Combinatorial Hopf algebras in quantum field theory I

Reviews in Mathematical Physics

Volumn 17, Issue 8, 2005, Pages 881-976

  Figueroa, Héctor ^a   Gracia Bondía, José M ^b  

^a UNIVERSIDAD DE COSTA RICA   (Costa Rica)

^b UNIVERSIDAD COMPLUTENSE DE MADRID   (Spain)

Author keywords

Combinatorics; Hopf algebras; Noncommutative geometry; Renormalization

Indexed keywords

EID: 24344472627     PISSN: 0129055X     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0129055X05002467     Document Type: Article

References (106)

1. S. A. Joni and G.-G. Rota, Contemp. Math. 6 (1982) 1.
2. D. Kreimer, Adv. Theor. Math. Phys. 2 (1998) 303.
3. A. Connes and D. Kreimer, Commun. Math. Phys. 210 (2000) 249.
4. A. Gonnes and D. Kreimer, Commun. Math. Phys. 216 (2001) 215.
5. A. Connes and H. Moscovici, Commun. Math. Phys. 198 (1998) 198.
6. Ch. Brouder, Quantum groups and interacting quantum fields, in Group 24: Physical and Mathematical Aspects of Symmetries, eds. J.-P. Gazeau, R. Kerner, J.-P. Antoine, S. Métens and J.-Y. Thibon (Institute of Physics Publ., Bristol, 2003), pp. 89-97.
7. Ch. Brouder, B. Fauser, A. Frabetti and R. Oeckl, J. Phys. A: Math. Gen. 37 (2004) 5895.
8. D. Kreimer, Phys. Reports 363 (2002) 387.
9. D. J. Broadhurst and D. Kreimer, Commun. Math. Phys. 215 (2000) 217.
10. D. Kreimer, private communication.
11. J. Cuntz, G. Skandalis and B. Tsygan, Cyclic Homology in Noncommutative Geometry (Springer, Berlin, 2004).
12. A. Connes and H. Moscovici, Mosc. Math. J. 4 (2004) 67.
13. W. Zimmermann, Commun. Math. Phys. 15 (1969) 208.
14. W. R. Schmitt, J. Combin. Theory A46 (1987) 264.
15. E. B. Manoukian, Penormalization (Academic Press, London, 1983).
16. H. Figueroa and J. M. Gracia-Bondía, Modern Phys. Lett. A16 (2001) 1427.
17. H. Figueroa, and J. M. Gracia-Bondía, Internat. J. Mod. Phys. A19 (2004) 2739.
18. R. P. Stanley, Enumerative Combinatorics, Vol. 1, Cambridge University Press, Cambridge, 1997.
19. M. Haiman and W. R. Schmitt, J. Combin. Theory A50 (1989) 172.
20. G. Hochschild, Introduction to Affine Algebraic Groups (Holden-Day, San Francisco, 1971).
21. S. Montgomery, Hopf Algebras and their Actions on Rings (American Mathematical Society, Providence, RI, 1993).
22. D. Manchon, Hopf Algebras from basics to applications to renormalization, math. QA/0408405, to appear in the Comptes Rendus des Rencontres Mathématiques de Glanon.
23. J. W. Milnor and J. C. Moore, Ann. Math. 81 (1965) 211.
24. J. M. Gracia,-Bondía, J. C. Várilly and H. Figueroa, Elements of Noncommutative Geometry (Birkhäuser, Boston, 2001).
25. T. Bröcker and T. tom Dieck, Representations of Compact Lie Groups (Springer, Berlin, 1985).
26. B. Peterson and E. J. Taft, Aequationes Mathematicae 20 (1980) 1.
27. L. Verde-Star, Internat. J. Theor. Phys. 40 (2001) 41.
28. G. Hochschild, La Structure des Groupes de Lie (Dunod, Paris, 1968).
29. J. C. Várilly, Hopf algebras in noncommutative geometry, in Geometrical and Topological Methods in Quantum Field Theory, eds. A. Cardona, H. Ocampo and S. Paycha (World Scientific, Singapore, 2003), pp. 1-85.
30. H. Hopf, Ann. Math. 42 (1941) 22.
31. A. Klimyk and K. Schmüdgen, Quantum Groups and their Representations (Springer, Berlin, 1997).
32. M. Aguiar, N. Bergeron and F. Sottile, Combinatorial Hopf algebras and generalized Dehn-Somerville relations, math.CO/0310016.
33. L. Foissy, Les algèbres de Hopf des arbres enracinés décorés, Ph.D. dissertation, Reims (2002).
34. R. Ree, Ann. Math. 68 (1958) 210.
35. K.-T. Chen, Bull. Amer. Math. Soc. 73 (1967) 975.
36. Ch. Reutenauer, Free Lie Algebrus (Clarendon Press, Oxford, 1993).
37. M. Aguiar and F. Sottile, Cocommutative Hopf algebras of permutations and trees, math.QA/0403101.
38. J. Kustermans and S. Vaes, Ann. Sci. Éc. Norm. Sup. 33 (2000) 837.
39. J. Leslie, Trans. Amer. Math. Soc. 333 (1992) 423.
40. F. Faà di Bruno, Annali di Scienze Matematiche e Fisiche di Tortoloni 6 (1855) 479;
41. Quart. J. Pure Appl. Math. 1 (1857) 359.
42. S. G. Krantz and H. R. Parks, A Primer of Real Analytic Functions (Birkhäuser, Boston, 1992).
43. A. Connes and D. Kreimer, Commun. Math. Phys. 199 (1998) 203.
44. A. Abdesselam, Feynman diagrams in algebraic combinatorics, math.co/0212121.
45. J. M. Gracia-Bondía and J. C. Várilly, J. Math. Phys. 35 (1994) 3340.
46. A. Salam and P. T. Matthews, Phys. Rev. 90 (1953) 690.
47. C. Itzykson and J.-B. Zuber, Quantum Field Theory (McGraw-Hill, New York, 1980).
48. R. P. Stanley, Enumerative Combinatorics, Vol. 2 (Cambridge University Press, Cambridge, 1999).
49. R. Ticciati, Quantum Field Theory for Mathematicians (Cambridge University Press, Cambridge, 1999).
50. L. Foissy, J. Algebra 255 (2002) 85.
51. W. Magnus, Commun. Pure Appl. Math. 7 (1954) 649.
52. D. P. Burum, Phys. Rev. B24 (1981) 3684.
53. W. R. Salzman, J. Chem. Phys. 82 (1985) 822.
54. J. F. Cariñena, K. Ebrahimi-Fard and J. M. Gracia-Bondía, Hopf algebras in dynamical systems theory, forthcoming.
55. J.-L. de Lagrange, Mém. Acad. Royale des Sciences et Belles-Lettres de Berlin 24 (1770) 251.
56. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
57. P. Henrici, J. Math. Anal. Appl. 8 (1964) 218.
58. M. Cohen, Contemp. Math. 134 (1992) 1.
59. M. Khalkhali and B. Rangipour, Introduction to Hopf-Cyclic cohomology, math.qa/0503244.
60. S. Majid, Foundations of Quantum Group Theory (Cambridge University Press, Cambridge, 1995).
61. S. Majid, A Quantum Groups Primer (Cambridge University Press Cambridge, 2002).
62. W. P. Johnson, Amer. Math. Monthly 109 (2002) 217.
63. A. D. D. Craik, Amer. Math. Monthly 112 (2005) 119.
64. G. 't Hooft and M. Veltman, Nucl. Phys. B44 (1972) 189.
65. C. G. Bollini and J. J. Giambiagi, Nuovo Cimento B12 (1972) 20.
66. J. M. Gracia-Bondía, and S. Lazzarini, Connes-Kreimer-Epstein- Glaser renormalization, hep-th/0006106.
67. W. Zimmermann, Remark on equivalent formulations for Bogoliubov's method of renormalization, in Renormalization Theory, eds. G. Velo and A. S. Wightman (D. Reidel, Dordrecht, 1976), pp. 161-170.
68. M. C. Bergere and Y.-M. Lam, J. Math. Phys. 17 (1976) 1546.
69. P. Doubilet, J. Alg. 28 (1974) 127.
70. W. R. Schmitt, J. Pure Appl. Algebra 96 (1994) 299.
71. A. Dür, Möbius Functions, Incidence Algebras and Power Series Representations, Lecture Notes in Mathematics 1202 (Springer, Berlin, 1986).
72. G.-C. Rota, Z. Wahrscheinlichskeitstheorie 2 (1964) 340.
73. A. Cayley, Phil. Magazine 13 (1857) 17;
74. also in Collected Mathematical Papers of Arthur Cayley, Vol. 3 (Cambridge University Press, Cambridge, 1889), pp. 242-246.
75. Ch. Brouder, Eur. Phys. J. C12 (2000) 521.
76. F. Girelli, T. Krajewski and P. Martinetti, J. Math. Phys. 45 (2004) 4679.
77. V. Turaev, J. Geom. Phys. 53 (2005) 461.
78. A. Murua, The Hopf algebra of rooted trees, free Lie algebras and Lie series, preprint, San Sebastian, 2003.
79. P. Henrici, Jahresber. Deutsch. Math.-Verein. 86 (1984) 115.
80. G. Popineau and R. Stora, A pedagogical remark on the main theorem of perturbative renormalization theory, unpublished preprint, CERN & LAPP-TH (1982).
81. A. Connes, Symétries galoisiennes et renormalisation, Séminaire Poincaré 2 (2002), pp. 75-91.
82. F. Girelli, T. Krajewski and P. Martinetti, Modern Phys. Lett. A 16 (2001) 299.
83. A. Connes and M. Marcolli, From physics to number theory via noncommutative geometry, Vol. II, Renormalization, the Riemann-Hilbert correspondence and motivic Galois theory, hep-th/0411114.
84. M. Berg and P. Cartier, Representations of the renormalization group as matrix Lie algebra, hep-th/0105315, Vol. 2,
85. to appear Chap. 17 in the book by C. DeWitt and P. Cartier, Functional Integration: Action and Symmetries (Cambridge University Press, Cambridge, 2005).
86. K. Rybnikov, Análisis Combinatorio (Mir, Moscow, 1988).
87. M. Anshelevich, E. G. Effros and M. Popa, Zimmermann type cancellation in the free Faà di Bruno algebra, math.co/0504436.
88. J. Dixmier, Algèbres Enveloppantes (Gauthier-Villars, Paris, 1974).
89. K. H. Hofmann and S. A. Morris, The Structure of Compact Groups (de Gruyter, Berlin, 1998).
90. C. Chryssomalakos, H. Quevedo, M. Rosenbaum and J. D. Vergara, Commun. Math. Phys. 225 (2002) 465.
91. F. Patras, Ann. Inst. Fourier (Grenoble) 43 (1993) 1067.
92. F. Patras, J. Algebra 170 (1994) 547.
93. J. L. Loday, Expos. Math. 12 (1994) 165.
94. J. L. Loday, Cyclic Homology (Springer, Berlin, 1998).
95. D. Kastler, Introduction à l'Électrodynamique Quantique (Dunod, Paris, 1960).
96. A. Mestre and R. Oeckl, Combinatorics of the n-point functions via Hopf algebra in quantum field theory, math-ph/0505066.
97. F. D. Grosshans, G.-C. Rota and J. A. Stein, Invariant Theory and Superalgebras (American Mathematical Society, Providence, RI, 1987).
98. R. Oeckl, Nuclear. Phys. B581 (2000) 559.
99. D. Kreimer, Ann. Phys. 303 (2003) 179;
100. ibid. 305 (2003) 79.
101. E. R. Caianello, Combinatorics and Renormalization in Quantum Field Theory (Benjamin, Reading, Massachusetts, 1973).
102. Ch. Brouder and W. Schmitt, Quantum groups and quantum field theory: 111. Renormalization, hep-th/0210097.
103. G.-C. Rota, Bull. Amer. Math. Soc. 75 (1969) 325;
104. ibid. 75 (1969) 330.
105. K. Ebrahimi-Fard, L. Guo and D. Kreimer, Ann. Inst. H. Poincaré 6 (2005) 369.
106. K. Ebrahimi-Fard, L. Guo and D. Kreimer, J. Phys. A37 (2004) 11037.