The Classical Groups and K-Theory

Author: Alexander J. Hahn,O.Timothy O'Meara

Publisher: Springer Science & Business Media

ISBN: 3662131528

Category: Mathematics

Page: 578

View: 8177

It is a great satisfaction for a mathematician to witness the growth and expansion of a theory in which he has taken some part during its early years. When H. Weyl coined the words "classical groups", foremost in his mind were their connections with invariant theory, which his famous book helped to revive. Although his approach in that book was deliberately algebraic, his interest in these groups directly derived from his pioneering study of the special case in which the scalars are real or complex numbers, where for the first time he injected Topology into Lie theory. But ever since the definition of Lie groups, the analogy between simple classical groups over finite fields and simple classical groups over IR or C had been observed, even if the concept of "simplicity" was not quite the same in both cases. With the discovery of the exceptional simple complex Lie algebras by Killing and E. Cartan, it was natural to look for corresponding groups over finite fields, and already around 1900 this was done by Dickson for the exceptional Lie algebras G and E • However, a deep reason for this 2 6 parallelism was missing, and it is only Chevalley who, in 1955 and 1961, discovered that to each complex simple Lie algebra corresponds, by a uniform process, a group scheme (fj over the ring Z of integers, from which, for any field K, could be derived a group (fj(K).

Algebra

Author: Jens Carsten Jantzen,Joachim Schwermer

Publisher: Springer-Verlag

ISBN: 3642405339

Category: Mathematics

Page: 458

View: 6410

Ausgehend von einer grundlegenden Einführung in Begriffe und Methoden der Algebra werden im Buch die wesentlichen Ergebnisse dargestellt und ein Einblick in viele Entwicklungen innerhalb der Algebra gegeben, die mit anderen Gebieten der Mathematik stark verflochten sind. Beginnend mit Begriffsbildungen wie Gruppe und Ring führt das Buch hin zu den Körpererweiterungen und der Galoistheorie. Danach werden zentrale Teile der Theorie der Moduln, Algebren und Ringe behandelt. Die Theorie der Divisionsalgebren und ihre Klassifikation mit Hilfe der Brauergruppe werden entwickelt. Es schließen sich Einführungen in die algebraischen Zahlentheorie und die Theorie der quadratischen Formen an. In zahlreichen Supplementen findet man Ausblicke auf weiterführende Themen. Betrachtet werden zum Beispiel allgemeine lineare Gruppen, Schiefpolynomringe, Darstellungen, Erweiterungen von Moduln, projektive Moduln und Frobenius-Algebren.

Quadratic and Hermitian Forms over Rings

Author: Max-Albert Knus

Publisher: Springer Science & Business Media

ISBN: 3642754015

Category: Mathematics

Page: 524

View: 8959

From its birth (in Babylon?) till 1936 the theory of quadratic forms dealt almost exclusively with forms over the real field, the complex field or the ring of integers. Only as late as 1937 were the foundations of a theory over an arbitrary field laid. This was in a famous paper by Ernst Witt. Still too early, apparently, because it took another 25 years for the ideas of Witt to be pursued, notably by Albrecht Pfister, and expanded into a full branch of algebra. Around 1960 the development of algebraic topology and algebraic K-theory led to the study of quadratic forms over commutative rings and hermitian forms over rings with involutions. Not surprisingly, in this more general setting, algebraic K-theory plays the role that linear algebra plays in the case of fields. This book exposes the theory of quadratic and hermitian forms over rings in a very general setting. It avoids, as far as possible, any restriction on the characteristic and takes full advantage of the functorial aspects of the theory. The advantage of doing so is not only aesthetical: on the one hand, some classical proofs gain in simplicity and transparency, the most notable examples being the results on low-dimensional spinor groups; on the other hand new results are obtained, which went unnoticed even for fields, as in the case of involutions on 16-dimensional central simple algebras. The first chapter gives an introduction to the basic definitions and properties of hermitian forms which are used throughout the book.

An Algebraic Introduction to K-Theory

Author: Bruce A. Magurn

Publisher: Cambridge University Press

ISBN: 9780521800785

Category: Mathematics

Page: 676

View: 3693

An introduction to algebraic K-theory with no prerequisite beyond a first semester of algebra.

Biographisch-literarisches Handworterbuch der exakten Naturwissenschaften, Johann Christian Poggendorff

Datenbank / Database

Author: Johann Ch. Poggendorff

Publisher: Wiley-VCH

ISBN: 9783527403066

Category: Science

Page: 216

View: 3543

"Somit wäre es denn zum einstweiligen Abschlusse gebracht das Werk, dem ich seit etwa fünfzehn Jahre alle meine Musse gewidmet habe. Ob es dieses Opfers werth gewesen? Darüber können die Ansichten verschieden sein; aber zu dem Glauben halte ich mich doch berechtigt, dass die Nützlichkeit, ja Nothwendikeit desselben nicht bestritten, vielmehr mit der Zeit immer mehr anerkannt werden wird." Mit diesen Worten am Beginn von "Vor- und Schlusswort des Verfassers" übergibt 1863 Johann Christian Poggendorff den ersten Band des Werkes, aus dem später "der Poggendorff" werden sollte, an die Leser. Der Name des Verfassers, Physiker und außerordentlicher Professor an der Universität Berlin, ist noch heute in der Mechanik als Schöpfer der Poggendorffschen Waage bekannt. Sein "Biographisch-literarisches Handwörterbuch der exakten Naturwissenschaften" ist ein traditionsreiches Nachschlagewerk, das seit 1863 "Nachweisungen über Lebensverhältnisse und Leistungen von Mathematikern, Astronomen, Physikern, Chemikern, Mineralogen, Geologen" und Wissenschaftlern weiterer Disziplinen bringt. Die etwa 29.000 Biobibliographien enthalten jeweils die Lebensdaten, einen kurzen wissenschaftlichen Lebenslauf, Ehrungen und Nekrologe sowie Sekundärliteratur und vollständige Schriftenverzeichnisse aller Publikationen (Bücher, Beiträge in Sammelwerken und Kongressberichten sowie Zeitschriftenaufsätze). Um die ungeheure Fülle von Angaben für eine zeitgemäße Nutzung zur Verfügung zu stellen, wurde eine Datenbank erstellt: Sämtliche Bände wurden elektronisch aufbereitet, biographische Angaben wurden verschlagwortet. Die Recherche nach Personen, Geburts- und Sterbeorten, Jahreszahlen, Fachgebieten u.v.m. wird durch komfortable Suchfunktionen erleichtert. Zu jedem Eintrag lassen sich die Originalseiten der ursprünglichen Ausgabe abrufen.

Handbook of Incidence Geometry

Buildings and Foundations

Author: Francis Buekenhout

Publisher: North-Holland

ISBN: 9780444883551

Category: Mathematics

Page: 1420

View: 9328

This Handbook deals with the foundations of incidence geometry, in relationship with division rings, rings, algebras, lattices, groups, topology, graphs, logic and its autonomous development from various viewpoints. Projective and affine geometry are covered in various ways. Major classes of rank 2 geometries such as generalized polygons and partial geometries are surveyed extensively. More than half of the book is devoted to buildings at various levels of generality, including a detailed and original introduction to the subject, a broad study of characterizations in terms of points and lines, applications to algebraic groups, extensions to topological geometry, a survey of results on diagram geometries and nearby generalizations such as matroids.

Memorandum

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Electrical engineering

Page: N.A

View: 6307

Mathematical Reviews

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Mathematics

Page: N.A

View: 5058

The Bloch-Kato Conjecture for the Riemann Zeta Function

Author: John Coates,A. Raghuram,Anupam Saikia,R. Sujatha

Publisher: Cambridge University Press

ISBN: 1107492963

Category: Literary Criticism

Page: 320

View: 4401

A graduate-level account of an important recent result concerning the Riemann zeta function.

Stratified Lie Groups and Potential Theory for Their Sub-Laplacians

Author: Andrea Bonfiglioli,Ermanno Lanconelli,Francesco Uguzzoni

Publisher: Springer Science & Business Media

ISBN: 3540718974

Category: Mathematics

Page: 802

View: 4415

This book provides an extensive treatment of Potential Theory for sub-Laplacians on stratified Lie groups. It also provides a largely self-contained presentation of stratified Lie groups, and of their Lie algebra of left-invariant vector fields. The presentation is accessible to graduate students and requires no specialized knowledge in algebra or differential geometry.

Reviews in K-theory, 1940-84

As Printed in Mathematical Reviews

Author: Bruce A. Magurn

Publisher: Amer Mathematical Society

ISBN: N.A

Category: Mathematics

Page: 811

View: 9929

Encyclopedia of mathematical physics

Author: Sheung Tsun Tsou

Publisher: Academic Pr

ISBN: 9780125126601

Category: Science

Page: 3500

View: 4672

The Encyclopedia of Mathematical Physics provides a complete resource for researchers, students and lecturers with an interest in mathematical physics. It enables readers to access basic information on topics peripheral to their own areas, to provide a repository of the core information in the area that can be used to refresh the researcher's own memory banks, and aid teachers in directing students to entries relevant to their course-work. The Encyclopedia does contain information that has been distilled, organised and presented as a complete reference tool to the user and a landmark to the body of knowledge that has accumulated in this domain. It also is a stimulus for new researchers working in mathematical physics or in areas using the methods originated from work in mathematical physics by providing them with focused high quality background information. * First comprehensive interdisciplinary coverage * Mathematical Physics explained to stimulate new developments and foster new applications of its methods to other fields * Written by an international group of experts * Contains several undergraduate-level introductory articles to facilitate acquisition of new expertise * Thematic index and extensive cross-referencing to provide easy access and quick search functionality * Also available online with active linking.