Algebra and Applications 2

This book is part of Algebra and Geometry, a subject within the SCIENCES collection published by ISTE and Wiley, and the second of three volumes specifically focusing on algebra and its applications. Algebra and Applications 2 centers on the increasing role played by combinatorial algebra and Hopf algebras, including an overview of the basic theories on non-associative algebras, operads and (combinatorial) Hopf algebras.

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. Alongside the focal topic of combinatorial algebra and Hopf algebras, non-associative algebraic structures in iterated integrals, chronological calculus, differential equations, numerical methods, control theory, non-commutative symmetric functions, Lie series, descent algebras, Butcher groups, chronological algebras, Magnus expansions and Rota-Baxter algebras are explored.

Algebra and Applications 2 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.

Auteur
Abdenacer Makhlouf is a Professor and head of the mathematics department at the University of Haute Alsace, France. His research covers structure, representation theory, deformation theory and cohomology of various types of algebras, including non-associative algebras, Hopf algebras and n-ary algebras.

Résumé
This book is part of Algebra and Geometry, a subject within the SCIENCES collection published by ISTE and Wiley, and the second of three volumes specifically focusing on algebra and its applications. Algebra and Applications 2 centers on the increasing role played by combinatorial algebra and Hopf algebras, including an overview of the basic theories on non-associative algebras, operads and (combinatorial) Hopf algebras.

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. Alongside the focal topic of combinatorial algebra and Hopf algebras, non-associative algebraic structures in iterated integrals, chronological calculus, differential equations, numerical methods, control theory, non-commutative symmetric functions, Lie series, descent algebras, Butcher groups, chronological algebras, Magnus expansions and RotaBaxter algebras are explored.

Algebra and Applications 2 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.

Contenu

Prefacexi
Abdenacer MAKHLOUF

Chapter 1. Algebraic Background for Numerical Methods, Control Theory and Renormalization **1
**Dominique MANCHON

1.1. Introduction 1

1.2. Hopf algebras: generalproperties 2

1.2.1. Algebras 2

1.2.2. Coalgebras 3

1.2.3. Convolution product 6

1.2.4. Bialgebras andHopf algebras 7

1.2.5. Some simple examples of Hopf algebras 8

1.2.6. Some basic properties of Hopf algebras 9

1.3. ConnectedHopf algebras 10

1.3.1. Connectedgradedbialgebras 10

1.3.2. An example: the Hopf algebra of decorated rooted trees 13

1.3.3. Connectedfiltered bialgebras 14

1.3.4. The convolution product 15

1.3.5. Characters 17

1.3.6. Group schemes and the CartierMilnorMooreQuillen theorem 19

1.3.7. Renormalization in connected filtered Hopf algebras 21

1.4. Pre-Lie algebras 24

1.4.1. Definition and general properties 24

1.4.2. The groupof formalflows 25

1.4.3. The pre-Lie PoincaréBirkhoffWitt theorem 26

1.5. Algebraicoperads 28

1.5.1. Manipulatingalgebraicoperations 28

1.5.2. The operad of multi-linear operations 29

1.5.3. A definition for linear operads 31

1.5.4. Afewexamplesof operads 32

1.6. Pre-Lie algebras (continued) 35

1.6.1. Pre-Lie algebras and augmented operads 35

1.6.2. A pedestrian approach to free pre-Lie algebra 36

1.6.3. Right-sided commutative Hopf algebras and theLodayRoncotheorem 38

1.6.4. Pre-Lie algebras of vectorfields 40

1.6.5. B-series, composition and substitution 42

1.7. Other related algebraic structures 44

1.7.1. NAPalgebras 44

1.7.2. Novikovalgebras 48

1.7.3. Assosymmetric algebras 48

1.7.4. Dendriformalgebras 48

1.7.5. Post-Lie algebras 49

1.8. References 50

Chapter 2. From Iterated Integrals and Chronological Calculus to Hopf and RotaBaxter Algebras **55
**Kurusch EBRAHIMI-FARD and Frédéric PATRAS

2.1. Introduction 55

2.2. Generalizediterated integrals 58

2.2.1. Permutations andsimplices 59

2.2.2. Descents,NCSFand theBCHformula 64

2.2.3. Rooted trees and nonlinear differential equations 67

2.2.4. Flows and Hopf algebraic structures 71

2.3. Advances in chronological calculus 74

2.3.1. Chronological calculus and half-shuffles 75

2.3.2. Chronological calculus and pre-Lie products 79

2.3.3. Time-ordered products and enveloping algebras 81

2.3.4. Formal flows and Hopf algebraic structures 83

2.4. RotaBaxter algebras 87

2.4.1. Origin 87

2.4.2. Definition and examples 91

2.4.3. Related algebraic structures 95

2.4.4. Atkinson's factorization and Bogoliubov's recursion 101

2.4.5. Spitzer's identity: commutative case 103

2.4.6. Free commutativeRotaBaxter algebras 107

2.4.7. Spitzer's identity: noncommutative case 108

2.4.8. FreeRotaBaxter algebras 111

2.5. References 113

Chapter 3. Noncommutative Symmetric Functions, Lie Series and Descent Algebras **119
**Jean-Yves THIBON

3.1. Introduction 119

3.2. Classical symmetric functions 120

3.2.1. Symmetric polynomials 120

3.2.2. The Hopf algebra of symmetric functions 122

3.2.3. The -ringnotation 124

3.2.4. Symmetric functions and formal power series 125

3.2.5. Duality 126

3.3. Noncommutativesymmetric functions 129

3.3.1. Basic definitions 129

3.3.2. Generators andlinear bases 131

3.3.3. Duality 133

3.3.4. Solomon'sdescent algebras 136

3.4. Lie series andLie idempotents 139

3.4.1. Permutational operators on tensor spaces 139 3.4.2. TheHausdorff series 139&lt...