On Some Types of Topological Groups

Authors

Kenkichi Iwasawa

Type: Article

Publication Date: 1949-07-01

Citations: 644

DOI: https://doi.org/10.2307/1969548

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Abstract

As is well-known the so-called fifth problem of Hilbert on continuous groups was solved by J. v. Neumann [14]2 for compact groups and by L. Pontrjagin [15] for abelian groups. More recently, it is reported, C. Chevalley [6] solved it for solvable groups.3 Now it seems, as H. Freudenthal [7] clarified for maximally almost periodic groups, that the essential source of the proof of Hilbert's problem for these groups lies in the fact that such groups can be approximated by Lie groups. Here we say that a locally compact group G can be approximated by Lie groups, if G contains a system of normal subgroups {Na} such that G/Na are Lie groups and that the intersection of all Na coincides with the identity e. For the brevity we call such a group a group of type (L) or an (L)-group. In the present paper we shall study the structure of such (L)-groups, and apply the result to solve the Hilbert's problem for a certain class of groups, which contains both compact and solvable groups as special cases. We shall be able to characterize a Lie group G, for which the factor group GIN of G modulo its radical N is compact, completely by its structure as a topological group. The outline of the paper is as follows. In ?1 we study the topological structure of the group of automorphisms of a compact group and prove theorems concerning compact normal subgroups of a connected topological group, which are to be used repeatedly in succeeding sections. In ?2 come some preliminary considerations on solvable groups, whereas finer structural theorems on these groups are, as special cases of (L)-groups, given later. In ?3 we prove some theorems on Lie groups. The theorems here stated are not all new, but we give them here for the sake of completeness, and thereby refine and modify these theorems so as to be applied appropriately in succeeding sections.4 After these preparations we study in ?4 the structure of (L)-groups. In particular, it is shown 'that the study of the local structure and the global topological structure

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Annals of Mathematics
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We establish the convexity of Mabuchi’s <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-energy functional along weak geodesics in the space of Kähler potentials on … We establish the convexity of Mabuchi’s <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper K"> <mml:semantics> <mml:mi>K</mml:mi> <mml:annotation encoding="application/x-tex">K</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-energy functional along weak geodesics in the space of Kähler potentials on a compact Kähler manifold, thus confirming a conjecture of Chen, and give some applications in Kähler geometry, including a proof of the uniqueness of constant scalar curvature metrics (or more generally extremal metrics) modulo automorphisms. The key ingredient is a new local positivity property of weak solutions to the homogeneous Monge-Ampère equation on a product domain, whose proof uses plurisubharmonic variation of Bergman kernels.

On Connectivity Properties of Finite-Dimensional Groups

1969-10-01

Sigmund N. Hudson

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Structural analysis and elementary representations of SL(4, R) and GL(4, R) and their covering groups

1986-04-01

V. K. Dobrev , Orlin Stoytchev

The structure of the groups SL(4,R) and GL(4,R) and their universal cover− ing groups SL(4,R) and GL(4,R), respectively, and Lie algebras sl(4,R) and gl(4,R), respectively, are studied. The parabolic subgroups … The structure of the groups SL(4,R) and GL(4,R) and their universal cover− ing groups SL(4,R) and GL(4,R), respectively, and Lie algebras sl(4,R) and gl(4,R), respectively, are studied. The parabolic subgroups and subalgebras are identified and the cuspidal parabolic subgroups singled out. The Iwasawa and Bruhat decompositions are given explicitly. All elementary representations (ER) of SL(4,R) are explicitly given in two equivalent realizations. Using the preceding detailed structural analysis the SL(4,R) constructions are used for the explicit realization of all ER of SL(4,R), GL(4,R), and GL(4,R). The results shall be applied (among other things) elsewhere for the construction of all irreducible representations of the above groups.

Compact Lorentz manifolds with local symmetry

2009-02-01

Karin Melnick

We prove a structure theorem for compact aspherical Lorentz manifolds with abundant local symmetry. If M is a compact, as- pherical, real-analytic, complete Lorentz manifold such that the isometry group … We prove a structure theorem for compact aspherical Lorentz manifolds with abundant local symmetry. If M is a compact, as- pherical, real-analytic, complete Lorentz manifold such that the isometry group of the universal cover has semisimple identity com- ponent, then the local isometry orbits in M are roughly fibers of a fiber bundle. A corollary is that if M has an open, dense, locally homogeneous subset, then M is locally homogeneous.

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Iwasawa decompositions of some infinite-dimensional Lie groups

2009-07-22

Daniel Beltiţă

We set up an abstract framework that allows the investigation of Iwasawa decompositions for involutive infinite-dimensional Lie groups modeled on Banach spaces. This provides a method to construct Iwasawa decompositions … We set up an abstract framework that allows the investigation of Iwasawa decompositions for involutive infinite-dimensional Lie groups modeled on Banach spaces. This provides a method to construct Iwasawa decompositions for classical real or complex Banach-Lie groups associated with the Schatten ideals $\mathfrak {S}_p(\mathcal {H})$ on a complex separable Hilbert space $\mathcal {H}$ if $1<p<\infty$.

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A Note on Haar Measure

1955-04-01

Harry F. Davis

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Superrigidity of lattices in solvable Lie groups

1995-12-01

Dave Witte

Zur Klassifikation topologischer Ebenen

1963-06-01

Helmut Salzmann

Non-compact Complex Lie Groups without Non-constant Holomorphic Functions

1965-01-01

Akihiko Morimoto

On the decomposition of an ( $L$ )-group

1950-06-01

YOZÔ MATSUSHIMA

On $t/\iota e$ .decomposition of an $(L)$ -group 265 As K. Iwasawa has shown, there exists in any connected locally compact group $G$ a unique maximal solvable connected closed normal … On $t/\iota e$ .decomposition of an $(L)$ -group 265 As K. Iwasawa has shown, there exists in any connected locally compact group $G$ a unique maximal solvable connected closed normal subgroup which is called the radical of $G^{S)}$ .Definition $I$ .A connected (L)-group is said to be semi-simple if its radical contains only the unit element.By a result of M. Got\^o4) any connected semi-simple (L)-group $G$ is the product of its maximal connected compact normal subgroup $C$ and a closed connected normal subgroup $L$ , which is a semi-simpl Lie group containing no compact connected normal subgroup other than the groupconsisting only of the unit element, such that $[L, C]=e$ and $L\cap G$ is a finite group5).We call such a decomposition of a connected semi- simple (L)-group a canonical decomposition.

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On the Brunn-Minkowski coefficient of a locally compact unimodular group

1969-01-01

M. McCrudden

Let G be a locally compact topological group, and let μ be the left Haar measure on G , with μ the corresponding outer measure. If R' denotes the non-negative … Let G be a locally compact topological group, and let μ be the left Haar measure on G , with μ the corresponding outer measure. If R' denotes the non-negative extended real numbers, B ( G ) the Borel subsets of G , and V = {μ( C ): C ∈ B ( G )}, then we can define Φ G : V × V → R' by where AB denotes the product set of A and B in G. Then clearly so that a knowledge of Φ G will give us some idea of how the outer measure of the product set AB compares with the measures of the sets A and B.

Homogeneous Ricci Solitons in Low Dimensions

2014-06-05

Romina M. Arroyo , R Lafuente

In this article, we classify expanding homogeneous Ricci solitons up to dimension 5, according to their presentation as homogeneous spaces.We obtain that they are all isometric to solvsolitons, and this, … In this article, we classify expanding homogeneous Ricci solitons up to dimension 5, according to their presentation as homogeneous spaces.We obtain that they are all isometric to solvsolitons, and this, in particular, implies that the generalized Alekseevskii conjecture holds in these dimensions.In addition, we prove that the conjecture holds in dimension 6 provided the transitive group is not semi-simple.

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A Brunn-Minkowski type inequality for Fano manifolds and some uniqueness theorems in Kähler geometry

2013-01-01

Bo Berndtsson

For $\phi$ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold $X$ we consider the volume of $X$ $$ \int_X e^{-\phi}. $$ We prove that the logarithm of … For $\phi$ a metric on the anticanonical bundle, $-K_X$, of a Fano manifold $X$ we consider the volume of $X$ $$ \int_X e^{-\phi}. $$ We prove that the logarithm of the volume is concave along bounded geodesics in the space of positively curved metrics on $-K_X$ and that the concavity is strict unless the geodesic comes from the flow of a holomorphic vector field on $X$. As a consequence we get a simplified proof of the Bando-Mabuchi uniqueness theorem for K\"ahler - Einstein metrics. A generalization of this theorem to 'twisted' K\"ahler-Einstein metrics and some classes of manifolds that satisfy weaker hypotheses than being Fano is also given. We moreover discuss a generalization of the main result to other bundles than $-K_X$, and finally use the same method to give a new proof of the theorem of Tian and Zhu of uniqueness of K\"ahler-Ricci solitons. This is an expanded version of an earlier preprint, "A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem", arXiv:1103.0923

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Category theoretical methods in topological algebra

1976-01-01

Karl H. Hofmann

Проблема Каждана - Мильмана для полупростых компактных групп Ли

2007-01-01

Александр Исаакович Штерн , A. I. Shtern

2007 г. январь -февраль т. 62, вып. 1 (373) УСПЕХИ МАТЕМАТИЧЕСКИХ НАУК УДК 517.986.6 Светлой памяти С. С. АверинцеваПроблема Каждана-Мильмана для полупростых компактных групп Ли А. И. ШтернЗа последние 25-30 … 2007 г. январь -февраль т. 62, вып. 1 (373) УСПЕХИ МАТЕМАТИЧЕСКИХ НАУК УДК 517.986.6 Светлой памяти С. С. АверинцеваПроблема Каждана-Мильмана для полупростых компактных групп Ли А. И. ШтернЗа последние 25-30 лет в теории отображений, близких к представлениям, -почти представлений, аппроксимативных представлений, квазипредставлений, псевдопредставлений и т. д. -накоплен большой материал и созданы технические приемы, имеющие нетривиальные приложения в алгебре и топологии, -от ограниченных когомологий до финслеровых метрик и инварианта Калаби в симплектической геометрии.В обзоре основные понятия и факты теории излагаются в связи с приводимым в данной работе доказательством "теоремы тривиальности" для конечномерных квазипредставлений компактных групп Ли: любое (не обязательно непрерывное) конечномерное унитарное квазипредставление с малым дефектом полупростой компактной группы Ли близко к обычному (непрерывному) представлению этой группы.Эта теорема, дающая полный ответ на вопрос Каждана-Мильмана 1982 г., является и частичным ответом на вопрос Громова 1995 г., а именно, хотя полупростая компактная группа в дискретной топологии не аменабельна, но все ее конечномерные унитарные квазипредставления являются возмущениями обычных представлений.Кроме того, указаны необходимые и достаточные условия справедливости аналога теоремы Ван дер Вардена (т.е.условия автоматической непрерывности всех локально ограниченных конечномерных представлений) для данной связной группы Ли и дано описание структуры всех конечномерных локально ограниченных квазипредставлений произвольных связных полупростых групп Ли.Обсуждаются и результаты, связанные с некоторыми другими направлениями исследований по теории отображений групп и алгебр, близких к представлениям, и их приложениями к геометрии и теории групп.Библиография: 225 названий.

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Connected components of definable groups and o-minimality I

2011-01-29

Annalisa Conversano , Anand Pillay

We study the connected components G^00, G^000 and their quotients for a group G definable in a saturated o-minimal expansion of a real closed field. We show that G^00/G^000 is … We study the connected components G^00, G^000 and their quotients for a group G definable in a saturated o-minimal expansion of a real closed field. We show that G^00/G^000 is naturally the quotient of a connected compact commutative Lie group by a dense finitely generated subgroup. We also highlight the role of universal covers of semisimple Lie groups.

Tessellations of homogeneous spaces of classical groups of real rank two

2001-01-01

Alessandra Iozzi , Dave Witte

Let H be a closed, connected subgroup of a connected, simple Lie group G with finite center. The homogeneous space G/H has a "tessellation" if there is a discrete subgroup … Let H be a closed, connected subgroup of a connected, simple Lie group G with finite center. The homogeneous space G/H has a "tessellation" if there is a discrete subgroup D of G, such that D acts properly discontinuously on G/H, and the double-coset space D\G/H is compact. Note that if either H or G/H is compact, then G/H has a tessellation; these are the obvious examples. It is not difficult to see that if G has real rank one, then only the obvious homogeneous spaces have tessellations. Thus, the first interesting case is when G has real rank two. In particular, R.Kulkarni and T.Kobayashi constructed examples that are not obvious when G = SO(2,2n) or SU(2,2n). H.Oh and D.Witte constructed additional examples in both of these cases, and obtained a complete classification when G = SO(2,2n). We simplify the work of Oh-Witte, and extend it to obtain a complete classification when G = SU(2,2n). This includes the construction of another family of examples. The main results are obtained from methods of Y.Benoist and T.Kobayashi: we fix a Cartan decomposition G = KAK, and study the intersection of KHK with A. Our exposition generally assumes only the standard theory of connected Lie groups, although basic properties of real algebraic groups are sometimes also employed; the specialized techniques that we use are developed from a fairly elementary level.

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Proper equivariant stable homotopy theory.

2019-08-02

Dieter Degrijse , Markus Hausmann , Wolfgang Lück +2 more

This monograph introduces a framework for genuine proper equivariant stable homotopy theory for Lie groups. The adjective `proper' alludes to the feature that equivalences are tested on compact subgroups, and … This monograph introduces a framework for genuine proper equivariant stable homotopy theory for Lie groups. The adjective `proper' alludes to the feature that equivalences are tested on compact subgroups, and that the objects are built from equivariant cells with compact isotropy groups; the adjective `genuine' indicates that the theory comes with appropriate transfers and Wirthmuller isomorphisms, and the resulting equivariant cohomology theories support the analog of an $RO(G)$-grading. Our model for genuine proper $G$-equivariant stable homotopy theory is the category of orthogonal $G$-spectra; the equivalences are those morphisms that induce isomorphisms of equivariant stable homotopy groups for all compact subgroups of $G$. This class of $\pi_*$-isomorphisms is part of a symmetric monoidal stable model structure and the associated tensor triangulated homotopy category is compactly generated. Every orthogonal $G$-spectrum represents an equivariant cohomology theory on the category of $G$-spaces, depending only on the `proper $G$-homotopy type', tested by fixed points under all compact subgroups. An important special case are infinite discrete groups. For these, our genuine equivariant theory is related to finiteness properties, in the sense of geometric group theory; for example, the $G$-sphere spectrum is a compact object in the equivariant homotopy category if the universal space for proper $G$-actions has a finite $G$-CW-model. For discrete groups, the represented equivariant cohomology theories on finite proper $G$-CW-complexes admit a more explicit description in terms of parameterized equivariant homotopy theory, suitably stabilized by $G$-vector bundles. Via this description, we can identify the previously defined $G$-cohomology theories of equivariant stable cohomotopy and equivariant K-theory as cohomology theories represented by specific orthogonal $G$-spectra.

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AUTOMATIC CONTINUITY OF ABSTRACT HOMOMORPHISMS BETWEEN LOCALLY COMPACT AND POLISH GROUPS

2019-07-08

Oskar Braun , Karl H. Hofmann , Linus Kramer

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Möbius Action of SL(2;$${\mathbb {R}})$$ on Different Homogeneous Spaces

2020-03-02

Debapriya Biswas , Sandipan Dutta

Linear representations of any dimensional Lorentz group and computation formulas for their matrix elements

1979-04-01

Takayoshi Maekawa

The representation matrix elements of SO(n,1) are discussed in a space spanned by the representation matrix elements of the maximal compact subgroup SO(n). A multiplier of the representation corresponding to … The representation matrix elements of SO(n,1) are discussed in a space spanned by the representation matrix elements of the maximal compact subgroup SO(n). A multiplier of the representation corresponding to the boost of SO(n,1) is completely determined by requiring the commutation relations of SO(n,1) for the differential operators of the multiplier representation and of the parameter group of SO(n). It is shown that the bases of the space, the representation matrix elements of SO(n), are classified by the group chains of the first and the second parameter groups of SO(n), whose differential operators commute with each other, and the characteristic numbers of SO(n,1) are the same as those of the first parameter group of SO(n−1) and a complex number appearing in the multiplier. By using the scalar product defined in the space, the matrix elements for the differential operators and the computation formulas for the representation corresponding to the boost of SO(n,1) are given for all unitary representations of SO(n,1) and useful formulas containing the d matrix elements of SO(n) are obtained. By making use of these results, even for the nonunitary representation of SO(n,1) the matrix elements for the differential operators and the computation formula for the representation corresponding to the boost are obtained by defining the matrix elements with respect to the bases of the space. It is also pointed out that the unitary representations (the complementary series) corresponding to some value of the parameter, which appear in the classification using only the matrix elements of the generators, should not be included in our classification table because of divergence of the normalization integral. The continuation to SO(n+1) and the contraction to ISO(n) from the principal series are discussed.

Maximally Almost Periodic Groups and a Theorem of Glicksberga

1996-12-01

Ta Sun Wu , LEMUEL RIGGINS

ABSTRACT: It is well known that if G is a locally compact Abelian group (LCA group) with Bohr compactification (β( G ), σ) then σ( G ) is normal in … ABSTRACT: It is well known that if G is a locally compact Abelian group (LCA group) with Bohr compactification (β( G ), σ) then σ( G ) is normal in β( G ) and, by a beautiful theorem of Glicksberg, we have that A ⊂σ( G ) is compact if and only if σ –1 ( A ) ⊂ G is compact. The aim of this paper is to study maximally almost periodic (MAP) groups which have these properties and the results obtained are as follows. (1) If G is a σ‐compact locally compact MAP group with Bohr compactification (β( G ), σ) and σ( G ) is normal in β( G ), then for each g εβ( G ), the automorphism induced by σ and conjugation by g is actually a topological isomorphism. (2) A finite extension of a L CA group is a MAP group and it has the property that A ⊂σ( G ) is compact if and only if σ –1 ( A ) ⊂ G is compact, and (3) A discrete MAP group G with Bohr compactification (β( G ), σ) satisfying both of the properties being considered must be Abelian by finite, i.e., a finite extension of an Abelian group.

Abelian $n$-division fields of elliptic curves and Brauer groups of product Kummer & abelian surfaces

2016-06-29

Anthony Várilly‐Alvarado , Bianca Viray

Let $Y$ be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of $\mathbb{Q}$. In 2008, Skorobogatov and Zarhin showed that the … Let $Y$ be a principal homogeneous space of an abelian surface, or a K3 surface, over a finitely generated extension of $\mathbb{Q}$. In 2008, Skorobogatov and Zarhin showed that the Brauer group modulo algebraic classes $\text{Br}\, Y/ \text{Br}_1\, Y$ is finite. We study this quotient for the family of surfaces that are geometrically isomorphic to a product of isogenous non-CM elliptic curves, as well as the related family of geometrically Kummer surfaces; both families can be characterized by their geometric Neron-Severi lattices. Over a field of characteristic $0$, we prove that the existence of a strong uniform bound on the size of the odd-torsion of $\text{Br}\, Y / \text{Br}_1\, Y$ is equivalent to the existence of a strong uniform bound on integers $n$ for which there exist non-CM elliptic curves with abelian $n$-division fields. Using the same methods we show that, for a fixed prime $p$, a number field $k$ of fixed degree $r$, and a fixed discriminant of the geometric Neron-Severi lattice, $(\text{Br}\, Y / \text{Br}_1\, Y)[p^\infty]$ is bounded by a constant that depends only on $p$, $r$, and the discriminant.

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Pro-Lie groups which are infinite-dimensional Lie groups

2008-11-17

Karl Heinz Hofmann , Karl‐Hermann Neeb

Abstract A pro-Lie group is a projective limit of a family of finite-dimensional Lie groups. In this paper we show that a pro-Lie group G is a Lie group in … Abstract A pro-Lie group is a projective limit of a family of finite-dimensional Lie groups. In this paper we show that a pro-Lie group G is a Lie group in the sense that its topology is compatible with a smooth manifold structure for which the group operations are smooth if and only if G is locally contractible. We also characterize the corresponding pro-Lie algebras in various ways. Furthermore, we characterize those pro-Lie groups which are locally exponential, that is, they are Lie groups with a smooth exponential function which maps a zero neighbourhood in the Lie algebra diffeomorphically onto an open identity neighbourhood of the group.

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Locally compact semigroups in which a subgroup with compact complement is dense

1963-01-01

Karl H. Hofmann

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On the Existence of Slices for Actions of Non-Compact Lie Groups

1961-03-01

Richard S. Palais

If G is a topological group then by a G-space we mean a completely regular space X together with a fixed action of G on X. If one restricts consideration … If G is a topological group then by a G-space we mean a completely regular space X together with a fixed action of G on X. If one restricts consideration to compact Lie groups then a substantial general theory of G-spaces can be developed. However if G is allowed to be anything more general than a compact Lie group, theorems about G-spaces become extremely scarce, and it is clear that if one hopes to recover any sort of theory, some restriction must be made on the way G is allowed to act. A clue as to the sort of restriction that should be made is to be found in one of the most fundamental facts in the theory of G-spaces when G is a compact Lie group; namely the result, proved in special cases by Gleason 12], Koszul [5], Montgomery and Yang [6] and finally, in full generality, by Mostow [8] that there is a through each point of a G-space (see 2.1.1 for definition). In fact it is clear from even a passing acquaintance with the methodology of proof in transformation group theory that if G is a Lie group and X a G-space with compact isotropy groups for which there exists a slice at each point, then many of the statements that apply when G is compact are valid in this case also. In ? 1 of this paper we define a G-space X (G any locally compact group) -to be a Cartan G-space if for each point of X there is a neighborhood U such that the set of g in G for which g U n U is not empty has compact closure. In case G acts freely on X (i.e., the isotropy group at each point is the identity) this turns out to be equivalent to H. Cartan's basic axiom PF for principal bundles in the Seminaire H. Cartan of 1948-49, which explains the choice of name. In ? 2 we show that if G is a Lie group then the Cartan G-spaces are precisely those G-spaces with compact isotropy groups for which there is a slice through every point. As remarked above this allows one to extend a substantial portion of the theory of G-space that holds when G is a compact Lie group to Cartan G-spaces (or the slightly more restrictive class of proper G-spaces, also introduced in ? 1) when G is an arbitrary Lie group. Part of this extension is carried out in ? 4, more or less by way of showing what can be done. In particular we prove a generalization of Mostow's equivariant embed-

A theory of spherical functions. I

1952-01-01

Roger Godement

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Espaces Homogènes De Stein Des Groupes De Lie Complexes

1960-02-01

YOZÔ MATSUSHIMA

A. Soit G un groupe de Lie complexe et connexe et soit K un sous-groupe compact maximal de G . Soient g et les algèbres de Lie de G et … A. Soit G un groupe de Lie complexe et connexe et soit K un sous-groupe compact maximal de G . Soient g et les algèbres de Lie de G et K respectivement. est une sous-algèbre réelle de l’algèbre complexe g. Soit ( i 2 = –1). Alors est une sous-algèbre complexe de g.

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Classes of nonabelian, noncompact, locally compact groups

1978-12-01

Theodore W. Palmer

Introduction.A great deal is known about locally compact abelian groups and about compact groups.Frequently the same result has been proved in both cases.Thus it is natural to look for a … Introduction.A great deal is known about locally compact abelian groups and about compact groups.Frequently the same result has been proved in both cases.Thus it is natural to look for a common generalization of these two quite different hypotheses-abelian and compact.This article surveys the literature on this idea.In some respects it may be regarded as an extension and updating of the second part of the important paper by Grosser and Moskowitz [23].However, it is only meant to provide orientation in this subject, and thus in order to simplify the presentation we frequently do not quote results in their maximum generality.We will also omit most proofs.The extensive bibliography and detailed references to it, will allow the reader to find these when he wishes.At the same time we will try to explain enough to keep the formal prerequisites to a minimum.An acquaintance with the simplest facts about locally compact groups and convolution multiplication in their L 1 -group algebras and about operator algebras on Hilbert space is all that is needed.We will use some standard results from [31] § 5 without comment.In this article locally compact groups are always assumed to satisfy the Hausdorff separation axiom.The identity element of a group is usually denoted by e.We use Z, R, C, T to denote the sets of integers, real numbers, complex numbers, and complex numbers of modulus 1 respectively, with their usual structures as topological groups or rings, etc.The paper is organized as follows.§ 2 contains definitions together with sufficient comments to orient the reader.More complete comments and more detailed references for all these matters are contained in § 4. § 3 contains four diagrams summarizing the known inclusions among the twenty classes which we discuss fully.These diagrams give

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Amenability and paradoxical decompositions for pseudogroups and for discrete metric spaces

2016-01-01

Tullio Ceccherini‐Silberstein , Rostislav Grigorchuk , Pierre de la Harpe

This is an expostion of various aspects of amenability and paradoxical decompositions for groups, group actions and metric spaces. First, we review the formalism of pseudogroups, which is well adapted … This is an expostion of various aspects of amenability and paradoxical decompositions for groups, group actions and metric spaces. First, we review the formalism of pseudogroups, which is well adapted to stating the alternative of Tarski, according to which a pseudogroup without invariant mean gives rise to paradoxical decompositions, and to defining a Følner condition. Using a Hall-Rado Theorem on matchings in graphs, we show then for pseudogroups that existence of an invariant mean is equivalent to the Følner condition; in the case of the pseudogroup of bounded perturbations of the identity on a locally finite metric space, these conditions are moreover equivalent to the negation of the Gromov's so-called doubling condition, to isoperimetric conditions, to Kesten's spectral condition for related simple random walks, and to various other conditions. We define also the minimal Tarski number of paradoxical decompositions associated to a non-amenable group action (an integer $\ge 4$), and we indicate numerical estimates (Sections II.4 and IV.2). The final chapter explores for metric spaces the notion of supramenability, due for groups to Rosenblatt.

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Isometries, rigidity, and universal covers

2005-01-01

Benson Farb , Shmuel Weinberger

We classify all closed, aspherical Riemannian manifolds M whose universal cover has indiscrete isometry group. One sample application is the theorem that any such M with word-hyperbolic fundamental group must … We classify all closed, aspherical Riemannian manifolds M whose universal cover has indiscrete isometry group. One sample application is the theorem that any such M with word-hyperbolic fundamental group must be isometric to a negatively curved, locally symmetric manifold. Another application is the classification of all contractible Riemannian manifolds covering both compact and (noncompact, complete) finite volume manifold. There are also applications to the Hopf Conjecture, a new proof of Kazhdan's Conjecture (Frankel's Theorem) in complex geometry, etc. Ideas in the proof come from Lie theory, the homological theory of transformation groups, harmonic maps, and large-scale geometry. An extension to the non-aspherical case is also given.

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Lie-like decompositions of groups definable in o-minimal structures

2009-01-01

Annalisa Conversano

There are strong analogies between groups definable in o-minimal structures and real Lie groups. Nevertheless, unlike the real case, not every definable group has maximal definably compact subgroups. We study … There are strong analogies between groups definable in o-minimal structures and real Lie groups. Nevertheless, unlike the real case, not every definable group has maximal definably compact subgroups. We study definable groups G which are not definably compact showing that they have a unique maximal normal definable torsion-free subgroup N; the quotient G/N always has maximal definably compact subgroups, and for every such a K there is a maximal definable torsion-free subgroup H such that G/N can be decomposed as G/N = KH, and the intersection between K and H is trivial. Thus G is definably homotopy equivalent to K. When G is solvable then G/N is already definably compact. In any case (even when G has no maximal definably compact subgroup) we find a definable Lie-like decomposition of G where the role of maximal tori is played by maximal 0-subgroups.

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The skew-hyperbolic motion group of the quaternion plane

1997-09-01

Markus Stroppel

Semigroups that are the union of a group on 𝐸³ and a plane

1971-01-01

Frank Knowles

In <italic>Semigroups on a half-space</italic>, Trans. Amer. Math. Soc. <bold>147</bold> (1970), 1-53, Horne considers semigroups that are the union of a group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> … In <italic>Semigroups on a half-space</italic>, Trans. Amer. Math. Soc. <bold>147</bold> (1970), 1-53, Horne considers semigroups that are the union of a group <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> and a plane <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> such that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G union upper L"> <mml:semantics> <mml:mrow> <mml:mi>G</mml:mi> <mml:mo>∪</mml:mo> <mml:mi>L</mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">G \cup L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a three-dimensional half-space and <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the interior. After proving a great many things about half-space semigroups, Horne introduces the notion of a radical and determines all possible multiplications in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for a half-space semigroup with empty radical. (It turns out that <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> has empty radical if and only if each <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>-orbit in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> contains an idempotent.) An example is provided for each configuration in <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. However, no attempt was made to show that the list of examples actually exhausted the possibilities for a half-space semigroup without radical. Another way of putting this problem is to determine when two different semigroups can have the same maximal group. In this paper we generalize Horne’s results, for a semigroup without zero, by showing that if <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is any locally compact semigroup in which <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper L"> <mml:semantics> <mml:mi>L</mml:mi> <mml:annotation encoding="application/x-tex">L</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is the boundary of <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper G"> <mml:semantics> <mml:mi>G</mml:mi> <mml:annotation encoding="application/x-tex">G</mml:annotation> </mml:semantics> </mml:math> </inline-formula>, then <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper S"> <mml:semantics> <mml:mi>S</mml:mi> <mml:annotation encoding="application/x-tex">S</mml:annotation> </mml:semantics> </mml:math> </inline-formula> is a half-space. Moreover, we are able to answer completely, for semigroups without radical and without a zero, the question posed above. It turns out that, with one addition (which we provide), Horne’s list of half-space semigroups without radical and without zero is complete.

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References (7)

Über Automorphismen der Iokal-kompakten abelschen Gruppen

1940-01-01

Makoto Abe

Es soll in der vorliegenden Note die Isomorphie yon dem Automor- phismenring einer lokal-kompakten abelschen Gruppe mit dem ihrer Charaktergruppe gezeigt werden.Es sei G eine lokal-kompakte (additive) abelsche Gruppe, die … Es soll in der vorliegenden Note die Isomorphie yon dem Automor- phismenring einer lokal-kompakten abelschen Gruppe mit dem ihrer Charaktergruppe gezeigt werden.Es sei G eine lokal-kompakte (additive) abelsche Gruppe, die dem zweiten Abzihlbarkeitsaxiom geniigt G* sei die Charaktergrupe

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Lie groups simply isomorphic with no linear group

1936-01-01

Garrett Birkhoff

BIRKHOFF 1. Introduction.One of the most interesting conjectures concerning finite continuous groups is the conjecture that every Lie group is topologically isomorphic with a group of matrices.The proof of this … BIRKHOFF 1. Introduction.One of the most interesting conjectures concerning finite continuous groups is the conjecture that every Lie group is topologically isomorphic with a group of matrices.The proof of this conjecture, even in the small, would establish the truth of the famous conjecture that every Lie group nucleus (or germ) is a piece of a Lie group.J This makes it of interest to know that there exist Lie groups in the large, simply isomorphic even in the purely algebraic sense-and a fortiori topologically isomorphic in the large-with no group of matrices.It is to the proof of this fact that the present note is devoted.2. The Basic Lemma.The proof ultimately rests on the following lemma.LEMMA 1.Let Y be any group of linear transformations.Suppose T contains elements S and T whose commutator R = S" 1 T~1ST is of prime order p, and satisfies SR=RS, TR = RT.Then T is of degree at least p.

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Über nilpotente topologische Gruppen, I

1945-01-01

Kenkiti Iwasawa

Uber nilpotente topolgische Gruppen, I Uber nilpotente topolgische Gruppen, I

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Topological groups in which multiplication of one side is differentiable

1946-01-01

I. E. Segal

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On the Topological Structure of Solvable Groups

1941-07-01

Claude Chevalley

E. Cartan has shown that a simply connected solvable Lie group is homeomorphic to some Cartesian space.' We want to investigate the structure of a solvable Lie group which is … E. Cartan has shown that a simply connected solvable Lie group is homeomorphic to some Cartesian space.' We want to investigate the structure of a solvable Lie group which is not necessarily simply connected. From a well known theorem,2 it follows that such a group may be considered as the factor group of a solvable simply connected group G by a discrete subgroup D of the center of G. Therefore what we have to do is to look for the possible situation of such a sub-group D in the group G.

The Theory of Topological Commutative Groups

1934-04-01

L. Pontrjagin

The purpose of the present paper is to make an exhaustive investigation into the structure of continuous, locally compact, commutative groups, satisfying the second axiom of countability.2 1. The purpose of the present paper is to make an exhaustive investigation into the structure of continuous, locally compact, commutative groups, satisfying the second axiom of countability.2 1.

Sur les groupes de transformations analytiques

1935-01-01

Henri Cartan