Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 18 (2022), 053, 42 pages      arXiv:2104.08895
Contribution to the Special Issue on Algebraic Structures in Perturbative Quantum Field Theory in honor of Dirk Kreimer for his 60th birthday

Pathlike Co/Bialgebras and their Antipodes with Applications to Bi- and Hopf Algebras Appearing in Topology, Number Theory and Physics

Ralph M. Kaufmann ab and Yang Mo a
a) Department of Mathematics, Purdue University, West Lafayette, IN, USA
b) Department of Physics and Astronomy, Purdue University, West Lafayette, IN, USA

Received April 18, 2021, in final form June 29, 2022; Published online July 11, 2022

We develop an algebraic theory of colored, semigrouplike-flavored and pathlike co-, bi- and Hopf algebras. This is the right framework in which to discuss antipodes for bialgebras naturally appearing in combinatorics, topology, number theory and physics. In particular, we can precisely give conditions for the invertibility of characters that is needed for renormalization in the formulation of Connes and Kreimer. These are met in the relevant examples. In order to construct antipodes, we discuss formal localization constructions and quantum deformations. These allow to define and explain the appearance of Brown style coactions. Using previous results, we can interpret all the relevant coalgebras as stemming from a categorical construction, tie the bialgebra structures to Feynman categories, and apply the developed theory in this setting.

Key words: Feynman category; bialgebra; Hopf algebra; antipodes; renomalization; characters; combinatorial coalgebra; graphs; trees; Rota-Baxter; colored structures.

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  1. Andruskiewitsch N., Ferrer Santos W., The beginnings of the theory of Hopf algebras, Acta Appl. Math. 108 (2009), 3-17.
  2. Artin M., Noncommutaive rings. Course notes, 1999, available at etingof/artinnotes.pdf.
  3. Atkinson F.V., Some aspects of Baxter's functional equation, J. Math. Anal. Appl. 7 (1963), 1-30.
  4. Baues H.J., The double bar and cobar constructions, Compositio Math. 43 (1981), 331-341.
  5. Berger C., Kaufmann R.M., Trees, graphs and aggregates: a categorical perspective on combinatorial surface topology, geometry, and algebra, arXiv:2201.10537.
  6. Berger C., Kaufmann R.M., Derived decorated Feynman categories, in preparation.
  7. Brown F., Motivic periods and the projective line minus three points, arXiv:1407.5165.
  8. Brown F., Feynman amplitudes, coaction principle, and cosmic Galois group, Commun. Number Theory Phys. 11 (2017), 453-556, arXiv:1512.06409.
  9. Butcher J.C., An algebraic theory of integration methods, Math. Comp. 26 (1972), 79-106.
  10. Cartier P., A primer of Hopf algebras, in Frontiers in Number Theory, Physics, and Geometry. II, Springer, Berlin, 2007, 537-615.
  11. Chen C.Y., Nichols W.D., A duality theorem for Hopf module algebras over Dedekind rings, Comm. Algebra 18 (1990), 3209-3221.
  12. Chen K.-T., Iterated integrals of differential forms and loop space homology, Ann. of Math. 97 (1973), 217-246.
  13. Connes A., Kreimer D., Hopf algebras, renormalization and noncommutative geometry, Comm. Math. Phys. 199 (1998), 203-242, arXiv:hep-th/9808042.
  14. Connes A., Kreimer D., Renormalization in quantum field theory and the Riemann-Hilbert problem. I. The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys. 210 (2000), 249-273, arXiv:hep-th/9912092.
  15. Connes A., Kreimer D., Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The $\beta$-function, diffeomorphisms and the renormalization group, Comm. Math. Phys. 216 (2001), 215-241, arXiv:hep-th/0003188.
  16. Dijkgraaf R., Pasquier V., Roche P., Quasi Hopf algebras, group cohomology and orbifold models, Nuclear Phys. B Proc. Suppl. 18 (1990), 60-72.
  17. Duffaut Espinosa L.A., Ebrahimi-Fard K., Gray W.S., A combinatorial Hopf algebra for nonlinear output feedback control systems, J. Algebra 453 (2016), 609-643, arXiv:1406.5396.
  18. Dür A., Möbius functions, incidence algebras and power series representations, Lecture Notes in Math., Vol. 1202, Springer-Verlag, Berlin, 1986.
  19. Ebrahimi-Fard K., Guo L., Kreimer D., Integrable renormalization. II. The general case, Ann. Henri Poincaré 6 (2005), 369-395, arXiv:hep-th/0403118.
  20. Ebrahimi-Fard K., Kreimer D., The Hopf algebra approach to Feynman diagram calculations, J. Phys. A: Math. Gen. 38 (2005), R385-R407, arXiv:hep-th/0510202.
  21. Fiedorowicz Z., Loday J.-L., Crossed simplicial groups and their associated homology, Trans. Amer. Math. Soc. 326 (1991), 57-87.
  22. Foissy L., Les algèbres de Hopf des arbres enracinés décorés. II, Bull. Sci. Math. 126 (2002), 249-288, arXiv:math.QA/0105212.
  23. Foissy L., Cointeracting bialgebras, Talk at Algebraic Structures in Perturbative Quantum Field Theory, IHES, November 2020.
  24. Gálvez-Carrillo I., Kaufmann R.M., Tonks A., Three Hopf algebras from number theory, physics & topology, and their common background I: operadic & simplicial aspects, Commun. Number Theory Phys. 14 (2020), 1-90, arXiv:1607.00196.
  25. Gálvez-Carrillo I., Kaufmann R.M., Tonks A., Three Hopf algebras from number theory, physics & topology, and their common background II: general categorical formulation, Commun. Number Theory Phys. 14 (2020), 91-169, arXiv:1607.00196.
  26. Gelfand S.I., Manin Yu.I., Methods of homological algebra, 2nd ed., Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2003.
  27. Goncharov A.B., Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Math. J. 128 (2005), 209-284, arXiv:math.AG/0208144.
  28. Guo L., An introduction to Rota-Baxter algebra, Surveys of Modern Mathematics, Vol. 4, International Press, Somerville, MA, Higher Education Press, Beijing, 2012.
  29. Joni S.A., Rota G.-C., Coalgebras and bialgebras in combinatorics, Stud. Appl. Math. 61 (1979), 93-139.
  30. Joyal A., On disks, dualities, and $\Theta$ categories, Preprint, 1997.
  31. Joyal A., Street R., Braided tensor categories, Adv. Math. 102 (1993), 20-78.
  32. Kassel C., Quantum groups, Graduate Texts in Mathematics, Vol. 155, Springer-Verlag, New York, 1995.
  33. Kaufmann R.M., Orbifolding Frobenius algebras, Internat. J. Math. 14 (2003), 573-617, arXiv:math.AG/0107163.
  34. Kaufmann R.M., On spineless cacti, Deligne's conjecture and Connes-Kreimer's Hopf algebra, Topology 46 (2007), 39-88.
  35. Kaufmann R.M., Lectures on Feynman categories, in 2016 MATRIX Annals, MATRIX Book Ser., Vol. 1, Springer, Cham, 2018, 375-438, arXiv:1702.06843.
  36. Kaufmann R.M., Feynman categories and representation theory, in Representations of Algebras, Geometry and Physics, Contemp. Math., Vol. 769, Amer. Math. Soc., Providence, RI, 2021, 11-84, arXiv:1911.10169.
  37. Kaufmann R.M., B-plus operators in field theory, Feynman categories and Hopf algebras, in preparation.
  38. Kaufmann R.M., Manin Yu., Zagier D., Higher Weil-Petersson volumes of moduli spaces of stable $n$-pointed curves, Comm. Math. Phys. 181 (1996), 763-787, arXiv:alg-geom/9604001.
  39. Kaufmann R.M., Medina-Mardones A.M., Cochain level May-Steenrod operations, Forum Math. 33 (2021), 1507-1526, arXiv:2010.02571.
  40. Kaufmann R.M., Pham D., The Drinfel'd double and twisting in stringy orbifold theory, Internat. J. Math. 20 (2009), 623-657, arXiv:0708.4006.
  41. Kaufmann R.M., Ward B.C., Feynman categories, Astérisque 387 (2017), vii+161 pages, arXiv:1312.1269.
  42. Kaufmann R.M., Yang M., Higher categorical Hopf algebras, in preparation.
  43. Kaufmann R.M., Zhang Y., Permutohedral structures on $E_2$-operads, Forum Math. 29 (2017), 1371-1411.
  44. Kreimer D., The core Hopf algebra, in Quanta of Maths, Clay Math. Proc., Vol. 11, Amer. Math. Soc., Providence, RI, 2010, 313-321, arXiv:0902.1223.
  45. Kreimer D., Cutkosky rules from outer space, arXiv:1607.04861.
  46. Kreimer D., Yeats K., Algebraic interplay between renormalization and monodromy, arXiv:2105.05948.
  47. Leroux P., Les catégories de Möbius, Cah. Topol. Géom. Différ. 16 (1975), 280-282.
  48. Loday J.-L., Cyclic homology, 2nd ed., Grundlehren der mathematischen Wissenschaften, Vol. 301, Springer-Verlag, Berlin, 1998.
  49. Mac Lane S., Categories for the working mathematician, 2nd ed., Graduate Texts in Mathematics, Vol. 5, Springer-Verlag, New York, 1978.
  50. Marcolli M., Ni X., Rota-Baxter algebras, singular hypersurfaces, and renormalization on Kausz compactifications, J. Singul. 15 (2016), 80-117, arXiv:1408.3754.
  51. May J.P., Ponto K., More concise algebraic topology. Localization, completion, and model categories, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 2012.
  52. Moerdijk I., On the Connes-Kreimer construction of Hopf algebras, in Homotopy Methods in Algebraic Topology (Boulder, CO, 1999), Contemp. Math., Vol. 271, Amer. Math. Soc., Providence, RI, 2001, 311-321, arXiv:math-ph/9907010.
  53. Montgomery S., Hopf algebras and their actions on rings, CBMS Regional Conference Series in Mathematics, Vol. 82, Amer. Math. Soc., Providence, RI, 1993.
  54. Munthe-Kaas H.Z., Wright W.M., On the Hopf algebraic structure of Lie group integrators, Found. Comput. Math. 8 (2008), 227-257, arXiv:math.AC/0603023.
  55. Nichols W., Sweedler M., Hopf algebras and combinatorics, in Umbral Calculus and Hopf Algebras (Norman, Okla., 1978), Contemp. Math., Vol. 6, Amer. Math. Soc., Providence, R.I., 1982, 49-84.
  56. Quillen D.G., Homotopical algebra, Lecture Notes in Math., Vol. 43, Springer-Verlag, Berlin - New York, 1967.
  57. Rota G.-C., On the foundations of combinatorial theory. I. Theory of Möbius functions, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340-368.
  58. Schmitt W.R., Antipodes and incidence coalgebras, J. Combin. Theory Ser. A 46 (1987), 264-290.
  59. Taft E.J., Wilson R.L., On antipodes in pointed Hopf algebras, J. Algebra 29 (1974), 27-32.
  60. Takeuchi M., Free Hopf algebras generated by coalgebras, J. Math. Soc. Japan 23 (1971), 561-582.
  61. Willerton S., The twisted Drinfeld double of a finite group via gerbes and finite groupoids, Algebr. Geom. Topol. 8 (2008), 1419-1457, arXiv:math.QA/0503266.

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