Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion

Abstract

We describe a unification of several apparently unrelated factorizations arisen from quantum field theory, vertex operator algebras, combinatorics and numerical methods in differential equations. The unification is given by a Birkhoff type decomposition that was obtained from the Baker-Campbell-Hausdorff formula in our study of the Hopf algebra approach of Connes and Kreimer to renormalization in perturbative quantum field theory. There we showed that the Birkhoff decomposition of Connes and Kreimer can be obtained from a certain Baker-Campbell-Hausdorff recursion formula in the presence of a Rota-Baxter operator. We will explain how the same decomposition generalizes the factorization of formal exponentials and uniformization for Lie algebras that arose in vertex operator algebra and conformal field theory, and the even-odd decomposition of combinatorial Hopf algebra characters as well as to the Lie algebra polar decomposition as used in the context of the approximation of matrix exponentials in ordinary differential equations.

accepted for publication in Comm. in Math. Phys

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Cited by in corpus (18)

• Rota-Baxter Algebras and Dendriform Algebras
• A Lie theoretic approach to renormalization
• Differential Type Operators and Gröbner-Shirshov Bases
• A Magnus- and Fer-type formula in dendriform algebras
• Rota-Baxter algebras and new combinatorial identities
• From dynamical systems to renormalization
• Dendriform Equations
• Post-Lie algebras and factorization theorems
• Post-Lie Algebras and Isospectral Flows
• Perturbative renormalisation for not-quite-connected bialgebras
• Rota-Baxter multiplicative 3-ary Hom-Nambu-Lie algebras
• Rota-Baxter operators on the polynomial algebras, integration and averaging operators
• Exact solution of the Schrodinger equation with the spin-boson Hamiltonian
• Polylogarithms and multiple zeta values from free Rota-Baxter algebras
• Polynomial functors and combinatorial Dyson-Schwinger equations
• Hopf algebras in dynamical systems theory
• Rota-Baxter modules toward derived functors
• Rayleigh-Schrödinger series and Birkhoff decomposition