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 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 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 On the K-theory of the loop space of a Lie group

1974 ◽  
Vol 76 (1) ◽  
pp. 1-20 ◽  
Author(s):  
Francis Clarke
Keyword(s):  
Lie Group ◽  
Loop Space ◽  
Simple Structure ◽  
Homology Theory ◽  
Compact Lie Group ◽  
Classifying Space ◽  
K Theory ◽  
Simply Connected ◽  
Torsion Free ◽  
Multiplicative Structures

Let G be a simply connected, semi-simple, compact Lie group, let K* denote Z/2-graded, representable K-theory, and K* the corresponding homology theory. The K-theory of G and of its classifying space BG are well known, (8),(1). In contrast with ordinary cohomology, K*(G) and K*(BG) are torsion-free and have simple multiplicative structures. If ΩG denotes the space of loops on G, it seems natural to conjecture that K*(ΩG) should have, in some sense, a more simple structure than H*(ΩG).

Eli’s Impact: A Case Study

2014 ◽  
Author(s):  
Charles Fefferman
Keyword(s):  
Fourier Series ◽  
Boltzmann Equation ◽  
Heat Kernel ◽  
Recent Work ◽  
Lie Group ◽  
Applied Mathematics ◽  
Compact Lie Group ◽  
Classical Study ◽  
The Right

This chapter illustrates the continuing powerful influence of Eli Stein's ideas. It starts by recalling his ideas on Littlewood–Paley theory, as well as several major developments in pure and applied mathematics, to which those ideas gave rise. Before Eli, Littlewood–Paley theory was one of the deepest parts of the classical study of Fourier series in one variable. Stein, however, found the right viewpoint to develop Littlewood–Paley theory and went on to develop Littlewood–Paley theory on any compact Lie group, and then in any setting in which there is a reasonable heat kernel. Afterward, the chapter discusses the remarkable recent work of Gressman and Strain on the Boltzmann equation, and explains in particular its connection to Stein's work.

Some Results in the Connective K-Theory of Lie Groups

1988 ◽  
Vol 31 (2) ◽  
pp. 194-199
Author(s):  
L. Magalhães
Keyword(s):  
Hopf Algebra ◽  
Lie Groups ◽  
Lie Group ◽  
Algebra Structure ◽  
Hopf Algebra Structure ◽  
K Theory ◽  
Torsion Free ◽  
Connected Lie Group

AbstractIn this paper we give a description of:(1) the Hopf algebra structure of k*(G; L) when G is a compact, connected Lie group and L is a ring of type Q(P) so that H*(G; L) is torsion free;(2) the algebra structure of k*(G2; L) for L = Z2 or Z.

A Bilinear Fractional Integral on Compact Lie Groups

2011 ◽  
Vol 54 (2) ◽  
pp. 207-216 ◽  
Author(s):  
Jiecheng Chen ◽  
Dashan Fan
Keyword(s):  
Lie Groups ◽  
Lie Group ◽  
Fractional Integral ◽  
Compact Lie Group ◽  
Compact Lie Groups ◽  
Boundedness Property ◽  
Recent Theorem

AbstractAs an analog of a well-known theoremon the bilinear fractional integral on ℝn by Kenig and Stein, we establish the similar boundedness property for a bilinear fractional integral on a compact Lie group. Our result is also a generalization of our recent theorem about the bilinear fractional integral on torus.

Exotic Holonomies ${\rm E}^{(a)}_7$

1997 ◽  
Vol 08 (05) ◽  
pp. 583-594 ◽  
Author(s):  
Quo-Shin Chi ◽  
Sergey Merkulov ◽  
Lorenz Schwachhöfer
Keyword(s):  
Symmetry Group ◽  
Lie Groups ◽  
Lie Group ◽  
Moduli Spaces ◽  
Local Symmetry ◽  
Positive Dimension ◽  
Affine Connections ◽  
Finite Dimensional ◽  
Torsion Free

It is proved that the Lie groups [Formula: see text] and [Formula: see text] represented in ℝ56 and the Lie group [Formula: see text] represented in ℝ112 occur as holonomies of torsion-free affine connections. It is also shown that the moduli spaces of torsion-free affine connections with these holonomies are finite dimensional, and that every such connection has a local symmetry group of positive dimension.

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 COMPACT LIE GROUP ACTIONS ON ASPHERICAL $A_k(\pi)$-MANIFOLDS

1993 ◽  
Vol 24 (4) ◽  
pp. 395-403
Author(s):  
DINGYI TANG
Keyword(s):  
Finite Group ◽  
Lie Group ◽  
Quotient Group ◽  
Group Actions ◽  
Compact Lie Group ◽  
Lie Group Actions ◽  
Torsion Free

Let M be an aspberical $A_k(\pi)$-manifold and $\pi'$-torsion-free, where $\pi'$ is some quotient group of $\pi$. We prove that (1) Suppose the Eu­ler characteristic $\mathcal{X}(M) \neq 0$ and $G$ is compact Lie group acting effectively on $M$, then $G$ is finite group (2) The semisimple degree of symmetry of $M$ $N_T^s \le (n - k)(n - k+1)/2$. We also unity many well-known results with simpler proofs.

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