Algebra and Applications 2

Combinatorial Algebra and Hopf Algebras
1. Auflage 2021
ISBN: 978-1-78945-018-7
Verlag: Wiley

Buch, Englisch, 336 Seiten, Format (B × H): 161 mm x 240 mm, Gewicht: 660 g

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.

Autoren/Hrsg.

Makhlouf, Abdenacer

Fachgebiete

• Sozialwissenschaften Pädagogik Lehrerausbildung, Unterricht & Didaktik Allgemeine Didaktik Naturwissenschaften, Mathematik (Unterricht & Didaktik)
• Mathematik | Informatik Mathematik Algebra Lineare und multilineare Algebra, Matrizentheorie
• Sozialwissenschaften Pädagogik Pädagogik Bildungssystem Curricula: Planung und Entwicklung

Inhaltsverzeichnis

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 Cartier–Milnor–Moore–Quillen 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 Poincaré–Birkhoff–Witt 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 theLoday–Roncotheorem 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 Rota–Baxter Algebras 55
Kurusch E

Über Autor(innen)

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.