◽  
Locally Compact ◽  
Locally Finite

AbstractWe establish a fixed point property for a certain class of locally compact groups, including almost connected Lie groups and compact groups of finite abelian width, which act by simplicial isometries on finite rank buildings with measurable stabilisers of points. As an application, we deduce amongst other things that all topological one-parameter subgroups of a real or complex Kac–Moody group are obtained by exponentiating ad-locally finite elements of the corresponding Lie algebra.

Ergodic actions of group extensions on von Neumann algebras

1989 ◽  
Vol 112 (1-2) ◽  
pp. 71-112
Author(s):  
Klaus Thomsen
Keyword(s):  
Fixed Point ◽  
Von Neumann Algebras ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Von Neumann ◽  
Finite Dimensional ◽  
Fixed Point Algebra ◽  
Atomic Centre

SynopsisWe consider automorphic actions on von Neumann algebras of a locally compact group E given as a topological extension 0 → A → E → G → 0, where A is compact abelian and second countable. Motivated by the wish to describe and classify ergodic actions of E when G is finite, we classify (up to conjugacy) first the ergodic actions of locally compact groups on finite-dimensional factors and then compact abelian actions with the property that the fixed-point algebra is of type I with atomic centre. We then handle the case of ergodic actions of E with the property that the action is already ergodic when restricted to A, and then, as a generalisation, the case of (not necessarily ergodic) actions of E with the property that the restriction to A is an action with abelian atomic fixed-point algebra. Both these cases are handled for general locally compact-countable G. Finally, we combine the obtained results to classify the ergodic actions of E when G is finite, provided that either the extension is central and Hom (G, T) = 0, or G is abelian and either cyclic or of an order not divisible by a square.

scholarly journals Qualitative uncertainty principle for the Gabor transform on certain locally compact groups

2018 ◽  
Vol 9 (3) ◽  
pp. 205-220
Author(s):  
Jyoti Sharma ◽  
Ajay Kumar
Keyword(s):  
Lie Groups ◽  
Uncertainty Principle ◽  
Discrete Group ◽  
Locally Compact Groups ◽  
Gabor Transform ◽  
Compact Groups ◽  
Nilpotent Lie Groups ◽  
Locally Compact ◽  
Qualitative Uncertainty ◽  
Low Dimensional

Abstract Several classes of locally compact groups have been shown to possess a qualitative uncertainty principle for the Gabor transform. These include Moore groups, the Heisenberg group {\mathbb{H}_{n}} , the group {\mathbb{H}_{n}\times D} (where D is a discrete group) and other low-dimensional nilpotent Lie groups.

scholarly journals Some properties of locally compact groups

1967 ◽  
Vol 7 (4) ◽  
pp. 433-454 ◽  
Author(s):  
Neil W. Rickert
Keyword(s):  
Compact Group ◽  
Lie Groups ◽  
Locally Compact Group ◽  
Factor Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Finite Dimensional ◽  
Arcwise Connected ◽  
New Formulation

In this paper a number of questions about locally compact groups are studied. The structure of finite dimensional connected locally compact groups is investigated, and a fairly simple representation of such groups is obtained. Using this it is proved that finite dimensional arcwise connected locally compact groups are Lie groups, and that in general arcwise connected locally compact groups are locally connected. Semi-simple locally compact groups are then investigated, and it is shown that under suitable restrictions these satisfy many of the properties of semi-simple Lie groups. For example, a factor group of a semi-simple locally compact group is semi-simple. A result of Zassenhaus, Auslander and Wang is reformulated, and in this new formulation it is shown to be true under more general conditions. This fact is used in the study of (C)-groups in the sense of K. Iwasawa.

scholarly journals Locally compact groups with every isometric action bounded or proper

2018 ◽  
Vol 12 (02) ◽  
pp. 267-292
Author(s):  
Romain Tessera ◽  
Alain Valette
Keyword(s):  
Lie Groups ◽  
Local Field ◽  
Algebraic Groups ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Isometric Action ◽  
Locally Compact ◽  
Linear Algebraic Groups ◽  
Isometric Actions

A locally compact group [Formula: see text] has property PL if every isometric [Formula: see text]-action either has bounded orbits or is (metrically) proper. For [Formula: see text], say that [Formula: see text] has property BPp if the same alternative holds for the smaller class of affine isometric actions on [Formula: see text]-spaces. We explore properties PL and BPp and prove that they are equivalent for some interesting classes of groups: abelian groups, amenable almost connected Lie groups, amenable linear algebraic groups over a local field of characteristic 0. The appendix provides new examples of groups with property PL, including nonlinear ones.

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