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Birkhoff Type Decompositions and the Baker–Campbell–Hausdorff Recursion

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Abstract

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.

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References

  1. Aguiar M. (2000) Prepoisson algebras. Lett. Math. Phys. 54(4): 263–277

    Article  MATH  MathSciNet  Google Scholar 

  2. Aguiar M., Bergeron N., Sottile F. (2006) Combinatorial Hopf algebras and generalized Dehn–Sommerville relations. Comp. Math. 142, 1–30

    Article  MATH  MathSciNet  Google Scholar 

  3. Atkinson F.V. (1963) Some aspects of Baxter’s functional equation. J. Math. Anal. Appl. 7, 1–30

    Article  MATH  MathSciNet  Google Scholar 

  4. Babelon O., Bernard D., Talon M. (2003) Introduction to classical integrable systems. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge

    Google Scholar 

  5. Barron K., Huang Y., Lepowsky J. (2000) Factorization of formal exponentials and uniformization. J. Algebra. 228, 551–579

    Article  MATH  MathSciNet  Google Scholar 

  6. Baxter G. (1960) An analytic problem whose solution follows from a simple algebraic identity. Pacific J. Math. 10, 731–742

    MathSciNet  Google Scholar 

  7. Belavin A.A., Drinfeld V.G. (1982) Solutions of the classical Yang-Baxter equation for simple Lie algebras. Funct. Anal. Appl. 16, 159–180

    Article  MathSciNet  Google Scholar 

  8. Bogoliubov N.N., Parasiuk O.S. (1957) On the multiplication of causal functions in the quantum theory of fields. Acta Math. 97, 227–266

    Article  MathSciNet  Google Scholar 

  9. Cartier P. (1972) On the structure of free Baxter algebras. Advances in Math. 9, 253–265

    Article  MATH  MathSciNet  Google Scholar 

  10. Connes A., Kreimer D. (1998) Hopf algebras, Renormalization and noncommutative geometry. Commun Math. Phys. 199, 203–242

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. Connes A., Kreimer D. (2000) Renormalization in quantum field theory and the Riemann–Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem. Commun. Math. Phys. 210(1): 249–273

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. Connes A., Kreimer D. (2001) Renormalization in quantum field theory and the Riemann–Hilbert problem. II. The β-function, diffeomorphisms and the renormalization group. Commun. Math. Phys. 216, 215–241

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. Collins J.C. (1984) Renormalization. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge

    Google Scholar 

  14. Ebrahimi-Fard K., Guo L., Kreimer D. (2004) Spitzer’s identity and the algebraic Birkhoff decomposition in pQFT. J. Phys. A: Math. Gen. 37, 11037–11052

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. Ebrahimi-Fard K., Guo L., Kreimer D. (2005) Integrable renormalization II: the general case. Ann. H. Poincaré. 6, 369–395

    Article  MATH  MathSciNet  Google Scholar 

  16. Ebrahimi-Fard K., Gracia-Bondía J.M., Guo L., Várilly J.C. (2006) Combinatorics of renormalization as matrix calculus. Phys. Lett. B. 632(4): 552–558

    Article  ADS  MathSciNet  Google Scholar 

  17. Ebrahimi-Fard, K., Guo, L. Matrix Representation of Renormalization in Perturbative Quantum Field Theory. http://arXiv.org/list/hep-th/0508155, 2005

  18. Ebrahimi-Fard K., Kreimer D. (2005) Hopf algebra approach to Feynman diagram calculations. J. Phys. A: Math. Gen. 38, R385–R406

    Article  MATH  ADS  MathSciNet  Google Scholar 

  19. Figueroa H., Gracia-Bondía J.M. (2005) Combinatorial Hopf algebras in quantum field theory I. Rev. Math. Phys. 17, 881–976

    Article  MATH  MathSciNet  Google Scholar 

  20. Godement, R. Introduction à la théorie des groupes de Lie. Reprint of the 1982 original. Berlin Heidelberg New York: Springer, 2004

  21. Hepp K. (1966) Proof of the Bogoliubov–Parasiuk theorem on renormalization. Commun. Math. Phys. 2, 301–326

    Article  ADS  Google Scholar 

  22. Kreimer D. (1998) On the Hopf algebra structure of perturbative quantum field theories. Adv. Theor. Math. Phys. 2, 303–334

    MATH  MathSciNet  Google Scholar 

  23. Kreimer D. (1999) Chen’s iterated integral represents the operator product expansion. Adv. Theor. Math. Phys. 3(3): 627–670

    MATH  MathSciNet  Google Scholar 

  24. Kreimer D. (2002) Combinatorics of (perturbative) quantum field theory. Phys. Rep. 363, 387–424

    Article  MATH  ADS  MathSciNet  Google Scholar 

  25. Kingman J.F.C. (1962) Spitzer’s identity and its use in probability theory. J. London Math. Soc. 37, 309–316

    Article  MATH  MathSciNet  Google Scholar 

  26. Loday J.-L. (1994) Série de Hausdorff, idempotents Eulériens et algèbres de Hopf. Expo. Math. 12, 165–178

    MATH  MathSciNet  Google Scholar 

  27. Magnus W. (1954) On the exponential solution of differential equations for a linear operator. Comm. Pure Appl. Math. 7, 649–673

    Article  MATH  MathSciNet  Google Scholar 

  28. Manchon, D. Hopf algebras, from basics to applications to renormalization. In: Comptes-rendus des Rencontres mathématiques de Glanon 2001, available at http://math.univ-bpclermont.fr/0107EManchon/biblio/bogofa2002.pdf, 2006

  29. Munthe-Kaas, H.Z., Quispel, G.R.W., Zanna, A. The polar decomposition of Lie groups with involutive automorphisms. Technical Report, 191, Dept. of Informatics, Univ. of Bergen, Norway 2000

  30. Munthe-Kaas H.Z., Quispel G.R.W., Zanna A. (2001) Generalized polar decompositions on Lie groups with involutive automorphisms. Found. Comput. Math. 1(3): 297–324

    MATH  MathSciNet  Google Scholar 

  31. Polishchuk A. (2002) Classical Yang–Baxter equation and the A ∞-constraint. Adv. Math. 168(1): 56–95

    Article  MATH  MathSciNet  Google Scholar 

  32. Reutenauer C. (1993) Free Lie algebras. Oxford University Press, Oxford

    MATH  Google Scholar 

  33. Rota, G.-C. Baxter algebras and combinatorial identities. I, II. Bull. Amer. Math. Soc. 75, 325–329 (1969); ibid. 75, 330–334 (1969)

    Google Scholar 

  34. Rota, G.-C., Smith, D. Fluctuation theory and Baxter algebras. Istituto Nazionale di Alta Matematica, IX, 179, (1972). Reprinted in: Gian-Carlo Rota on Combinatorics: Introductory papers and commentaries. J.P.S. Kung, ed., Contemp. Mathematicians, Boston, MA: Birkhäuser Boston, 1995

  35. Rota, G.-C. Baxter operators, an introduction. In: Gian-Carlo Rota on Combinatorics, Introductory papers and commentaries. J.P.S. Kung ed., Contemp.~Mathematicians, Boston, MA: Birkhäuser Boston, 1995

  36. Rota, G.-C. Ten mathematics problems I will never solve. Invited address at the joint meeting of the American Mathematical Society and the Mexican Mathematical Society, Oaxaca, Mexico, December 6, 1997. DMV Mittellungen, Heft 2, 45–52 (1998)

  37. Semenov-Tian-Shansky M.A. (1983) What is a classical r-matrix? Funct. Anal. Appl. 17(4): 254–272

    Google Scholar 

  38. Semenov-Tian-Shansky, M.A. Integrable systems and factorization problems. Lectures given at the Faro International Summer School on Factorization and Integrable Systems (Sept. 2000), Basel-Boston: Birkhäuser, 2003

  39. Spitzer F. (1956) A combinatorial lemma and its application to probability theory. Trans. Amer. Math. Soc. 82, 323–339

    Article  MATH  MathSciNet  Google Scholar 

  40. Varadarajan V.S. (1984) Lie groups, Lie algebras, and Their Representations. Springer, Berlin Heidelberg New Yrok

    MATH  Google Scholar 

  41. Wendel J.G. (1962) A brief proof of a theorem of Baxter. Math. Scand. 11, 107–108

    MATH  MathSciNet  Google Scholar 

  42. Zanna A. (2004) Recurrence relations and convergence theory of the generalized polar decomposition on Lie groups. Math. Comp. 73(246): 761–776

    Article  MATH  ADS  MathSciNet  Google Scholar 

  43. Zimmermann W. (1969) Convergence of Bogoliubov’s method of renormalization in momentum space. Commun. Math. Phys. 15, 208–234

    Article  MATH  ADS  Google Scholar 

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Authors and Affiliations

  1. I.H.É.S., Le Bois-Marie, 35, Route de Chartres, F-91440, Bures-sur-Yvette, France

    Kurusch Ebrahimi-Fard

  2. Department of Mathematics and Computer Science, Rutgers University, Newark, NJ, 07102, U.S.A.

    Li Guo

  3. Université Blaise Pascal, C.N.R.S.-UMR 6620, 63177, Aubière, France

    Dominique Manchon

Authors
  1. Kurusch Ebrahimi-Fard
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  2. Li Guo
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  3. Dominique Manchon
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Correspondence to Kurusch Ebrahimi-Fard.

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Communicated by A. Connes

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Ebrahimi-Fard, K., Guo, L. & Manchon, D. Birkhoff Type Decompositions and the Baker–Campbell–Hausdorff Recursion. Commun. Math. Phys. 267, 821–845 (2006). https://doi.org/10.1007/s00220-006-0080-7

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