Quantum Lie Theory A Multilinear Approach
This is an introduction to the mathematics behind the phrase “quantum Lie algebra”. The numerous attempts over the last 15-20 years to define a quantum Lie algebra as an elegant algebraic object with a binary “quantum” Lie bracket have not been widely acc
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Vladislav Kharchenko
Quantum Lie Theory A Multilinear Approach
Lecture Notes in Mathematics Editors-in-Chief: J.-M. Morel, Cachan B. Teissier, Paris Advisory Board: Camillo De Lellis, Zurich Mario di Bernardo, Bristol Alessio Figalli, Austin Davar Khoshnevisan, Salt Lake City Ioannis Kontoyiannis, Athens Gabor Lugosi, Barcelona Mark Podolskij, Aarhus Sylvia Serfaty, Paris and NY Catharina Stroppel, Bonn Anna Wienhard, Heidelberg
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More information about this series at http://www.springer.com/series/304
Vladislav Kharchenko
Quantum Lie Theory A Multilinear Approach
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Vladislav Kharchenko Universidad Nacional Autónoma de México Cuautitlán Izcalli Estado de México, Mexico
ISSN 0075-8434 Lecture Notes in Mathematics ISBN 978-3-319-22703-0 DOI 10.1007/978-3-319-22704-7
ISSN 1617-9692 (electronic) ISBN 978-3-319-22704-7 (eBook)
Library of Congress Control Number: 2015958730 Mathematics Subject Classification: 17B37, 20G42, 16T20, 16T05, 17A50, 17B75, 17B81, 81R50 Springer Cham Heidelberg New York Dordrecht London © Springer International Publishing Switzerland 2015 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. Printed on acid-free paper Springer International Publishing AG Switzerland is part of Springer Science+Business Media (www.springer.com)
Je dédie ce livre à Elená, Vadim, Ilya, Ludivine, et Andrei mes petits monstres mais aussi ma source d’inspiration. Moim malenkim monstram: Al ëne, Vadimu, Ilxe, Ldmile i Andre.
Preface
The numerous attempts over the last 15–20 years to define a quantum Lie algebra as an elegant algebraic object with a binary “quantum” Lie bracket have not been evidently and widely accepted. Nevertheless, the q-deformations of the enveloping algebras introduced independently by Drinfeld and Jimbo have profoundly impacted the development of both the modern theory of quantum groups and the much older mathematical theory of Hopf algebras. Although the definition of the Drinfeld– Jimbo quantization is not simple, a clear common pro
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