A noncommutative Bohnenblust-Spitzer identity for Rota-Baxter algebras solves Bogoliubov's recursion

Abstract

The Bogoliubov recursion is a particular procedure appearing in the process of renormalization in perturbative quantum field theory. It provides convergent expressions for otherwise divergent integrals. We develop here a theory of functional identities for noncommutative Rota-Baxter algebras which is shown to encode, among others, this process in the context of Connes-Kreimer's Hopf algebra of renormalization. Our results generalize the seminal Cartier-Rota theory of classical Spitzer-type identities for commutative Rota-Baxter algebras. In the classical, commutative, case, these identities can be understood as deriving from the theory of symmetric functions. Here, we show that an analogous property holds for noncommutative Rota-Baxter algebras. That is, we show that functional identities in the noncommutative setting can be derived from the theory of noncommutative symmetric functions. Lie idempotents, and particularly the Dynkin idempotent play a crucial role in the process. Their action on the pro-unipotent groups such as those of perturbative renormalization is described in detail along the way.

improved version, accepted for publication in the Journal of Noncommutative Geometry

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• The Magnus expansion and some of its applications
• New identities in dendriform algebras
• Exponential renormalization
• Deformations and homotopy theory of relative Rota-Baxter Lie algebras
• Rota-Baxter algebras and new combinatorial identities
• From dynamical systems to renormalization
• Dendriform Equations
• Factorizable Lie bialgebras, quadratic Rota-Baxter Lie algebras and Rota-Baxter Lie bialgebras
• Twisted dendriform algebras and the pre-Lie Magnus expansion
• Representations and cohomologies of relative Rota-Baxter Lie algebras and applications
• Time-ordering and a generalized Magnus expansion
• The pre-Lie structure of the time-ordered exponential
• From iterated integrals and chronological calculus to Hopf and Rota-Baxter algebras
• The combinatorics of Bogoliubov's recursion in renormalization
• The combinatorics of Green's functions in planar field theories
• A survey on deformations, cohomologies and homotopies of relative Rota-Baxter Lie algebras
• Renormalization, isogenies and rational symmetries of differential equations
• Quasi-idempotent Rota-Baxter operators arising from quasi-idempotent elements
• A Zassenhaus-type algorithm solves the Bogoliubov recursion
• Ecalle's averages, Rota-Baxter algebras and the construction of moulds
• Logarithmic Derivatives and Generalized Dynkin Operators
• Formal integration of complete Rota-Baxter Lie algebras