scholarly journals O-minimal spectra, infinitesimal subgroups and cohomology

2007 ◽  
Vol 72 (4) ◽  
pp. 1177-1193 ◽  
Author(s):  
Alessandro Berarducci
Keyword(s):  
Compact Group ◽  
Lie Groups ◽  
Lie Group ◽  
Compact Groups ◽  
Natural Homomorphism ◽  
Ordered Field ◽  
Special Cases ◽  
Logic Topology ◽  
Cech Cohomology ◽  
Exact Sequences

AbstractBy recent work on some conjectures of Pillay, each definably compact group G in a saturated o-minimal expansion of an ordered field has a normal “infinitesimal subgroup” G00 such that the quotient G/G00, equipped with the “logic topology”, is a compact (real) Lie group. Our first result is that the functor G ↦ G/G00 sends exact sequences of definably compact groups into exact sequences of Lie groups. We then study the connections between the Lie group G/G00 and the o-minimal spectrum of G. We prove that G/G00 is a topological quotient of . We thus obtain a natural homomorphism Ψ* from the cohomology of G/G00 to the (Čech-)cohomology of . We show that if G00 satisfies a suitable contractibility conjecture then is acyclic in Čech cohomology and Ψ is an isomorphism. Finally we prove the conjecture in some special cases.

scholarly journals Cohomology of groups in o-minimal structures: acyclicity of the infinitesimal subgroup

2009 ◽  
Vol 74 (3) ◽  
pp. 891-900 ◽  
Author(s):  
Alessandro Berarducci
Keyword(s):  
Compact Group ◽  
Recent Work ◽  
Lie Groups ◽  
Lie Group ◽  
Compact Groups ◽  
Compact Lie Group ◽  
Minimal Structure ◽  
Divisible Subgroup ◽  
Torsion Free ◽  
Cohomology Of Groups

AbstractBy recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal structure is an extension of a compact Lie group by a torsion free normal divisible subgroup, called its infinitesimal subgroup. We show that the infinitesimal subgroup is cohomologically acyclic. This implies that the functorial correspondence between definably compact groups and Lie groups preserves the cohomology.

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 Moments of probability measures on a group

1981 ◽  
Vol 4 (1) ◽  
pp. 1-37 ◽  
Author(s):  
Herbert Heyer
Keyword(s):  
Topological Group ◽  
Banach Spaces ◽  
Lie Groups ◽  
Finite Groups ◽  
Lie Group ◽  
Probability Measures ◽  
Moment Conditions ◽  
New Developments ◽  
Large Numbers ◽  
Special Cases

New developments and results in the theory of expectatiors and variances for random variables with range in a topological group are presented in the following order (i) Introduction (2) Basic notions (3) The three series theorem in Banach spaces (4) Moment Conditions (5) Expectations and variances (6) A general three series theorem (7) The special cases of finite groups and Lie groups (8)The strong laws of large numbers on a Lie group (9) Further studies on moments of probability measures.

scholarly journals A topological conjugacy of invariant flows on some class of Lie groups

2019 ◽  
Vol 38 (3) ◽  
pp. 151-160
Author(s):  
Alexandre J. Santana ◽  
Simão N. Stelmastchuk
Keyword(s):  
Lie Groups ◽  
Lie Algebra ◽  
Lie Group ◽  
Topological Conjugacy ◽  
Special Cases

The aim of this paper is to classify invariant flows on Lie group $G$ whose Lie algebra $\mathfrak{g}$ is associative or semisimple. Specifically, we present this classification from the hyperbolicity of the lift flows on $G \times \mathfrak{g}$. Then we apply this construction to some special cases as ${\rm Gl}(2,{\Bbb R})$ and affine Lie group.

scholarly journals Non-tall compact groups admit infinite Sidon sets

1977 ◽  
Vol 23 (4) ◽  
pp. 467-475 ◽  
Author(s):  
M. F. Hutchinson
Keyword(s):  
Compact Group ◽  
Lie Group ◽  
Unitary Representations ◽  
Compact Groups ◽  
Sufficient Condition ◽  
Compact Lie Group ◽  
Sidon Sets ◽  
Sidon Set ◽  
Irreducible Unitary Representations

AbstractRiesz polynomials are employed to give a sufficient condition for a non-abelian compact group G to have an infinite uniformly approximable Sidon set. This condition is satisfied if G admits infinitely many pairwise inequivalent continuous irreducible unitary representations of the same degree. Consequently a compact Lie group admits an infinite Sidon set if and only if it is not semi-simple.

scholarly journals Diffusive wavelets on groups and homogeneous spaces

2011 ◽  
Vol 141 (3) ◽  
pp. 497-520 ◽  
Author(s):  
Svend Ebert ◽  
Jens Wirth
Keyword(s):  
Fourier Transform ◽  
Lie Groups ◽  
Lie Group ◽  
Homogeneous Spaces ◽  
General Concept ◽  
Compact Groups ◽  
Compact Lie Group ◽  
Weyl Decomposition ◽  
Group Fourier Transform ◽  
Basic Ideas

We explain the basic ideas behind the concept of diffusive wavelets on spheres in the language of representation theory of Lie groups and within the framework of the group Fourier transform given by the Peter-Weyl decomposition of L2() for a compact Lie group . After developing a general concept for compact groups and their homogeneous spaces, we give concrete examples for tori, which reflect the situation on ℝn, and for 2 and 3 spheres.

scholarly journals Approximation of nilpotent orbits for simple Lie groups

2021 ◽  
Vol 56 (2) ◽  
pp. 287-327
Author(s):  
Lucas Fresse ◽  
◽  
Salah Mehdi ◽  
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Finite Union ◽  
Nilpotent Orbits ◽  
Simple Lie Groups ◽  
Special Cases ◽  
Adjoint Orbits

We propose a systematic and topological study of limits \(\lim_{\nu\to 0^+}G_\mathbb{R}\cdot(\nu x)\) of continuous families of adjoint orbits for a non-compact simple real Lie group \(G_\mathbb{R}\). This limit is always a finite union of nilpotent orbits. We describe explicitly these nilpotent orbits in terms of Richardson orbits in the case of hyperbolic semisimple elements. We also show that one can approximate minimal nilpotent orbits or even nilpotent orbits by elliptic semisimple orbits. The special cases of \(\mathrm{SL}_n(\mathbb{R})\) and \(\mathrm{SU}(p,q)\) are computed in detail.

Contraction subgroups and semistable measures on p-adic Lie groups

1991 ◽  
Vol 110 (2) ◽  
pp. 299-306 ◽  
Author(s):  
S. G. Dani ◽  
Riddhi Shah
Keyword(s):  
Compact Group ◽  
Lie Groups ◽  
Locally Compact Group ◽  
Locally Compact Groups ◽  
Probability Measures ◽  
Compact Groups ◽  
Locally Compact ◽  
Significant Class

Continuous one-parameter semigroups {μt}t≥0 of probability measures on a locally compact group which are semistable with respect to some automorphism τ of the group, namely such that τ(μt) = μct for all t ≥ 0, for a fixed c ∈ (0, 1), have attracted considerable attention of various researchers in recent years (cf. [3], [5] and other references cited therein). A detailed study of semistable measures on (real) Lie groups is carried out in [5]. In this context it is of interest to study semistable measures on the class of p-adic Lie groups, which is another significant class of locally compact groups.

scholarly journals Some groups with T1 primitive ideal spaces

1985 ◽  
Vol 38 (1) ◽  
pp. 55-64 ◽  
Author(s):  
A. L. Carey ◽  
W. Moran
Keyword(s):  
Normal Subgroup ◽  
Lie Groups ◽  
Lie Group ◽  
Locally Compact Group ◽  
Primitive Ideal ◽  
Ideal Space ◽  
Solvable Lie Groups ◽  
Locally Compact ◽  
Simply Connected ◽  
Connected Lie Group

AbstractLet G be a second countable locally compact group possessing a normal subgroup N with G/N abelian. We prove that if G/N is discrete then G has T1 primitive ideal space if and only if the G-quasiorbits in Prim N are closed. This condition on G-quasiorbits arose in Pukanzky's work on connected and simply connected solvable Lie groups where it is equivalent to the condition of Auslander and Moore that G be type R on N (-nilradical). Using an abstract version of Pukanzky's arguments due to Green and Pedersen we establish that if G is a connected and simply connected Lie group then Prim G is T1 whenever G-quasiorbits in [G, G] are closed.

scholarly journals Compact groups with a set of positive Haar measure satisfying a nilpotent law

2021 ◽  
pp. 1-4
Author(s):  
ALIREZA ABDOLLAHI ◽  
MEISAM SOLEIMANI MALEKAN
Keyword(s):  
Compact Group ◽  
Haar Measure ◽  
Positive Answer ◽  
Nilpotent Subgroup ◽  
Compact Groups

Abstract The following question is proposed by Martino, Tointon, Valiunas and Ventura in [4, question 1·20]: Let G be a compact group, and suppose that \[\mathcal{N}_k(G) = \{(x_1,\dots,x_{k+1}) \in G^{k+1} \;|\; [x_1,\dots, x_{k+1}] = 1\}\] has positive Haar measure in $G^{k+1}$ . Does G have an open k-step nilpotent subgroup? We give a positive answer for $k = 2$ .

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