Classical Hopf Algebras and Their Applications (Algebra and Applications, 29)

✍ Scribed by Pierre Cartier, Frédéric Patras

• Publisher: Springer
• Year: 2021
• Tongue: English
• Leaves: 277
• Edition: 1st ed. 2021
• Category: Library

✦ Synopsis

This book is dedicated to the structure and combinatorics of classical Hopf algebras. Its main focus is on commutative and cocommutative Hopf algebras, such as algebras of representative functions on groups and enveloping algebras of Lie algebras, as explored in the works of Borel, Cartier, Hopf and others in the 1940s and 50s.

The modern and systematic treatment uses the approach of natural operations, illuminating the structure of Hopf algebras by means of their endomorphisms and their combinatorics. Emphasizing notions such as pseudo-coproducts, characteristic endomorphisms, descent algebras and Lie idempotents, the text also covers the important case of enveloping algebras of pre-Lie algebras. A wide range of applications are surveyed, highlighting the main ideas and fundamental results.

Suitable as a textbook for masters or doctoral level programs, this book will be of interest to algebraists and anyone working in one of the fields of application of Hopf algebras.

✦ Table of Contents

Preface
Conventions
Symbols
Contents
1 Introduction
1.1 Linearization
1.2 Coalgebras
1.3 Gebras
1.4 Natural Endomorphisms
1.5 Applications
1.6 Structure of the Book
Part I General Theory
2 Coalgebras, Duality
2.1 Preliminaries on Vector Spaces and Algebras
2.2 Coalgebras: Definition and First Properties
2.3 Primitive and Group-Like Elements
2.4 Tensors
2.5 Endomorphisms
2.6 The Structure of Coalgebras
2.7 Representative Functions
2.8 Comodules
2.9 Representations and Comodules
2.10 Algebra Endomorphisms and Pseudo-coproducts
2.11 Coalgebra Endomorphisms and Quasi-coproducts
2.12 Duals of Algebras and Convolution
2.13 Graded and Conilpotent Coalgebras
2.14 Bibliographical Indications
References
3 Hopf Algebras and Groups
3.1 Bialgebras, Hopf Algebras
3.2 Modules and Comodules
3.3 Characteristic Endomorphisms and the Dynkin Operator
3.4 Hopf Algebras and Groups
3.5 Algebraic Groups
3.6 Unipotent and Pro-unipotent Groups
3.7 Enveloping Algebras, Groups, Tangent Spaces
3.8 Filtered and Complete Hopf Algebras
3.9 Signed Hopf Algebras
3.10 Module Algebras and Coalgebras
3.11 Bibliographical Indications
References
4 Structure Theorems
4.1 Dilations, Unipotent Bialgebras, and Weight Decompositions
4.2 Enveloping Algebras
4.3 Cocommutative Unipotent Hopf Algebras
4.4 Commutative Unipotent Hopf Algebras
4.5 Cocommutative Hopf Algebras
4.6 Complete Cocommutative Hopf Algebras
4.7 Remarks and Complements
4.8 Bibliographical Indications
References
5 Graded Hopf Algebras and the Descent Gebra
5.1 Descent Gebras of Graded Bialgebras
5.2 Lie Idempotents
5.3 Logarithmic Derivatives
5.4 The Descent Gebra
5.5 Combinatorial Descents
5.6 Bibliographical Indications
References
6 Pre-lie Algebras
6.1 The Basic Concept
6.2 Symmetric Brace Algebras
6.3 Free Pre-Lie Algebras and Gebras of Trees
6.4 Left-Linear Groups and Faà di Bruno
6.5 Exponentials and Logarithms
6.6 The Agrachev–Gamkrelidze Group Law
6.7 Other Examples
6.8 Brace Algebras
6.9 Right-Handed Tensor Hopf Algebras
6.10 Commutative Shuffles and Quasi-shuffles
6.11 Bibliographical Indications
References
Part II Applications
7 Group Theory
7.1 Compact Lie Groups are Algebraic
7.2 Algebraic Envelopes
7.3 Free Groups and Free Lie Algebras
7.4 Tannaka Duality
7.5 Bibliographical Indications
References
8 Algebraic Topology
8.1 Homology of Groups and H-Spaces
8.2 Hopf Algebras with Divided Powers
8.3 Eilenberg–MacLane Spaces and the Bar Construction
8.4 The Steenrod Hopf Algebra and Its Dual
8.5 Bibliographical Indications
References
9 Combinatorial Hopf Algebras, Twisted Structures, and Species
9.1 Vector Species and S–Modules
9.2 Hopf Species
9.3 The Hopf Species of Decorated Forests
9.4 Twisted Hopf Algebras
9.5 The Tensor Gebra as a Twisted Hopf Algebra
9.6 From Twisted to Classical Hopf Algebras
9.7 The Gebra of Permutations
9.8 The Structure of Twisted Hopf Algebras
9.9 Bibliographical Indications
References
10 Renormalization
10.1 Wick Products
10.2 Diagrammatics
10.3 The Hopf Algebra of Feynman Graphs
10.4 Exponential Renormalization
10.5 Bibliographical Indications
References
Appendix A Categories
A.1 The Language of Categories
A.2 Equalizers, Generators and Products
A.3 Monoidal Categories
A.4 Abelian Categories
Appendix B Operads
B.1 Operads
B.2 Polynomial Functors
B.3 Symmetric Operads
B.4 Algebras over Operads
Index