Abdenacer Makhlouf - Algebra and Applications 1

1 Cover

2 Title Page SCIENCES Mathematics, Field Director – Nikolaos Limnios Algebra and Geometry, Subject Head – Abdenacer Makhlouf

3 Copyright 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

4 Foreword

5 1 Jordan Superalgebras 1 Jordan Superalgebras Consuelo MARTINEZ1 and Efim ZELMANOV2 1Department of Mathematics, University of Oviedo, Spain 2Department of Mathematics, University of California San Diego, USA 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

6 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

7 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

8 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

9 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

10 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

11 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

12 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

13 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

14 List of Authors

15 Index

16 End User License Agreement

1 Chapter 2 Figure 2.1. Multiplication table of the split Cayley algebra Figure 2.2. Multiplication table of the split Okubo algebra

2 Chapter 6Figure 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...

1 Cover

2 Table of Contents

3 Title Page SCIENCES Mathematics, Field Director – Nikolaos Limnios Algebra and Geometry, Subject Head – Abdenacer Makhlouf

4 Copyright 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

5 Foreword

6 Begin Reading

7 List of Authors

8 Index

9 End User License Agreement

1 v

2 iii

3 iv

4 xi

5 1

6 2

7 3

8 4

9 5

10 6

11 7

12 8

13 9

14 10

15 11

16 12

17 13

18 14

19 15

20 16

21 17

22 18

23 19

24 20

25 21

26 22

27 23

28 24

29 25

30 27

31 28

32 29

33 30

34 31

35 32

36 33

37 34

38 35

39 36

40 37

41 38

42 39

43 40

44 41

45 42

46 43

47 44

48 45

49 46

50 47

51 48

52 49

53 50

54 51

55 52

56 53

57 54

58 55

59 56

60 57

61 59

62 60

63 61

64 62

65 63

66 64

67 65

68 66

69 67

70 68

71 69

72 70

73 71

74 72

75 73

76 74

77 75