Birkhoff Type Decompositions and the Baker–Campbell–Hausdorff Recursion

We describe a unification of several apparently unrelated factorizations arising 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 the Lie algebra polar decomposition as used in the context of the approximation of matrix exponentials in ordinary differential equations.

[1]  J. G. Wendel.  Brief Proof of a Theorem of Baxter. , 1962 .

[2]  Denis Bernard,et al.  Introduction to Classical Integrable Systems: Introduction , 2003 .

[3]  K. Ebrahimi-Fard,et al.  Matrix Representation of Renormalization in Perturbative Quantum Field Theory , 2005, hep-th/0508155.

[4]  Seok-Jin Kang,et al.  Lie Algebras and Their Representations , 1996 .

[5]  Alain Connes,et al.  Hopf Algebras, Renormalization and Noncommutative Geometry , 1998 .

[6]  A. A. Belavin,et al.  Solutions of the classical Yang - Baxter equation for simple Lie algebras , 1982 .

[7]  Antonella Zanna,et al.  Generalized Polar Decompositions on Lie Groups with Involutive Automorphisms , 2001, Found. Comput. Math..

[8]  W. Zlmmbrmann.  Convergence of Bogoliubov’s Method of Renormalization in Momentum Space , 2000 .

[9]  Alain Connes,et al.  Renormalization in Quantum Field Theory and the Riemann--Hilbert Problem II: The β-Function, Diffeomorphisms and the Renormalization Group , 2001 .

[10]  A. Polishchuk.  Classical Yang–Baxter Equation and the A∞-Constraint , 2002 .

[11]  Combinatorics of renormalization as matrix calculus , 2005, hep-th/0508154.

[12]  Li Guo.  Matrix Representation of Renormalization in Perturbative Quantum Field Theory , 2005 .

[13]  Le Bois-Marie,et al.  Integrable Renormalization II : the general case , 2004 .

[14]  Frank Sottile,et al.  Combinatorial Hopf algebras and generalized Dehn–Sommerville relations , 2003, Compositio Mathematica.

[15]  Denis Bernard,et al.  Introduction to classical integrable systems , 2003 .

[16]  F. Atkinson,et al.  Some aspects of Baxter's functional equation , 1963 .

[17]  D. Manchon.  Hopf algebras, from basics to applications to renormalization , 2004, math/0408405.

[18]  Roger Godement.  Introduction a la Th?eorie des Groupes de Lie , 1982 .

[19]  Klaus Hepp,et al.  Proof of the Bogoliubov-Parasiuk theorem on renormalization , 1966 .

[20]  G. Baxter,et al.  AN ANALYTIC PROBLEM WHOSE SOLUTION FOLLOWS FROM A SIMPLE ALGEBRAIC IDENTITY , 1960 .

[21]  Antonella Zanna,et al.  Recurrence relations and convergence theory of the generalized polar decomposition on Lie groups , 2003, Math. Comput..

[22]  M. Semenov-Tian-Shansky.  Integrable Systems and Factorization Problems , 2002, nlin/0209057.

[23]  C. Reutenauer.  Free Lie Algebras , 1993 .

[24]  Alain Connes,et al.  Renormalization in quantum field theory and the Riemann-Hilbert problem , 1999 .

[25]  F. Spitzer.  A Combinatorial Lemma and its Application to Probability Theory , 1956 .

[26]  Antonella Zanna,et al.  The polar decomposition on Lie groups with involutive automorphisms , 2000 .

[27]  Gian-Carlo Rota,et al.  Baxter algebras and combinatorial identities. II , 1969 .

[28]  N. N. Bogoliubow,et al.  Über die Multiplikation der Kausalfunktionen in der Quantentheorie der Felder , 1957 .

[29]  Dirk Kreimer,et al.  On the Hopf algebra structure of perturbative quantum field theories , 1997 .

[30]  W. Magnus.  On the exponential solution of differential equations for a linear operator , 1954 .

[31]  Factorization of Formal Exponentials and Uniformization , 1999, math/9908151.

[32]  J. Kingman.  Spitzer's Identity and its use in Probability Theory , 1962 .

[33]  TOPICAL REVIEW: The Hopf algebra approach to Feynman diagram calculations , 2005, hep-th/0510202.

[34]  Hector Figueroa,et al.  Combinatorial Hopf algebras in quantum field theory. I , 2005 .

[35]  Li Guo,et al.  Spitzer's identity and the algebraic Birkhoff decomposition in pQFT , 2004, hep-th/0407082.

[36]  Joseph P. S. Kung,et al.  Gian-Carlo Rota on combinatorics : introductory papers and commentaries , 1995 .

[37]  M. A. Semenov-Tyan-Shanskii.  What is a classical r-matrix? , 1983 .

[38]  Dirk Kreimer.  Chen’s iterated integral represents the operator product expansion , 1999 .

[39]  Alain Connes,et al.  Renormalization in Quantum Field Theory and the Riemann–Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem , 2000 .

[40]  D. Kreimer,et al.  Hopf algebra approach to Feynman diagram calculations , 2005 .

[41]  Pierre Cartier,et al.  On the structure of free baxter algebras , 1972 .

[42]  Marcelo Aguiar,et al.  Pre-Poisson Algebras , 2000 .

[43]  Gian-Carlo Rota,et al.  Ten Mathematics Problems I will never solve , 1998 .

[44]  Pss Pss,et al.  Mexico , 2002 .