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.

On nth roots and infinitely divisible elements in a connected Lie group

1981 ◽  
Vol 89 (2) ◽  
pp. 293-299 ◽  
Author(s):  
M. McCrudden
Keyword(s):  
Topological Group ◽  
Finite Number ◽  
Lie Groups ◽  
Lie Group ◽  
Connected Components ◽  
Compact Lie Group ◽  
Infinitely Divisible ◽  
Closed Set ◽  
Image Position ◽  
Connected Lie Group

For any group G, x ∈ G and n ∈ ℕ (the natural numbers), leti.e. the set of all nth roots of x in G. If G is a Hausdorff topological group, then Rn(x, G) is a closed set in G, but may otherwise be quite complicated. However, as we have observed in (4), if G is a compact Lie group, then Rn(x, G) always has a finite number of connected components, and this result has led us to wonder about the connectedness properties of Rn(x, G) for other Lie groups G. Here is the result.

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.

Construction of Levi processes on path spaces of Lie groups

2016 ◽  
Vol 19 (01) ◽  
pp. 1650002
Author(s):  
A. A. Kalinichenko
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Weak Limit ◽  
Convolution Semigroup ◽  
Path Space ◽  
Probability Measures ◽  
Compact Lie Group ◽  
Piecewise Constant ◽  
Finite Dimensional

Given a compact Lie group and a conjugate-invariant Levi process on it, generated by the operator [Formula: see text], we construct the Levi process on the path space of [Formula: see text], associated with the convolution semigroup [Formula: see text] of probability measures, where [Formula: see text] is the distribution of the Levi process on [Formula: see text] generated by [Formula: see text]. The constructed process is obtained as the weak limit of piecewise constant paths, which, as well as proving its existence and properties, provides finite-dimensional approximations of Chernoff type to the integrals with respect to its distribution.

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.

Invariance groups and convergence of types of measures on Lie groups

1992 ◽  
Vol 112 (1) ◽  
pp. 91-108 ◽  
Author(s):  
S. G. Dani
Keyword(s):  
Probability Measure ◽  
Lie Groups ◽  
Lie Group ◽  
Weak Topology ◽  
Probability Measures ◽  
Affine Subspace ◽  
Invariance Groups ◽  
Connected Lie Group

Let G be a connected Lie group and let {λi} be a sequence of probability measures on G converging (in the usual weak topology) to a probability measure λ. Suppose that {αi} is a sequence of affine automorphisms of G such that the sequence {αi,(λi)} also converges, say to a probability measure μ. What does this imply about the sequence {αi}? It is a classical observation that if G = ℝn for some n, and neither of λ and μ is supported on a proper affine subspace of ℝn, then under the above condition, {αi} is relatively compact in the group of all affine automorphisms of ℝn.

Condition that All Irreducible Representations of a Compact Lie Group, if Restricted to a Subgroup, Contain no Representation More than Once

1972 ◽  
Vol 24 (3) ◽  
pp. 432-438 ◽  
Author(s):  
Fredric E. Goldrich ◽  
Eugene P. Wigner
Keyword(s):  
Irreducible Representation ◽  
Lie Groups ◽  
Finite Groups ◽  
Lie Group ◽  
Unitary Group ◽  
Unit Vector ◽  
Doctoral Dissertation ◽  
Irreducible Representations ◽  
Compact Lie Group ◽  
Compact Lie Groups

One of the results of the theory of the irreducible representations of the unitary group in n dimensions Un is that these representations, if restricted to the subgroup Un-1 leaving a vector (let us say the unit vector e1 along the first coordinate axis) invariant, do not contain any irreducible representation of this Un-1 more than once (see [1, Chapter X and Equation (10.21)]; the irreducible representations of the unitary group were first determined by I. Schur in his doctoral dissertation (Berlin, 1901)). Some time ago, a criterion for this situation was derived for finite groups [3] and the purpose of the present article is to prove the aforementioned result for compact Lie groups, and to apply it to the theory of the representations of Un.

scholarly journals Characterizing Lie groups by controlling their zero-dimensional subgroups

2018 ◽  
Vol 30 (2) ◽  
pp. 295-320
Author(s):  
Dikran Dikranjan ◽  
Dmitri Shakhmatov
Keyword(s):  
Topological Group ◽  
Lie Groups ◽  
Lie Group ◽  
Compact Lie Group ◽  
Locally Compact ◽  
Abelian Topological Group ◽  
Local Compactness ◽  
Sequential Completeness ◽  
Local Minimality ◽  
Countable Compactness

AbstractWe provide characterizations of Lie groups as compact-like groups in which all closed zero-dimensional metric (compact) subgroups are discrete. The “compact-like” properties we consider include (local) compactness, (local) ω-boundedness, (local) countable compactness, (local) precompactness, (local) minimality and sequential completeness. Below is A sample of our characterizations is as follows:(i) A topological group is a Lie group if and only if it is locally compact and has no infinite compact metric zero-dimensional subgroups.(ii) An abelian topological groupGis a Lie group if and only ifGis locally minimal, locally precompact and all closed metric zero-dimensional subgroups ofGare discrete.(iii) An abelian topological group is a compact Lie group if and only if it is minimal and has no infinite closed metric zero-dimensional subgroups.(iv) An infinite topological group is a compact Lie group if and only if it is sequentially complete, precompact, locally minimal, contains a non-empty open connected subset and all its compact metric zero-dimensional subgroups are finite.

Battlefields from Event to Heritage

2020 ◽  
Author(s):  
John Carman ◽  
Patricia Carman
Keyword(s):  
Twentieth Century ◽  
Security Studies ◽  
New Developments ◽  
Critical Security Studies ◽  
The Past ◽  
New Approaches ◽  
Large Numbers ◽  
The World ◽  
War And Conflict ◽  
Modern War

What is—or makes a place—a ‘historic battlefield’? From one perspective the answer is a simple one—it is a place where large numbers of people came together in an organized manner to fight one another at some point in the past. But from another perspective it is far more difficult to identify. Quite why any such location is a place of battle—rather than any other kind of event—and why it is especially historic is more difficult to identify. This book sets out an answer to the question of what a historic battlefield is in the modern imagination, drawing upon examples from prehistory to the twentieth century. Considering battlefields through a series of different lenses, treating battles as events in the past and battlefields as places in the present, the book exposes the complexity of the concept of historic battlefield and how it forms part of a Western understanding of the world. Taking its lead from new developments in battlefield study—especially archaeological approaches—the book establishes a link to and a means by which these new approaches can contribute to more radical thinking about war and conflict, especially to Critical Military and Critical Security Studies. The book goes beyond the study of battles as separate and unique events to consider what they mean to us and why we need them to have particular characteristics. It will be of interest to archaeologists, historians, and students of modern war in all its forms.

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.

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