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Topic: Topological abelian group

Related Topics

Abelian group

abelian categories

abelian varieties

free abelian group

finitely generated abelian groups

rank of an abelian group

abelian extensions

pre Abelian categories

category of abelian groups

Topological abelian group

arithmetic of abelian varieties

Abelian and tauberian theorems

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	Citebase - Duality for topological abelian group stacks and T-duality
		We extend Pontrjagin duality from topological abelian groups to certain locally compact group stacks.
		is the sheaf represented by a locally compact group G and T is the circle.
		-bundles in terms of Pontrjagin duality of abelian group stacks.
	www.citebase.org /abstract?id=oai:arXiv.org:math/0701428 (173 words)

	PlanetMath: topological group
		A subgroup of a topological group is itself a topological group, with the subspace topology.
		Consequently, the fundamental group of a topological group is abelian.
		A topological group that is Hausdorff and locally compact possesses a natural measure, called the Haar measure.
	planetmath.org /encyclopedia/TopologicalGroup2.html (347 words)

	Group theory
		Group theory is that branch of mathematics concerned with the study of groups.
		Groups are used throughout mathematics and the sciences, often to capture the internal symmetry of other structures, in the form of automorphism groups.
		In algebraic topology, groups are used to describe invariants of topological spaces (the name of the torsion subgroup of an infinite group shows the legacy of this field of endeavor).
	www.ebroadcast.com.au /lookup/encyclopedia/gr/Group_theory.html (633 words)

	Abelian group
		In abstract algebra, an abelian group is a group (G, ) that is commutative, i.e., in which a b = b * a holds for all elements a and b in G.
		If a group is abelian, we usually write the operation as + instead of *, the identity element as 0 (often called the zero element in this context) and the inverse of the element a as -a.
		The abelian groups, together with group homomorphisms, form a category, the prototype of an abelian category.
	www.ebroadcast.com.au /lookup/encyclopedia/ab/Abelian.html (430 words)

	PlanetMath: group
		Groups often arise as the symmetry groups of other mathematical objects; the study of such situations uses group actions.
		See Also: subgroup, cyclic group, simple group, symmetric group, free group, ring, field, group homomorphism, Lagrange's theorem, identity element, proper subgroup, groupoid, fundamental group, topological group (obsolete), Lie group, Proof: The orbit of any element of a group is a subgroup, locally cyclic group, existence of Hilbert class field, abelian group,
		This is version 17 of group, born on 2001-08-29, modified 2006-03-14.
	planetmath.org /encyclopedia/Group.html (312 words)

	Topological group - Wikipedia, the free encyclopedia
		In mathematics, a topological group is a group G together with a topology on G such that the group and topological structures are compatible.
		Topological groups allow one to study the notion of continuous symmetries in the form of continuous group actions.
		Topological groups, together with their homomorphisms, form a category.
	en.wikipedia.org /wiki/Topological_group (1059 words)

	Abelian group Summary
		Such groups are generally easier to understand, although large infinite abelian groups remain a subject of current research.
		The matrices, in contrast, do not form an abelian group under multiplication, for even when restricted to sets of invertible matrices, matrix multplication is generally non-communtative.
		This is a special application of the fundamental theorem of finitely generated abelian groups in the case when G has torsion-free rank equal to 0.
	www.bookrags.com /Abelian_group (2196 words)

	math space: group
		Group theory originated with the work of Évariste Galois in 1830, which concerned the problem of when an algebraic equation is soluble by radicals.
		Group theory allows for the properties of such structures to be investigated in a general setting.
		Groups are thus essential abstractions in branches of physics involving symmetry principles, such as relativity, quantum mechanics, and particle physics.
	deepakonline.spaces.live.com /Blog/cns!8821195381A28EA0!134.entry (2545 words)

	R&E 24 Abstracts
		We study the topological semigroups that admit the adjunction of a non-isolated absorbing element and the structure and permanence properties of the class AA of topological semigroups admitting this type of adjunctions.
		We discuss lattice theoretic properties of group topologies (maximal and minimal topologies, complementation etc.) as well as their relation to topologies generated by compact representations (homomorphisms in the circle group in the abelian case).
		The theory of Lie groups in finite dimensions relies on the strong connection between the two structures of the group: the finite dimenional manifold structure and the topological group structure.
	www.heldermann.de /R&E/rae24abs.htm (1668 words)

	List of Publications
		Non-completeness measure of convergence abelian groups, General topology and its relations to modern analysis and algebra, VI (Prague, 1986), 125--134, R \and E Res.
		Functorial topologies on abelian groups, Proceedings of SUNLAG 2000, a spring stage on Algebra, Logic and Geometry, March 2000, Seconda Univeritá di Napoli, 2002, pp.
		Topological groups and their generators, in: M. Curzio and F. De Giovanni Editors, Topics in Infinite Groups, Quaderni di Matematica, vol.
	www.dimi.uniud.it /~dikranja/publications.html (2269 words)

	Sets, Groups, Rings and Algebras
		A group is an algebraic system consisting of a set, an identity element, one operation and its inverse operation.
		Group Axioms: let a, b and c be elements of a group G1: Closure.
		An Ideal, I, is a subset of a Ring, R, with the properties: 1) I is a subgroup of the additive group of R and 2) for every i in I and every r in R, ir and ri are in I. Example: The set of all multiples of any integer is an Ideal.
	www.csee.umbc.edu /help/theory/group_def.shtml (1295 words)

	Springer Online Reference Works
		Topological groups for which the duality theorem is valid are called reflexive.
		Locally compact groups are not the only reflexive groups, since any reflexive Banach space, regarded as a topological group, is reflexive [8].
		There is an analogue of Pontryagin duality for non-commutative groups (the duality theorem of Tannaka–Krein) (see, [6], [7]).
	eom.springer.de /p/p073760.htm (599 words)

	HogBlog: Fundamental Groups of Topological Groups are Abelian
		Topologically these are great because they're compact smooth (even real-analytic) manifolds, and things like their fundamental groups are easy to extract from (for SO and SU) the fibrations we get by considering their actions on spheres.
		One compact topological group that isn't a Lie group is the p-adic integers, for any prime p: they actually form a compact topological ring.
		Another example of compact topological groups would be compact linear algebraic groups: consider the orthogonal group of nxn matrices over an arbitrary field k, even of prime characteristic, and give it the Zariski topology, making it an affine variety.
	www.koschei.net /blog/archives/000434.html (984 words)

	Atlas: Duality for Convergence Abelian Groups by M. Montserrat Bruguera
		Examples of reflexive groups which are not locally compact are known from the late forties.
		If G is a LCA group, the continuous convergence structure in \GammaG is precisely the convergence given by the compact open topology [3], thus, the ''convergence dual'' and the ordinary dual are identical.
		Topological abelian groups are, in an obvious way, convergence groups, therefore it is natural to compare reflexivity and BB-reflexivity for them.
	atlas-conferences.com /c/a/a/h/10.htm (848 words)

	22: Topological groups, Lie groups
		Lie groups are an important special branch of group theory.
		Thus Lie groups and other topological groups lie at the convergence of the different areas of pure mathematics.
		Topological groups are covered well, Lie groups per se hardly at all.
	www.math.niu.edu /~rusin/known-math/index/22-XX.html (348 words)

	ITFA - Topological interactions in gauge theories (Site not responding. Last check: 2007-10-15)
		Much of the relevant physics described by gauge theories is non-perturbative in nature; for example the spectrum, the study of confinement and the study of topological defects and their physical properties.
		These feature topological (magnetic flux) sectors which are characterized by certain non-abelian quantum numbers (conjugacy classes to be specific), whereas the allowed dyonic sectors carry electric charges falling into representations of the stabilizer group of the conjugacy class.
		One of the exotic properties of the theory is the emergence of the topological concept of "cheshire charge", a nonlocalizable manifestation of electric and/or magnetic charge.
	www.science.uva.nl /research/itf/topological.php (1240 words)

	[math/0701428] Duality for topological abelian group stacks and T-duality
		Title: Duality for topological abelian group stacks and T-duality
		In order to show that a connected abelian group is admissible on the site of locally compact spaces we must in addition assume that it is locally topologically divisible.
		This condition is used in the proof of Lemma 4.62.
	arxiv.org /abs/math.AT/0701428 (123 words)

	Сибирский Математический Журнал, Том 42 (2001), Номер 3, с. 550-560 (Site not responding. Last check: 2007-10-15)
		A topology τ on a group G is complemented if there exists an indiscrete topology τ' on G such that U∩V={0} for suitable neighborhoods of zero U and V in the topologies τ and τ.
		Locally compact groups with complemented topologies have been described.
		A group all of whose continuous homomorphic images are complete is proved to be compact.
	www.univie.ac.at /EMIS/journals/SMZ/2001/03/550.htm (147 words)

	[No title]
		Now we note that if X is either an abelian topological monoid or an infinite loopspace, then 1 X+ is in the category Alg.
		But, since N is the free abelian (topological) monoid on one generator, we can identify Map Ab (N, A) with A. Thus we are asking that SP 1 (K, N) satisfy Map Ab(SP 1 (K, N), A) = Map T(K, A).
		Note that the abelian topological monoid satisfying only the relations of type (i) and (ii) is SP1 (K ^ A).
	hopf.math.purdue.edu /Kuhn/kuhn-mc.txt (4631 words)

	Dept. Math/Stats: 2004 Calendar of Events
		The most important theorem in the theory of LCA groups is Pontryagin's Duality which states that any LCA group is isomorphic in the category of Hausdorff topological groups with its bidual when the dual groups are the groups of continuous group morphisms into the unitary circle with the compact-open topology.
		Relating an LCA group with its Bohr topology is a very important theorem known as Glicksberg's Theorem that states that the LCA topology and the Bohr topology associated to it determine the same compact subsets of the considered group.
		It is known that epimorphisms in the category of Hausdorff topological groups need not have dense image and the cowellpowerdness problem asks wether the class of extensions of a topological group for which the embeding is epimorphism is a set or a proper class.
	www.math.yorku.ca /CalendarOfEvents/2004calendar.html (16508 words)

	The concentration phenomenon and topological groups by Vladimir Pestov
		Many `infinite-dimensional' topological groups G of importance have the fixed point on compacta property, or are extremely amenable, that is, every continuous action of G on a compact space has a fixed point.
		20,21], the group of measurable maps from the interval to the circle group with the L
		Every extremely amenable abelian topological group G is minimally almost periodic, that is, admits no non-trivial continuous characters.
	at.yorku.ca /t/a/i/c/35.htm (1855 words)

	OctDec2004
		ABSTRACT: A group with topology (X,,T) is called a `semitopological group' if multiplication is separately continuous, a `paratopological group' if multiplication is jointly continuous and, of course, a `topological* group' if inversion is continuous as well.
		In two classic papers of 1957, Robert Ellis proved, for locally compact Hausdorff spaces, that every paratopological group is a topological group and then, a few months later, that every semitopological group is a topological group.
		A primary problem in the area is that of developing algorithmic tests for topological or topological embedding equivalence.
	home.att.net /~topann/OctDec2004.html (924 words)

	Prepints
		topology of a topological group (G, r) is the topology that the group inherites spects a
		Constructive groups were introduced by Sternfeld in (5) as metrizable groups G whose group of G-valued functions satisfies a (suitably defined) sort of Stone-Weirestrass theorem.
		In the case of connected locally compact groups the homotopy theory of compact groups (essentially applied to simple compact Lie groups and compact connected Abelian groups) and the structure theory of locally compact groups turn to be appropriate tools.
	www.deptmat.uji.es /www/prepints.html (4923 words)

	Michael G. Tka\v{c}enko (Site not responding. Last check: 2007-10-15)
		It is well known that every compact connected Abelian group of weight less than or equal to the continuum is monothetic, i.
		For a topological group G, let b(G) be the least cardinal \kappa such that for every neighborhood of the identity in G there exists a subset C \subseteq G with C Theorem 2.
		A separable totally bounded topological group of countable pseudocharacter has a suitable set.
	www.utm.edu /staff/jschomme/topology/c/a/a/l/19.htm (329 words)

	[No title] (Site not responding. Last check: 2007-10-15)
		2.Continuous group cohomology is the same as Borel group cohomology if the group is locally compact and the coefficients are vector spaces.
		3.The algebraic group cohomology you mentioned is the group cohomology of the group made discrete (forgetting the topology) as I understand and thus is the cohomology of the classifying space of the group made discrete.
		4.Topological group extension of locally compact groups is Borel split, thus short exact sequence of coefficients which are locally compact leads to a Bockstein type long exact sequence of cohomology.
	www.lehigh.edu /dmd1/public/www-data/pza.txt (120 words)

	Whaley Group Research
		One of the biggest obstacles to quantum information processing is the rapid decay of quantum coherence in most physical systems due to noise and other uncontrolled interactions with the environment.
		We are especially interested in the interplay between topological constraints imposed by an external potential, interparticle interactions (often in the strongly interacting limit) and coherence.
		Recent work includes a detailed analysis of the effects of strong interactions on number statistics in a double well condensate and a study of the properties of weakly linked superfluids in both simple and complex (superposition states) of flow in a ring geometry.
	www.cchem.berkeley.edu /kbwgrp/research.html (659 words)

	AMCA: Pontryagin duality for topological Abelian groups by Salvador Hernandez (Site not responding. Last check: 2007-10-15)
		AMCA: Pontryagin duality for topological Abelian groups by Salvador Hernandez
		A topological abelian group G satisfies Pontryagin duality, or is Pontryagin reflexive for short, if the natural homomorphism of G to its bidual group is a topological isomorphism.
		The aim of this talk is to report on some recent results related to Pontryagin duality theory of topological Abelian groups.
	at.yorku.ca /c/a/f/q/07.htm (156 words)

	FELIX KLEIN'S GEOMETRIC ABSTRACTION BY GROUPS (Site not responding. Last check: 2007-10-15)
		There is an abstract group g acting on S which has a representation as a group of linear substitutions on the coordintes.
		Thus, generally, a geometry is defined by a space (set) S, a set R of relations, and the group g of transformations of S that leave R invariant, since the set R is implied by the action of g on S, Thus, most generally the coordinates can be conceptually eliminated.
		Example: S is a Euclidean space and g is IO(n) which is the semidirect product of the full group of rotations O(n) and an abelian group, (a group of translations) isomorphic to R^n.
	graham.main.nc.us /~bhammel/MATH/geomdef.html (546 words)

	Citebase - Abelian topological groups with host algebras
		The concept of a host algebra generalises that of a group C-algebra to groups* which are not locally compact in the sense that its non-degenerate representations are in one-to-one correspondence with representations of the group under consideration.
		Our main negative result is essentially that a topological abelian group has a full host algebra (covering all its continuous unitary representations) if and only if it embeds densely into a locally compact group.
		On the positive side, we show that the canonical symplectic form on a countably dimensional complex vector space leads to an abelian group with multiplier for which a full host algebra exists.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0605413 (209 words)

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