Last edited by Mozilkree
Monday, May 4, 2020 | History

3 edition of Classifications of Abelian Groups and Pontrjagin Duality (Algebra, Logic & Applications) found in the catalog.

# Classifications of Abelian Groups and Pontrjagin Duality (Algebra, Logic & Applications)

## by Peter Loth

Written in English

Subjects:
• Groups & group theory,
• Mathematics,
• Science/Mathematics,
• Geometry - General,
• Mathematics / Geometry / General,
• Algebra - Intermediate

• The Physical Object
FormatHardcover
Number of Pages180
ID Numbers
Open LibraryOL9106607M
ISBN 109056991698
ISBN 109789056991692

Basic characteristics of Abelian groups Simply presented groups Warfield groups Infinitary logic looks at groups Groups with partial decomposition bases Characters and Pontrjagin duality of locally compact abelian groups Classifications of compact abelian groups. Series Title: De Gruyter studies in mathematics, pontryagin topological groups pdf Hausdorff Abelian groups, Pontryagin duality and the notes provide a brief introduction to topological groups with a pdws pdf special. Charactres and Pontryagin-van Kampen duality to number theory, physics and. Create a book Download as PDF Printable mathematics, specifically in.

Dualities in Mathematics: Locally compact abelian groups Part III: Pontryagin Duality Prakash Panangaden1 1School of Computer Science McGill University Spring School, Oxford 20 - 22 May Pontryaginâ€“van Kampen duality theorem says that this functor is an involution i.e., Ì‚ G âˆ¼ = G for every G âˆˆ er, is functor sends compact groups to discrete ones and vice versa, i.e., it defines a duality between the subcategory C of mpact abelian groups and the subcategory D of discrete abelian by: 2.

Pontryagin duality (again) Showing of 47 messages. Pontryagin duality (again) Timothy Murphy: 9/30/08 AM: What is the simplest proof that G** = G for any locally compact abelian group, ie how does one show that any continuous homomorphism G* -> T arises from some g in G as chi -> chi(g)? I've looked at a couple of books, and they both. Lev Semyonovich Pontryagin (Russian: Лев Семёнович Понтрягин, also written Pontriagin or Pontrjagin) (3 September – 3 May ) was a Soviet was born in Moscow and lost his eyesight due to a primus stove explosion when he was Despite his blindness he was able to become one of the greatest mathematicians of the 20th century, partially with the Born: 3 September , Moscow, Russian Empire.

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### Classifications of Abelian Groups and Pontrjagin Duality (Algebra, Logic & Applications) by Peter Loth Download PDF EPUB FB2

In the second part of this text, the author studies certain classes of compact abelian groups using Pontrjagin duality. After providing the necessary tools for dualization, the structure of the compact groups dual to the totally projective p-groups, balanced projective groups, and Warfield groups Cited by: 7.

In the second part of this text, the author studies certain classes of compact abelian groups using Pontrjagin duality. After providing the necessary tools for dualization, the structure of the compact groups dual to the totally projective p-groups, balanced projective groups, and Warfield groups is established.

This text introduces the theory of abelian groups focusing on the classification problem. The structure of totally projective p-groups is determined and Hill's version of Ulm's Theorem is proved. The author also studies certain classes of compact abelian groups using Pontrjagin duality.

Pontryagin duality in the class of precompact Abelian groups and the Baire property. A topological Abelian group G is Pontryagin reflexive, or P-reflexive for short, if the natural homomorphism of G to its bidual group is a topological isomorphism. The Pontryagin duality theorem for locally compact abelian groups (briefly, LCA groups) has been the starting point for many different routes of research in Mathematics.

A topological group, G, is a topological space which is also a group. is a group, in fact another locally compact abelian group. Pontryagin duality states that for. On the construction and topological invariance of the Pontryagin. groups of such co. invariance of the Pontryagin classes from a topological transversality.

Pontryagin duality goes like this. Suppose A A is a locally compact Hausdorff topological abelian group. Let A * A^* be the set of characters: that is, continuous homomorphisms f: A → U (1) f: A \to \mathrm{U}(1).

A * A^* becomes an abelian group thanks to pointwise multiplication of characters. It becomes a topological group with the compact-open topology — that is, the topology of. Deﬁnition We call a sheaf of abelian groups F∈ShAbS dualizable, if the eval-uation map evF: F→D(D(F)) is an isomorphism of sheaves.

The sheaf theoretic reformulation of Pontrjagin duality is now: Theorem (Sheaf theoretic version of Pontrjagin duality) If G is a lo-cally compact group. We present a wide class of reflexive, precompact, non-compact, Abelian topological groups G determined by three requirements.

They must have the Baire property, satisfy the open refinement condition, and contain no infinite compact combination of properties guarantees that all compact subsets of the dual group G ∧ are finite.

We also show that many (non-reflexive) precompact Cited by: It is discussed in Pontryagin's book on topological groups (see Example 72) and the first volume on abstract harmonic analysis by Hewitt and Ross (p.

we study the class $\mathcal D_{\Pi}$ consisting of all Hausdorff Abelian groups topologically isomorphic to a product of a compact group with a countable product of copies of $\mathbb R. Title: Pontryagin duality in the class of precompact Abelian groups and the Baire property Authors: Montserrat Bruguera, Mikhail Tkachenko (Submitted on 24 Jan )Cited by: 1. These lecture notes begin with an introduction to topological groups and proceed to a proof of the important Pontryagin-van Kampen duality theorem and a detailed exposition of the structure of locally compact abelian groups. Measure theory and Banach algebra are entirely avoided and only a small amount of group theory and topology is by: Abstract. In this chapter we are mainly interested in the study of abelian locally compact groups A, their dual groups $$\hat{A}$$ together with various associated group algebras. Using the Gelfand-Naimark Theorem as a tool, we shall then give a proof of the Plancherel Theorem, which asserts that the Fourier transform extends to a unitary equivalence of the Hilbert spaces $$L^2(A)$$ and \(L^2 Author: Anton Deitmar, Siegfried Echterhoff. I need to show that the category of torsion abelian groups is dual to the category of profinite abelian. (I'm trying to solve as many exercises as I can about Galois Cohomology in this book, not always with success). Pontryagin duality for torsion abelian groups. Isomorphism between Topological Groups. Abelian Groups: Structures and Classifications. The research monograph Abelian Groups: Structures and Classifications, coauthored with Dr. Carol Jacoby, covers in a comprehensive manner the current state of classification theory with respect to infinite abelian groups. A wide variety of ways to characterize different classes of abelian groups by invariants, isomorphisms and duality principles. Classification of compact connected abelian groups. Ask Question The classification of torsion-free abelian groups is not easy but there's a large literature on it.$\endgroup$– YCor Nov 13 '18 at 2$\begingroup$Yes any compact connected Lie group is a torus. By Pontryagin duality this is a restatement of the fact that any. One important application of Pontryagin duality is the following characterization of compact abelian topological groups: Theorem. A locally compact abelian group G is compact if and only if the dual group ^ is discrete. Conversely, G is discrete if and only if ^ is compact. This monograph covers in a comprehensive manner the current state of classification theory with respect to infinite abelian groups. A wide variety of ways to characterise different classes of abelian groups by invariants, isomorphisms and duality principles are discussed. These lecture notes begin with an introduction to topological groups and proceed to a proof of the important Pontryagin-van Kampen duality theorem and a detailed exposition of the structure of locally compact abelian groups. Measure theory and Banach algebra are entirely avoided and only a small amount of group theory and topology is required, dealing with the subject in an elementary fashion. The following is some learned commentary on Theorem 25 in here: Sidney A. Morris, Pontryagin Duality and the Structure of Locally Compact Abelian Groups, London Math. Soc. Lecture No Cambridge U. Press, The theorem is supposed to shed light on the structure of abelian topological groups that are locally compact and Hausdorff — the groups that allow for a massive generalization.The Fourier transform on locally compact abelian groups is formulated in terms of Pontrjagin duals (see below). Also see: Michael Barr, On duality of topological abelian groups. which provides a perhaps better context for Pontryagin duality than the category of locally compact Hausdorff abelian groups (also known as ‘LCA groups’).We present a wide class of reflexive, precompact, non-compact, Abelian topological groups$G\$ determined by three requirements.

They must have the Baire.