Joachim Cuntz, Georges Skandalis, Boris Tsygan

Cyclic Homology in Non-Commutative Geometry

• Broschiertes Buch

Produktbeschreibung

Cyclic homology was introduced in the early eighties independently by Connes and Tsygan. They came from different directions. Connes wanted to associate homological invariants to K-homology classes and to describe the index pair ing with K-theory in that way, while Tsygan was motivated by algebraic K-theory and Lie algebra cohomology. At the same time Karoubi had done work on characteristic classes that led him to study related structures, without however arriving at cyclic homology properly speaking. Many of the principal properties of cyclic homology were already developed in the fundamental article of Connes and in the long paper by Feigin-Tsygan. In the sequel, cyclic homology was recognized quickly by many specialists as a new intriguing structure in homological algebra, with unusual features. In a first phase it was tried to treat this structure as well as possible within the traditional framework of homological algebra. The cyclic homology groups were computed in many examples and new important properties such as prod uct structures, excision for H-unital ideals, or connections with cyclic objects and simplicial topology, were established. An excellent account of the state of the theory after that phase is given in the book of Loday.

Produktdetails

• Encyclopaedia of Mathematical Sciences 121
• Verlag: Springer / Springer Berlin Heidelberg / Springer, Berlin
• Artikelnr. des Verlages: 978-3-642-07337-3
• Softcover reprint of hardcover 1st ed. 2004
• Seitenzahl: 156
• Erscheinungstermin: 23. Januar 2011
• Englisch
• Abmessung: 235mm x 155mm x 9mm
• Gewicht: 248g
• ISBN-13: 9783642073373
• ISBN-10: 3642073379
• Artikelnr.: 32056603

Inhaltsangabe

I. Cyclic Theory, Bivariant K-Theory and the Bivariant Chern-Connes Character by J. Cuntz: 1. Cyclic Theory; 2. Cyclic Theory for Locally Convex Algebras; 3. Bivariant K-Theory; 4. Infinite-Dimensional Cyclic Theories; A. Locally Convex Algebras; B. Standard Extensions.- II. Noncommutative Geometry, the Transverse Signature Operator, and Hopf Algebras (after A. Connes and H. Moscovici) by G. Skandalis: 1. Preliminaries; 2. The Local Index Formula; 3. The Diff-Invariant Signature Operator; 4. The 'Transverse' Hopf Algebra.- III. Cyclic Homology by B. Tsygan: 1. Introduction; 2. Hochschild and Cyclic Homology of Algebras; 3. The Cyclic Complex C^{lambda}_{bullet}; 4. Non-Commutative Differential Calculus; 5. Cyclic Objects; 6. Examples; 7. Index Theorems; 8. Riemann-Roch Theorem for D-Modules.

Rezensionen

From the reviews:

"This volume of the 'Encyclopedia of Mathematical Sciences' is a very important and useful contribution to the literature on cyclic homology and noncommutative geometry. ... This book contains three expository articles, covering very important recent results." (Alexander Gorokhovsky, Mathematical Reviews, 2005 k)

Aus den Rezensionen: "... Das vorliegende Buch gibt eine Zusammenfassung der wichtigsten, klassischen und insbesondere auch aktuellen Resultate und Techniken der zyklischen Theorie. ... Alle drei Artikel enthalten auf knappem Raum eine große Fülle von Material. ... Zu den wichtigeren Resultaten wird die grundlegende Beweisidee angegeben. Für detaillierte Beweise wird in aller Regel auf die Originalliteratur verwiesen. Auf diese Weise wird das Buch zu einem sehr nützlichen Nachschlagewerk ... Auch als Einstieg, um einen Überblick zu erhalten, was zyklische Theorie bedeutet, ist es hervorragend geeignet ..." (T. Schick, in: Jahresbericht der Deutschen Mathematiker-Vereinigung, 2007, Vol. 109, Issue 2, S. 19 f.)