# and quasi-deformations - quasi-lie algebras and quasi-deformations. algebraic structures associated

Post on 11-Feb-2020

1 views

Embed Size (px)

TRANSCRIPT

QUASI-LIE ALGEBRAS AND

QUASI-DEFORMATIONS ALGEBRAIC STRUCTURES ASSOCIATED WITH TWISTED

DERIVATIONS

DANIEL LARSSON

Faculty of Engineering Centre for Mathematical Sciences

Mathematics

Mathematics Centre for Mathematical Sciences Lund University Box 118 SE-221 00 Lund Sweden

http://www.maths.lth.se/

Doctoral Theses in Mathematical Sciences 2006:1 ISSN 1404-0034

ISBN 91-628-6739-3 LUTFMA-1020-2006

c© Daniel Larsson, 2006

Printed in Sweden by KFS, Lund 2006

Organization Document name Centre for Mathematical Sciences Lund Institute of Technology

DOCTORATE THESIS IN MATHEMATICAL SCIENCES

Mathematics Date of issue Box 118 February 2006 SE-221 00 LUND Document Number LUTFMA-1020-2006 Author(s) Supervisors Daniel Larsson Sergei Silvestrov, Gunnar Sparr Sponsering organisation Lund University, Marie Curie/Liegrits network,

University of Antwerp, STINT, Crafoord Foundation Title and subtitle Quasi-Lie Algebras and Quasi-Deformations. Algebraic Structures Associated with Twisted Derivations Abstract This thesis introduces a new deformation scheme for Lie algebras, which we refer to as “quasi-deformations” to clearly distinguish it from the classical Grothendieck-Schlessinger and Gerstenhaber deformation schemes. The main difference is that quasi-deformations are not in general category-preserving, i.e., quasi-deforming a Lie algebra gives an object in the larger category of “quasi-Lie algebras”, a notion which is also introduced in this thesis. The quasi-deformation scheme can be loosely described as follows: represent a Lie algebra by derivations acting on a commutative, associative algebra with unity and replace these derivations with twisted versions. An algebra structure is then imposed, thus arriving at the quasi-deformed algebra. Therefore the quasi-deformation takes place on the level of representations: we “deform” the representation, which is then “pulled-back” to an algebra structure. The different Chapters of this thesis is concerned with different aspects of this quasi-deformation scheme, for instance: Burchnall-Chaundy theory for the q-deformed Heisenberg algebra (Chapter II), the (quasi-Lie) algebraic structure on the vector space of twisted derivations (Chapter III), deformed Witt, Virasoro and loop algebras (Chapter III and IV), Central extension theory (Chapter III and IV), the Lie algebra sl(2) and some associated quadratic algebras. Key words Lie Algebras, Quasi-Lie Algebras, Quasi-Deformations Classifiction system and/or index terms (if any) 2000 Mathematics Subject Classification 16W55 (primary), 17B75, 17B68 (secondary) Supplementary bibliography information ISSN and key title ISBN 1404-0034 91-628-6739-3 Language Number of pages Recipient’s notes English 155 Security classification Distribution by Mathematics, Centre for Mathematical Sciences, Lund Institute of Technology, P.O. Box 118, S-221 00 Lund, Sweden I, the undersigned, being the copyright owner of the abstract of the above-mentioned dissertation, hereby grant to all reference sources permission to publish and disseminate the abstract of the above-mentioned dissertation. Signature__________________________________________________________ Date________________________________

Freedom is the freedom to say two plus two make four. If this is granted, all else follows.

Winston Smith Nineteeneightyfour

(George Orwell)

Post hoc – ergo propter hoc.

iv

Acknowledgments

The set Important of important people surrounding any given person can be decom- posed into two, possibly empty (in which case we call it a sad life) subsets: Professional and Friends of hopefully finite or countable cardinality.

In the present author’s case Important is comprised of:

Professional: Sergei Silvestrov, Jonas Hartwig, Fred Van Oystaeyen, Arnfinn Laudal, Gunnar Sparr, Victor Ufnarovski, Clas Löfwall, Gunnar Sigurdsson, Lars Hellström, Magnus Fontes, Anki Ottosson and Jaak Peetre.

Friends: Max, Lotta, Mum, Dad, Grandma, Grandpa, Mattias, Andreas, Robban, Mal- ice, Nina, Miško, Goro and Lena.

Details:∫ Sergei: Obviously this thesis would not have been written without the guidance, lively, animated and heated discussions and cooperation with my supervisor Do- cent Sergei Silvestrov whose eager attitude toward me and mathematics saved many a day. We managed to find common ground in the area of algebra with my incli- nation toward the intersection between algebra and geometry and his toward oper- ator/representation theory and there found a little gold mine to explore. There is still a lot of work ahead, Sergei!∫ Gunnar: I am grateful to assistant supervisor Professor Gunnar Sparr for advice and support of this work.∫ Fred : A very big thanks goes out to Professor Fred Van Oystaeyen for sharing some of his uncanny knowledge of all areas of algebra and in particular graded ring theory and non-commutative geometry.∫ Arnfinn, Victor and Clas: Thank you, Professor Arnfinn Laudal, Docent Victor Ufnarovski and Professor Clas Löfwall for answers (to sometimes stupid questions, I admit), inspiration, friendship, suggestions and knowledge.∫ Jonas, Gunnar and Lars: Fellow students of Sergei: Jonas Hartwig, Gunnar Sig- urdsson and Dr. Lars Hellström. In one way or another we suffered the same and gained the same. Thanks for all discussions and help on various things that only you (and possibly me) know.∫ Jaak: Thank you, Professor Jaak Peetre for your advice, encouraging words and belief in me from, when was it, 1993 (?) to present.

1

∫ Magnus: Thank you, Docent Magnus Fontes for support of this work.∫ Anki: For help with all kinds of practical details, small or big, and also for always greeting me with a happy face, I thank Anki Ottosson.

We all have to eat and be clothed, so I gratefully acknowledge:

• The Liegrits Marie Curie network, Mittag-Leffler Institute (Stockholm, Sweden), Swedish Foundation for International Cooperation in Research and Higher Edu- cation (STINT), the Crafoord Foundation and the Non-commutative Geometry (NOG) program (ESF, European Science Foundation) for financial support.

• The Algebra and Geometry group, Department of Mathematics @ the University of Antwerp, Belgium, for support and excellent research environment.

And �nally:

♥ Friends: You all know what you’ve done keeping me sane when balancing on the edge. A mere ’thank you’ could never be enough, but what more can I offer? Thank you all so much.

2

Contents

1 Introduction 7

2 Burchnall–Chaundy theory of q-Difference operators and q-deformed Heisen- berg algebras 31 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.2 A Burchnall–Chaundy type theorem . . . . . . . . . . . . . . . . . . . 34

3 Deformations of Lie Algebras using σ-Derivations 45 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 Some general considerations . . . . . . . . . . . . . . . . . . . . . . . 48

3.2.1 Generalized derivations on commutative algebras and UFD’s . . 48 3.2.2 A bracket on σ-derivations . . . . . . . . . . . . . . . . . . . 52 3.2.3 hom-Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . 60 3.2.4 Extensions of hom-Lie algebras . . . . . . . . . . . . . . . . . 63

3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.3.1 A q-deformed Witt algebra . . . . . . . . . . . . . . . . . . . 68 3.3.2 Non-linearly deformed Witt algebras . . . . . . . . . . . . . . 71 3.3.3 An example with a shifted difference . . . . . . . . . . . . . . 80

3.4 A bracket on σ-differential operators . . . . . . . . . . . . . . . . . . . 82 3.5 A deformation of the Virasoro algebra . . . . . . . . . . . . . . . . . . 85

3.5.1 Uniqueness of the extension . . . . . . . . . . . . . . . . . . . 85 3.5.2 Existence of a non-trivial extension . . . . . . . . . . . . . . . 89

4 Quasi-hom-Lie Algebras, Central Extensions and 2-Cocycle-Like Identities 97 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.2 Definitions and notations . . . . . . . . . . . . . . . . . . . . . . . . 100 4.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.4 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

4.4.1 Equivalence between extensions . . . . . . . . . . . . . . . . . 113 4.4.2 Existence of extensions . . . . . . . . . . . . . . . . . . . . . 116 4.4.3 Central extensions of the (α, β, ω)-deformed loop algebra . . . 119

3

CONTENTS

5 Quasi-deformations of sl2(F) using twisted derivations 127 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 5.2 Qhl-algebras associated with σ-derivations . . . . . . . . . . . . . . . . 129 5.3 Quasi-Deformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

5.3.1 Quasi-Deformations with base algebra A = F[t] . . . . . . . . 132 5.3.2 Deformations with base algebra F[t]/(t3) . . . . . . . . . . . . 139

4

CONTENTS

Preface

The present thesis covers five different papers (A, B, C, D and E) which more or less correspond to the four non-introductory Chapters of the thesis. The sole exception is Paper D which is molded into Chapters 3 and 4. The papers are:

A. Larsson, D., Silvestrov, S.D., Burchnall-Chaundy Theory for q-Difference Operators and q-Deformed H

Recommended