Hopf algebras, from basics to applications to renormalization

Abstract

An extended version of a series of lectures given at Bogota in december 2002. It consists in a presentation of some aspects of Connes' and Kreimer's work on renormalization in the context of general connected Hopf algebras, in particular Birkhoff decomposition and, in the graded case, the scattering-type formula.

Minor corrections, BCH approach precised in subsection II.6, references added. plain TeX, 6 eps figures, some xypic diagrams

References in corpus (3)

• Renormalization in quantum field theory and the Riemann-Hilbert problem II: the $\beta$-function, diffeomorphisms and the renormalization group
• Insertion and Elimination: the doubly infinite Lie algebra of Feynman graphs
• Noncommutative renormalization for massless QED

Cited by in corpus (47)

• A Lie theoretic approach to renormalization
• Hopf Algebras in Combinatorics
• Comments on noncommutative gravity
• Integrable Renormalization II: the general case
• A noncommutative Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogoliubov's recursion
• Hopf algebra approach to Feynman diagram calculations
• Rota-Baxter Algebras in Renormalization of Perturbative Quantum Field Theory
• Combinatorial Hopf algebra for the Ben Geloun-Rivasseau tensor field theory
• An algebraic scheme associated with the noncommutative KP hierarchy and some of its extensions
• Combinatorial Hopf algebraic description of the multiscale renormalization in quantum field theory
• Integrable Renormalization I: the Ladder Case
• Hopf algebraic Renormalization of Kreimer's toy model
• Groupoids and Faa di Bruno formulae for Green functions in bialgebras of trees
• Noncommutative Bell polynomials, quasideterminants and incidence Hopf algebras
• Algebraic Birkhoff decomposition and its applications
• Perturbative renormalisation for not-quite-connected bialgebras
• Double posets and the antipode of QSym
• Renormalization of Multiple $q$-Zeta Values
• Combinatorial theory of permutation-invariant random matrices I: partitions, geometry and renormalization
• On an extension of Knuth's rotation correspondence to reduced planar trees
• The Magnus expansion, trees and Knuth's rotation correspondence
• Hopf algebra of multi-decorated rooted forests, free matching Rota-Baxter algebras and Gröbner-Shirshov bases
• The Hopf algebra of (q)multiple polylogarithms with non-positive arguments
• The combinatorics of Bogoliubov's recursion in renormalization
• Polynomial functors and combinatorial Dyson-Schwinger equations
• Clean Single-Valued Polylogarithms
• Combinatorial Hopf algebra for interconnected nonlinear systems
• Post-Lie Algebras, Factorization Theorems and Isospectral-Flows
• A tourist's guide to regularity structures and singular stochastic PDEs
• Renormalization in Combinatorially Non-Local Field Theories: the BPHZ Momentum Scheme
• Linear compactness and combinatorial bialgebras
• Feynman Diagrams and Lax Pair Equations
• Left counital Hopf algebra structures on free commutative Nijenhuis algebras
• Hopf algebras and Dyson-Schwinger equations
• An algebraic formulation of the locality principle in renormalisation
• Feynman diagrams and their algebraic lattices
• Multiplicative and semi-multiplicative functions on non-crossing partitions, and relations to cumulants
• Universal Hopf Algebra of Renormalization and Hopf Algebras of Rooted Trees
• Renormalised iterated integrals of symbols with linear constraints
• Several locality semigroups, path semigroups and partial semigroups
• Locally free representations of quivers over commutative Frobenius algebras
• (Weak) incidence bialgebras of monoidal categories
• Equivariant Morita-Takeuchi Theory
• The signed permutation group on Feynman graphs
• From orthocomplementations to locality
• On the square of the antipode in a connected filtered Hopf algebra
• Commutative modified Rota-Baxter algebras, shuffle products and Hopf algebras