FAA Di Bruno Hopf Algebras, Dyson-schwinger Equations, and Lie-butcher Series (IRMA Lectures in Mathematics and Theoretical Physics)

Preface
Contents
José M. Gracia-Bondía
Foreword
	References
Loïc Foissy
Pre-Lie algebras and systems of Dyson–Schwinger equations
	Contents
	Introduction
	Feynman graphs
		Definition
		Insertion
		Algebraic structures on Feynman graphs
		Dyson–Schwinger equations
	Rooted trees
		The Connes–Kreimer Hopf algebra
		Decorated rooted trees
		Completion of a graded Hopf algebra
	Pre-Lie algebras
		Definition and examples
		Enveloping algebra of a pre-Lie algebra
		Examples
		From rooted trees to Faà di Bruno
		Duality
		From the Faà di Bruno Lie algebra to Dyson–Schwinger equations
	Combinatorial Dyson–Schwinger equations
		Definition
		Pre-Lie structure associated to a Hopf Dyson–Schwinger equation
		Definition of the structure coefficients
		Main theorem for single equations
	Systems of Dyson–Schwinger equations
		Definition
		General results
		Operations on Hopf SDSE
		The graph associated to a Dyson–Schwinger system
		Structure of the graph of a Hopf SDSE
	Quasi-cyclic SDSE
		Structure of the cycles
		Connected Hopf SDSE with a quasi-cycle
	Fundamental systems
		Level of a vertex
		Definition of fundamental SDSE
		Fundamental systems are Hopf
		Self-dependent vertices
		Hopf SDSE containing a 2-cycle
		Systems with only vertices of level 0
		Vertices of level 1
		Vertices of level 2
	Comments and examples of fundamental systems
		Graph of a fundamental system
		Examples of fundamental systems
		Dual pre-Lie algebras
	References
Alessandra Frabetti and Dominique Manchon
Five interpretations of Faà di Bruno's formula
	Contents
	Introduction
	Faà di Bruno's formula
		Francesco Faà di Bruno
		Faà di Bruno's and Lagrange's formulae
		Some points of history
	Group interpretation
		Formal diffeomorphisms
	Hopf algebra interpretation
		Hopf algebras
		Affine group schemes and proalgebraic groups
			The Hopf algebra of symmetric functions
		The Faà di Bruno Hopf algebra
		Two more examples
			The Hopf algebra of rooted trees
			Feynman graphs and diffeographisms [22]
		The non-commutative Faà di Bruno Hopf algebra
		Proalgebraic groups on non-commutative algebras
			Cogroups [6]
			The cogroup of invertible series
		Open questions about the non-commutative Faà di Bruno Hopf algebra
	Lie algebra interpretation
		Vector fields, Witt and Virasoro Lie algebras
		Lie algebra of algebraic and proalgebraic groups
			Lie algebra for the Faà di Bruno Hopf algebra
			Lie algebra for the Hopf algebra of rooted trees
	Combinatorial interpretation
		Incidence Hopf algebras
			The binomial and the divided power Hopf algebras
			The Faà di Bruno Hopf algebra
			The Hopf algebra of rooted trees
		Combinatorial Hopf algebras
			The non-commutative Faà di Bruno Hopf algebra and the brace bracket
			The Faà di Bruno Hopf algebra and the pre-Lie bracket
			The shuffle Hopf algebra
	Operadic interpretation
		Manipulating algebraic operations
		Algebraic operads
		Pre-Lie algebras, Lie algebras and groups associated to operads
			The Faà di Bruno Hopf algebra and the associative operad
			The QED charge Hopf algebra on planar binary trees and the duplicial operad
	References
W. Steven Gray and Luis A. Duffaut Espinosa
A Faà di Bruno Hopf algebra for analytic nonlinear feedback control systems
	Contents
	Introduction
	Fliess operators and their interconnections
		Fliess operators
		System interconnections
	Hopf algebra fundamentals
	A Faà di Bruno Hopf algebra for a group of Fliess operators
		Group of Fliess operators
		Construction of the Faà di Bruno Hopf algebra
	Explicit formula for the feedback product
	Conclusions and future research
	References
Alexander Lundervold and Hans Z. Munthe-Kaas
On algebraic structures of numerical integration on vector spaces and manifolds
	Contents
	Introduction
	Numerical integration on vector spaces
		Numerical methods and structure-preservation
		Trees and Butcher series
		Hopf algebras and the composition of Butcher series
		Substitution and backward error analysis for Butcher series
		Pre-Lie Butcher series
	Geometric numerical integration on manifolds
		Setting the stage: homogeneous manifolds and differential equations
		Lie group integrators in applications
		Geometry meets algebra
		Ordered trees and D-algebras
		Lie–Butcher series
		Composition of Lie–Butcher series
		Lie–Butcher series and flows on manifolds
		Substitution and backward error analysis for Lie–Butcher series
	References
Emmanuel Vieillard-Baron
Simple and contracting arborification
	Contents
	Introduction
	Encoding of the action of a comould associated to a derivation: the pre-coarborification
		Notations
		Decorated trees
		Pre-coarborification
		Precoarborified product
		Precoarborified composition
	Arborification and contracting arborification
		Some definitions and lemmae
		Some reminders about mould symmetries
		Simple or contracted arborification of a mould
		The main theorem
		Atomicity and separativity
	Simple or contracted arborified product
	Simple or contracted arborified composition
	Three applications of arborified composition
		Arborified product inverse of a given mould
		Backwards arborification
		Arborified composition inverse of a mould
	Several examples of mould arborification
		The symmetrel mould S bullet
		The symmetrel mould S
		The symmetral exponential mould exp
		The alternal mould T
	Another proof of the main theorem
	Conclusion
	References
Craig D. Roberts
Strong QCD and Dyson–Schwinger equations
	Contents
	Introduction
		Quantum chromodynamics
		Hadron physics
		Euclidean metric
		Quarks and contemporary nuclear physics
	Emergent phenomena
		Dyson–Schwinger equations
		Exploring dynamical mass generation
		Dynamical mass and confinement
		Confinement
		Aperçu
	Gap and Bethe–Salpeter equations
	Exact results in hadron physics
		Pseudoscalar meson mass formula
		Flavourless pseudoscalar mesons
	Condensates are confined within hadrons
	Many facets of DCSB
		DCSB in the Bethe–Salpeter kernel
		Quark anomalous magnetic moments
		normalnormala_1-normalnormal mass splitting
	Probing the hadronic interior
		Elastic form factors
			Pion form factor
		Deep inelastic scattering
			Pion and kaon valence-quark distributions
	Charting the interaction between light–quarks
	Describing baryons and mesons simultaneously
		Faddeev equation
		Nucleon form factors
		Valence-quark distributions at x=1
	Epilogue
	References