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- Herausgeber: John Wiley & Sons
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
This book is part of Algebra and Geometry, a subject within the SCIENCES collection published by ISTE and Wiley, and the first of three volumes specifically focusing on algebra and its applications. Algebra and Applications 1 centers on non-associative algebras and includes an introduction to derived categories. The chapters are written by recognized experts in the field, providing insight into new trends, as well as a comprehensive introduction to the theory. The book incorporates self-contained surveys with the main results, applications and perspectives. The chapters in this volume cover a wide variety of algebraic structures and their related topics. Jordan superalgebras, Lie algebras, composition algebras, graded division algebras, non-associative C*- algebras, H*-algebras, Krichever-Novikov type algebras, preLie algebras and related structures, geometric structures on 3-Lie algebras and derived categories are all explored. Algebra and Applications 1 is of great interest to graduate students and researchers. Each chapter combines some of the features of both a graduate level textbook and of research level surveys.
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Seitenzahl: 639
Veröffentlichungsjahr: 2021
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Table of Contents
Cover
Title Page
Copyright
Foreword
1 Jordan Superalgebras
1.1 Introduction
1.2 Tits–Kantor–Koecher construction
1.3 Basic examples (classical superalgebras)
1.4 Brackets
1.5 Cheng–Kac superalgebras
1.6 Finite dimensional simple Jordan superalgebras
1.7 Finite dimensional representations
1.8 Jordan superconformal algebras
1.9 References
2 Composition Algebras
2.1 Introduction
2.2 Quaternions and octonions
2.3 Unital composition algebras
2.4 Symmetric composition algebras
2.5 Triality
2.6 Concluding remarks
2.7 Acknowledgments
2.8 References
3 Graded-Division Algebras
3.1 Introduction
3.2 Background on gradings
3.3 Graded-division algebras over algebraically closed fields
3.4 Real graded-division associative algebras
3.5 Real loop algebras with a non-split centroid
3.6 Alternative algebras
3.7 Gradings of fields
3.8 References
4 Non-associative
C
*-algebras
4.1 Introduction
4.2
JB
-algebras
4.3 The non-associative Vidav–Palmer and Gelfand–Naimark theorems
4.4
JB
*-triples
4.5 Past, present and future of non-associative
C
*-algebras
4.6 Acknowledgments
4.7 References
5 Structure of
H
*-algebras
5.1 Introduction
5.2 Preliminaries: aspects of the general theory
5.3 Ultraproducts of
H
*-algebras
5.4 Quadratic
H
*-algebras
5.5 Associative
H
*-algebras
5.6 Flexible
H
*-algebras
5.7 Non-commutative Jordan
H
*-algebras
5.8 Jordan
H
*-algebras
5.9 Moufang
H
*-algebras
5.10 Lie
H
*-algebras
5.11 Topics closely related to Lie
H
*-algebras
5.12 Two-graded
H
*-algebras
5.13 Other topics: beyond the
H
*-algebras
5.14 Acknowledgments
5.15 References
6 Krichever–Novikov Type Algebras: Definitions and Results
6.1 Introduction
6.2 The Virasoro algebra and its relatives
6.3 The geometric picture
6.4 Algebraic structures
6.5 Almost-graded structure
6.6 Central extensions
6.7 Examples and generalizations
6.8 Lax operator algebras
6.9 Fermionic Fock space
6.10 Sugawara representation
6.11 Application to moduli space
6.12 Acknowledgments
6.13 References
7 An Introduction to Pre-Lie Algebras
7.1 Introduction
7.2 Some appearances of pre-Lie algebras
7.3 Some basic results and constructions of pre-Lie algebras
7.4 Pre-Lie algebras and CYBE
7.5 A larger framework: Lie analogues of Loday algebras
7.6 References
8 Symplectic, Product and Complex Structures on 3-Lie Algebras
8.1 Introduction
8.2 Preliminaries
8.3 Representations of 3-pre-Lie algebras
8.4 Symplectic structures and phase spaces of
3
-Lie algebras
8.5 Product structures on
3
-Lie algebras
8.6 Complex structures on
3
-Lie algebras
8.7 Complex product structures on
3
-Lie algebras
8.8 Para-Kähler structures on
3
-Lie algebras
8.9 Pseudo-Kähler structures on
3
-Lie algebras
8.10 References
9 Derived Categories
9.1 Introduction
9.2 Grothendieck’s definition
9.3 Verdier’s definition
9.4 Triangulated structure
9.5 Derived functors
9.6 Derived Morita theory
9.7 Dg categories
9.8 References
List of Authors
Index
End User License Agreement
List of Illustrations
Chapter 2
Figure 2.1.
Multiplication table of the split Cayley algebra
Figure 2.2.
Multiplication table of the split Okubo algebra
Chapter 6
Figure 6.1. Riemann surface of genus zero with one incoming and one outgoing poi...
Figure 6.2. Riemann surface of genus two with one incoming and one outgoing poin...
Figure 6.3. Riemann surface of genus two with two incoming points and one outgoi...
Guide
Cover
Table of Contents
Title Page
Copyright
Foreword
Begin Reading
List of Authors
Index
End User License Agreement
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SCIENCES
Mathematics, Field Director – Nikolaos Limnios
Algebra and Geometry, Subject Head – Abdenacer Makhlouf
Algebra and Applications 1
Non-associative Algebras and Categories
Coordinated by
Abdenacer Makhlouf
First published 2020 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.
Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:
ISTE Ltd
27-37 St George’s Road
London SW19 4EU
UK
www.iste.co.uk
John Wiley & Sons, Inc.
111 River Street
Hoboken, NJ 07030
USA
www.wiley.com
© ISTE Ltd 2021
The rights of Abdenacer Makhlouf to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.
Library of Congress Control Number: 2020938694
British Library Cataloguing-in-Publication Data
A CIP record for this book is available from the British Library
ISBN 978-1-78945–017-0
ERC code:
PE1 Mathematics
PE1_2 Algebra
PE1_5 Lie groups, Lie algebras
PE1_12 Mathematical physics
Foreword
Abdenacer MAKHLOUF
IRIMAS-Department of Mathematics, University of Haute Alsace, Mulhouse, France
We set out to compile several volumes pertaining to Algebra and Applications in order to present new research trends in algebra and related topics. The subject of algebra has grown spectacularly over the last several decades; algebra reasoning and combinatorial aspects turn out to be very efficient in solving various problems in different domains. Our objective is to provide an insight into the fast development of new concepts and theories. The chapters encompass surveys of basic theories on non-associative algebras, such as Jordan and Lie theories, using modern tools in addition to more recent algebraic structures, such as Hopf algebras, which are related to quantum groups and mathematical physics.
We provide self-contained chapters on various topics in algebra, each combining some of the features of both a graduate-level textbook and a research-level survey. They include an introduction with motivations and historical remarks, the basic concepts, main results and perspectives. Furthermore, the authors provide comments on the relevance of the results, as well as relations to other results and applications.
This first volume deals with non-associative and graded algebras (Jordan algebras, Lie theory, composition algebras, division algebras, pre-Lie algebras, Krichever–Novikov type algebras, C*-algebras and H*-algebras) and provides an introduction to derived categories.
I would like to express my deep gratitude to all the contributors of this volume and ISTE Ltd for their support.
2Composition Algebras
Alberto ELDUQUE
Department of Mathematics, University of Zaragoza, Spain
