Generalized Lie Theory in Mathematics, Physics and Beyond Sergei Silvestrov • Eugen Paal Viktor Abramov • Alexander Stolin Editors Generalized Lie Theory in Mathematics, Physics and Beyond ABC Sergei Silvestrov (Editor in Chief) Viktor Abramov Centre for Mathematical Sciences Institute of Pure Mathematics Division of Mathematics University of Tartu Lund Institute of Technology J. Liivi 2 Lund University Box 118 50409 Tartu 221 00 Lund Estonia Sweden [email protected] [email protected] Alexander Stolin Eugen Paal Mathematical Sciences Department of Mathematics Chalmers University of Technology Tallinn University of Technology and Göteborg University Ehitajate tee 5 Department of Mathematical Sciences 19086 Tallinn Estonia 412 96 Göteborg [email protected] Sweden [email protected] ISBN: 978-3-540-85331-2 e-ISBN: 978-3-540-85332-9 Library of Congress Control Number: 2008933563 Math.Subject Classification (2000): 17-06, 16-06, 81-06, 17A-XX, 17B-XX, 81R-XX °c 2009 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, 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. Cover design: WMX Design GmbH, Heidelberg Printed on acid-free paper springer.com Preface The aim of this book is to extend the understanding of the fundamental role of generalizations of Lie and related non-commutative and non-associative structures in Mathematics and Physics. This is a thematic volume devoted to the interplay between several rapidly expan- ding research fields in contemporary Mathematics and Physics, such as generaliza- tions of the main structures of Lie theory aimed at quantization and discrete and non-commutative extensions of differential calculus and geometry, non-associative structures, actions of groups and semi-groups, non-commutative dynamics, non- commutative geometry and applications in Physics and beyond. The specific fields covered by this volume include: Applications of Lie, non-associative and non-commutative associative structures • to generalizations of classical and quantum mechanics and non-linear integrable systems, operadic and group theoretical methods; Generalizations and quasi-deformations of Lie algebras such as color and • super Lie algebras, quasi-Lie algebras, Hom–Lie algebras, infinite-dimensional Lie algebras of vector fields associated to Riemann surfaces, quasi-Lie algebras of Witt type and their central extensions and deformations important for inte- grable systems, for conformal field theory and for string theory; Non-commutative deformation theory, moduli spaces and interplay with non- • commutative geometry, algebraic geometry and commutative algebra, q-deformed differential calculi and extensions of homological methods and structures; Crossed product algebras and actions of groups and semi-groups, graded rings • and algebras, quantum algebras, twisted generalizations of coalgebras and Hopf algebra structures such as Hom-coalgebras, Hom-Hopf algebras, and super Hopf algebras and their applications to bosonisation, parastatistics, parabosonic and parafermionic algebras, orthoalgebas and root systems in quantum mechanics; Commutative subalgebras in non-commutative algebras and their connections • to algebraic curves, important for the extension of algebra–geometric methods to discrete versions of non-linear equations and representation theory, Lie and generalized Lie methods for differential equations and geometry. v vi Preface The volume will stimulate further advances in the topics represented as well as in related directions and applications. This book will be a source of inspiration for a broad spectrum of researchers and for research students, as the contributions lie at the intersection of the research directions and interests of several large research communities and research groups on modern Mathematics and Physics. This volume consists of 5 parts comprising 25 chapters, which were contributed by 32 researchers from 12 different countries. All contributions in the volume have been refereed. Lund, Sweden Sergei Silvestrov June 2008 Eugen Paal Viktor Abramov Alexander Stolin Acknowledgements This volume is an outcome of Baltic-Nordic and international cooperation initiatives by Centre for Mathematical Sciences at Lund University, Department of Mathema- tics at Chalmers University of Technology and Göteborg University in Sweden and the Departments of Mathematics at Tallinn University of Technology and Univer- sity of Tartu in Estonia. This cooperation, originally aimed at strengthening contacts between Estonia and Sweden in Mathematics and Physics, has rapidly expanded to Baltic-Nordic and European level resulting by now in several major jointly orga- nized international research events and projects. Contributions in the volume were reported at the international workshop of the Baltic-Nordic “Algebra, Geometry and Mathematical Physics” Network, that took place at the Centre for Mathematical Sciences in Lund University, Lund, Sweden on October 12–14, 2006. This conference was co-organized by Lund University (Sergei Silvestrov, Chair), Tallinn University of Technology (Eugen Paal), University of Tartu (Viktor Abramov), Chalmers University of Technology and Göteborg Univer- sity (Alexander Stolin). This workshop was the first such major international con- ference that was organized in Sweden in the framework of this cooperation. More than fifty researchers from different countries took part and gave talks. This volume was created as a result of this successful thematically focused research event. In Lund University, the conference and preparation of this volume was organized and supported as an international initiative associated to activities and projects of Noncommutative Geometry and Noncommutative Analysis Seminar lead by Sergei Silvestrov at the Center of Mathematical Sciences. The administrative and logistic support and facilities for the conference provided by the Centre for Mathematical Sciences at Lund University is gratefully acknowledged. This conference and the efforts in preparation of this volume in Lund were also supported from the Swedish side by The Crafoord Foundation, The Swedish Foundation of International Coop- eration in Research and Higher Education (STINT), The Royal Swedish Academy of Sciences and The Swedish Research Council. The co-organizers of the conference Viktor Abramov and Eugen Paal gratefully acknowledge financial support of their research and participation in preparation of this volume and in the work of the AGMF Conference in Lund rendered by the vii viii Acknowledgements Estonian Science Foundation under the research grants ETF6206, ETF5634 and ETF6912. Excellent Web support for the conference and for the AGMF network as well has been provided by Astralgo Science in Tallinn. We are also grateful to Johan Öinert from the Centre for Mathematical Sciences, Lund University for his invaluable technical assistance during the preparation of the volume for publication. Contents Part I Non-Associative and Non-Commutative Structures for Physics 1 Moufang Transformations and Noether Currents ................. 3 Eugen Paal 1.1 Introduction . ........................................... 3 1.2 Moufang Loops and Mal’tsev Algebras . ..................... 4 1.3 Birepresentations ......................................... 4 1.4 Moufang–Noether Currents and ETC ......................... 6 References . .................................................. 8 2 Weakly Nonassociative Algebras, Riccati and KP Hierarchies ...... 9 Aristophanes Dimakis and Folkert Müller-Hoissen 2.1 Introduction . ........................................... 9 2.2 NonassociativityandKP................................... 10 2.3 A Class of WNA Algebras and a Matrix Riccati Hierarchy . 13 2.4 WNA Algebras and Solutions of the Discrete KP Hierarchy . 17 2.5 From WNA to Gelfand–Dickey–Sato . ..................... 20 2.6 Conclusions . ........................................... 23 References . .................................................. 24 3 Applications of Transvectants ................................. 29 Chris Athorne 3.1 Introduction . ........................................... 29 3.2 Transvectants............................................. 30 3.3 Hirota................................................... 31 3.4 Padé.................................................... 33 3.5 Hyperelliptic . ........................................... 34 References . .................................................. 36 ix x Contents 4 Automorphisms of Finite Orthoalgebras, Exceptional Root Systems and Quantum Mechanics ..................................... 39 Artur E. Ruuge and Fred Van Oystaeyen 4.1 Introduction . ........................................... 39 4.2 Saturated Configurations . ................................ 41 4.3 Non-Colourable Configurations