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.
Author(s): Abdenacer Makhlouf
Series: Mathematics: Algebra and Geometry
Publisher: Wiley-ISTE
Year: 2021
Language: English
Pages: 328
City: London
Cover
Half-Title Page
Title Page
Copyright Page
Contents
Preface
1. Algebraic Background for Numerical Methods, Control Theory and Renormalization
1.1. Introduction
1.2. Hopf algebras: general properties
1.2.1. Algebras
1.2.2. Coalgebras
1.2.3. Convolution product
1.2.4. Bialgebras and Hopf algebras
1.2.5. Some simple examples of Hopf algebras
1.2.6. Some basic properties of Hopf algebras
1.3. Connected Hopf algebras
1.3.1. Connected graded bialgebras
1.3.2. An example: the Hopf algebra of decorated rooted trees
1.3.3. Connected filtered bialgebras
1.3.4. The convolution product
1.3.5. Characters
1.3.6. Group schemes and the Cartier–Milnor–Moore–Quillen theorem
1.3.7. Renormalization in connected filtered Hopf algebras
1.4. Pre-Lie algebras
1.4.1. Definition and general properties
1.4.2. The group of formal flows
1.4.3. The pre-Lie Poincaré–Birkhoff–Witt theorem
1.5. Algebraic operads
1.5.1. Manipulating algebraic operations
1.5.2. The operad of multi-linear operations
1.5.3. A definition for linear operads
1.5.4. A few examples of operads
1.6. Pre-Lie algebras (continued)
1.6.1. Pre-Lie algebras and augmented operads
1.6.2. A pedestrian approach to free pre-Lie algebra
1.6.3. Right-sided commutative Hopf algebras and the Loday–Ronco theorem
1.6.4. Pre-Lie algebras of vector fields
1.6.5. B-series, composition and substitution
1.7. Other related algebraic structures
1.7.1. NAP algebras
1.7.2. Novikov algebras
1.7.3. Assosymmetric algebras
1.7.4. Dendriform algebras
1.7.5. Post-Lie algebras
1.8. References
2. From Iterated Integrals and Chronological Calculus to Hopf and Rota–Baxter Algebras
2.1. Introduction
2.2. Generalized iterated integrals
2.2.1. Permutations and simplices
2.2.2. Descents, NCSF and the BCH formula
2.2.3. Rooted trees and nonlinear differential equations
2.2.4. Flows and Hopf algebraic structures
2.3. Advances in chronological calculus
2.3.1. Chronological calculus and half-shuffles
2.3.2. Chronological calculus and pre-Lie products
2.3.3. Time-ordered products and enveloping algebras
2.3.4. Formal flows and Hopf algebraic structures
2.4. Rota–Baxter algebras
2.4.1. Origin
2.4.2. Definition and examples
2.4.3. Related algebraic structures
2.4.4. Atkinson’s factorization and Bogoliubov’s recursion
2.4.5. Spitzer’s identity: commutative case
2.4.6. Free commutative Rota–Baxter algebras
2.4.7. Spitzer’s identity: noncommutative case
2.4.8. Free Rota–Baxter algebras
2.5. References
3. Noncommutative Symmetric Functions, Lie Series and Descent Algebras
3.1. Introduction
3.2. Classical symmetric functions
3.2.1. Symmetric polynomials
3.2.2. The Hopf algebra of symmetric functions
3.2.3. The λ-ring notation
3.2.4. Symmetric functions and formal power series
3.2.5. Duality
3.3. Noncommutative symmetric functions
3.3.1. Basic definitions
3.3.2. Generators and linear bases
3.3.3. Duality
3.3.4. Solomon’s descent algebras
3.4. Lie series and Lie idempotents
3.4.1. Permutational operators on tensor spaces
3.4.2. The Hausdorff series
3.4.3. Lie idempotents in the descent algebra
3.5. Lie idempotents as noncommutative symmetric functions
3.5.1. Noncommutative power-sums
3.5.2. The Magnus expansion
3.5.3. The continuous BCH expansion
3.5.4. Another proof of the Magnus expansion
3.5.5. The (1 - q) -transform
3.5.6. Hopf algebras enter the scene
3.5.7. A one-parameter family of Lie idempotents
3.5.8. The iterated q-bracketing and its diagonalization
3.6. Decompositions of the descent algebras
3.6.1. Complete families of minimal orthogonal idempotents
3.6.2. Eulerian idempotents
3.6.3. Generalized Eulerian idempotents
3.7. Decompositions of the tensor algebra
3.8. General deformations
3.9. Lie quasi-idempotents as Lie polynomials
3.9.1. The left derivative
3.9.2. Multilinear Lie polynomials
3.9.3. Decompositions on other bases
3.10. Permutations and free quasi-symmetric functions
3.10.1. Free quasi-symmetric functions
3.11. Packed words and word quasi-symmetric functions
3.12. References
4. From Runge–Kutta Methods to Hopf Algebras of Rooted Trees
4.1. Numerical integration methods for ordinary differential equations
4.1.1. Introduction
4.1.2. Runge–Kutta methods
4.2. Algebraic theory of Runge–Kutta methods
4.2.1. The order conditions of RK methods
4.2.2. The independence of order conditions
4.2.3. Proof of necessary and sufficient order conditions
4.2.4. Composition of RK methods, rooted trees and forests
4.2.5. The Butcher group
4.2.6. Equivalence classes of RK methods
4.2.7. Bibliographical comments
4.3. B-series and related formal expansions
4.3.1. B-series
4.3.2. Backward error analysis, the exponential and the logarithm
4.3.3. Series of linear differential operators
4.3.4. The Lie algebra of the Butcher group
4.3.5. The pre-Lie algebra structure on ɡ
4.3.6. Bibliographical comments
4.4. Hopf algebras of rooted trees
4.4.1. The commutative Hopf algebra of rooted trees
4.4.2. The dual algebra Н* and the dual Hopf algebra H°
4.4.3. B-series and series of differential operators revisited
4.4.4. A universal property of the commutative Hopf algebra of rooted trees
4.4.5. The substitution law
4.4.6. Bibliographical comments
4.5. References
5. Combinatorial Algebra in Controllability and Optimal Control
5.1. Introduction
5.1.1. Motivation: idealized examples
5.1.2. Controlled dynamical systems
5.1.3. Fundamental questions in control
5.2. Analytic foundations
5.2.1. State-space models and vector fields on manifolds
5.2.2. Chronological calculus
5.2.3. Piecewise constant controls and the Baker–Campbell–Hausdorff formula
5.2.4. Picard iteration and formal series solutions
5.2.5. The Chen–Fliess series and abstractions
5.3. Controllability and optimality
5.3.1. Reachable sets and accessibility
5.3.2. Small-time local controllability
5.3.3. Nilpotent approximating systems
5.3.4. Optimality and the maximum principle
5.3.5. Control variations and approximating cones
5.4. Product expansions and realizations
5.4.1. Variation of parameters and exponential products
5.4.2. Computations using Zinbiel products
5.4.3. Exponential products and normal forms for nilpotent systems
5.4.4. Logarithm of the Chen series
5.5. References
6. Algebra is Geometry is Algebra – Interactions Between Hopf Algebras, Infinite Dimensional Geometry and Application
6.1. The Butcher group and the Connes–Kreimer algebra
6.1.1. The Butcher group and B-series from numerical analysis
6.1.2. Beyond the Butcher group
6.2. Character groups of graded and connected Hopf algebras
6.2.1. The exponential and logarithm
6.3. Controlled groups of characters
6.3.1. Conventions for this section
6.3.2. Combinatorial Hopf algebras and the inverse factorial character
6.4. Appendix: Calculus in locally convex spaces
6.4.1. Cr -Manifolds and Cr -mappings between them
6.5. References
List of Authors
Index
EULA
