Integrable Renormalization II: the general case

Abstract

We extend the results we obtained in an earlier work. The cocommutative case of rooted ladder trees is generalized to a full Hopf algebra of (decorated) rooted trees. For Hopf algebra characters with target space of Rota-Baxter type, the Birkhoff decomposition of renormalization theory is derived by using the Rota-Baxter double construction, respectively Atkinson's theorem. We also outline the extension to the Hopf algebra of Feynman graphs via decorated rooted trees.

26 pages, 1 figure

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• Quantum Analogy of Poisson Geometry, Related Dendriform Algebras and Rota-Baxter Operators
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• From Physics to Number Theory via Noncommutative Geometry, Part II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory
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• Rota-Baxter algebras and new combinatorial identities
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• From quantum quasi-shuffle algebras to braided Rota-Baxter algebras
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• Multiple zeta values and Rota--Baxter algebras
• Entropy algebras and Birkhoff factorization
• Algebraic Birkhoff decomposition and its applications
• Renormalization of multiple zeta values
• Matrix Representation of Renormalization in Perturbative Quantum Field Theory
• The uses of Connes and Kreimer's algebraic formulation of renormalization theory
• Construction of Rota-Baxter algebras via Hopf module algebras
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• The combinatorics of Bogoliubov's recursion in renormalization
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• Weighted infinitesimal unitary bialgebras on matrix algebras and weighted associative Yang-Baxter equations
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• O-operators on associative algebras and dendriform algebras
• Hopf algebra structure of generalized quasi-symmetric functions in partially commutative variables
• A Hopf algebra deformation approach to renormalization
• Gröbner-Shirshov bases for Rota-Baxter algebras
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• Feynman Diagrams and Lax Pair Equations
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• Pathlike Co/Bialgebras and their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics
• Noncommutative Poisson bialgebras
• Enumeration of Rota-Baxter Words
• Dyson-Schwinger equations in the theory of computation
• Universal Hopf Algebra of Renormalization and Hopf Algebras of Rooted Trees
• Hopf Algebraic Structures in the Cutting Rules
• A mathematical perspective on the phenomenology of non-perturbative Quantum Field Theory
• Inhomogeneous linear equation in Rota-Baxter algebra
• On the algebra of quasi-shuffles
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