Advertisement

Overview of (pro-)Lie Group Structures on Hopf Algebra Character Groups

Conference paper
  • 363 Downloads
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 267)

Abstract

Character groups of Hopf algebras appear in a variety of mathematical and physical contexts. To name just a few, they arise in non-commutative geometry, renormalisation of quantum field theory, and numerical analysis. In the present article we review recent results on the structure of character groups of Hopf algebras as infinite-dimensional (pro-)Lie groups. It turns out that under mild assumptions on the Hopf algebra or the target algebra the character groups possess strong structural properties. Moreover, these properties are of interest in applications of these groups outside of Lie theory. We emphasise this point in the context of two main examples:
  • the Butcher group from numerical analysis and

  • character groups which arise from the Connes–Kreimer theory of renormalisation of quantum field theories.

Keywords

Infinite-dimensional Lie group Hopf algebra Locally convex algebra Butcher group Weakly complete space pro-Lie group Regular Lie group 

MSC 2010

22E65 (primary) 16T05 43A40 58B25 46H30 22A05 (Secondary) 

Notes

Acknowledgements

The research on this paper was partially supported by the project Topology in Norway (NRC project 213458) and Structure Preserving Integrators, Discrete Integrable Systems and Algebraic Combinatorics (NRC project 231632). We thank J. M. Sanz-Serna for pointing out references to results from numerical analysis which the authors were unaware of. Finally, we thank the anonymous referees for many useful comments which helped to improve the manuscript.

References

  1. 1.
    Connes, A., Marcolli, M.: Noncommutative geometry, quantum fields and motives. In: American Mathematical Society Colloquium Publications, vol. 55. American Mathematical Society, Providence, RI; Hindustan Book Agency, New Delhi (2008)Google Scholar
  2. 2.
    Brouder, C.: Trees, renormalization and differential equations. BIT Num. Anal. 44, 425–438 (2004)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bogfjellmo, G., Schmeding, A.: The Lie group structure of the Butcher group. Found. Comput. Math. 17(1), 127–159 (2017)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bogfjellmo, G., Dahmen, R., Schmeding, A.: Character groups of Hopf algebras as infinite-dimensional Lie groups. Ann. Inst. Fourier (Grenoble) 66(5), 2101–2155 (2016)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Ebrahimi-Fard, K., Lundervold, A., Munthe-Kaas, H.Z.: On the Lie enveloping algebra of a post-Lie algebra. J. Lie Theory 25(4), 1139–1165 (2015)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Hofmann, K.H., Morris, S.A.: The Lie theory of connected pro-Lie groups. In: EMS Tracts in Mathematics, vol. 2. EMS, Zürich (2007)Google Scholar
  7. 7.
    Cartier, P.: A primer of Hopf algebras. Frontiers in number theory, physics, and geometry. II, pp. 537–615. Springer, Berlin (2007)Google Scholar
  8. 8.
    Kassel, C.: Quantum groups. In: Graduate Texts in Mathematics, vol. 155. Springer-Verlag, New York (1995)Google Scholar
  9. 9.
    Majid, S.: Foundations of quantum group theory. Cambridge University Press, Cambridge (1995)CrossRefGoogle Scholar
  10. 10.
    Manchon, D.: Hopf algebras in renormalisation. In: Handbook of Algebra, vol. 5, pp. 365–427. Elsevier/North-Holland, Amsterdam (2008)Google Scholar
  11. 11.
    Sweedler, M.E.: Hopf algebras. In: Mathematics Lecture Note Series. W. A. Benjamin Inc, New York (1969)Google Scholar
  12. 12.
    Waterhouse, W.C.: Introduction to affine group schemes. In: Graduate Texts in Mathematics, vol. 66. Springer-Verlag, New York-Berlin (1979)CrossRefGoogle Scholar
  13. 13.
    Connes, A., Kreimer, D.: Hopf algebras, renormalization and noncommutative geometry. Commun. Math. Phys. 199, 203–242 (1998)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chartier, P., Hairer, E., Vilmart, G.: Algebraic structures of B-series. Found. Comput. Math. 10(4), 407–427 (2010)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Munthe-Kaas, H.Z., Wright, W.M.: On the Hopf algebraic structure of Lie group integrators. Found. Comput. Math. 8(2), 227–257 (2008)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Reutenauer, C.: Free Lie algebras. In: London Mathematical Society Monographs. New Series, vol. 7. The Clarendon Press, Oxford University Press, New York. Oxford Science Publications (1993)Google Scholar
  17. 17.
    Lundervold, A., Munthe-Kaas, H.: Hopf algebras of formal diffeomorphisms and numerical integration on manifolds. In: Combinatorics and Physics. Contemporary Mathematics, vol. 539, pp. 295–324. American Mathematical Society, Providence, RI (2011)Google Scholar
  18. 18.
    Murua, A., Sanz-Serna, J.M.: Word series for dynamical systems and their numerical integrators. Found. Comput. Math. (2015).  https://doi.org/10.1007/s10208-015-9295-3CrossRefzbMATHGoogle Scholar
  19. 19.
    Knapp, A.W.: Lie groups, Lie algebras, and cohomology. In: Mathematical Notes, vol. 34. Princeton University Press, Princeton, NJ (1988)Google Scholar
  20. 20.
    Jarchow, H.: Locally convex spaces. In: Lecture Notes in Mathematics, vol. 417. Teubner, Stuttgart (1981)Google Scholar
  21. 21.
    Schaefer, H.H.: Topological vector spaces. Springer-Verlag, New York-Berlin. Third printing corrected, Graduate Texts in Mathematics, vol. 3 (1971)CrossRefGoogle Scholar
  22. 22.
    Michor, P.W.: Manifolds of differentiable mappings. In: Shiva Mathematics Series, vol. 3. Shiva Publishing Ltd., Nantwich (1980)Google Scholar
  23. 23.
    Keller, H.: Differential calculus in locally convex spaces. In: Lecture Notes in Mathematics, vol. 417. Springer Verlag, Berlin (1974)Google Scholar
  24. 24.
    Glöckner, H.: Infinite-dimensional Lie groups without completeness restrictions. In: Geometry and analysis on finite- and infinite-dimensional Lie groups (Będlewo, 2000), Banach Center Publications, vol. 55, pp. 43–59. Polish Academy of Sciences, Warsaw (2002)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Kriegl, A., Michor, P.W.: The convenient setting of global analysis. In: Mathematical Surveys and Monographs, vol. 53. American Mathematical Society, Providence, RI (1997)Google Scholar
  26. 26.
    Bertram, W., Glöckner, H., Neeb, K.-H.: Differential calculus over general base fields and rings. Expo. Math. 22(3), 213–282 (2004)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Milnor, J.: Remarks on infinite-dimensional Lie groups. In: Relativity groups and topology, II (Les Houches, 1983), pp. 1007–1057. North-Holland, Amsterdam (1984)Google Scholar
  28. 28.
    Glöckner, H.: Instructive examples of smooth, complex differentiable and complex analytic mappings into locally convex spaces. J. Math. Kyoto Univ. 47(3), 631–642 (2007)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Alzaareer, H., Schmeding, A.: Differentiable mappings on products with different degrees of differentiability in the two factors. Expo. Math. 33(2), 184–222 (2015)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Neeb, K.-H.: Towards a Lie theory of locally convex groups. Jpn. J. Math. 1(2), 291–468 (2006)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Trèves, F.: Topological vector spaces, distributions and kernels. Dover Publications, Inc., Mineola, NY. Unabridged republication of the 1967 original (2006)Google Scholar
  32. 32.
    Glöckner, H.: Algebras whose groups of units are Lie groups. Studia Math. 153(2), 147–177 (2002)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Glöckner, H., Neeb, K.-H.: When unit groups of continuous inverse algebras are regular Lie groups. Studia Math. 211(2), 95–109 (2012)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Milnor, J.W., Moore, J.C.: On the structure of Hopf algebras. Ann. Math. 2(81), 211–264 (1965)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Ebrahimi-Fard, K., Gracia-Bondía, J.M., Patras, F.: A Lie theoretic approach to renormalization. Comm. Math. Phys. 276(2), 519–549 (2007)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Dahmen, R., Schmeding, A.: The Lie group of real analytic diffeomorphisms is not real analytic. Studia Math. 229(2), 141–172 (2015)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Floret, K.: Lokalkonvexe Sequenzen mit kompakten Abbildungen. J. Reine Angew. Math. 247, 155–195 (1971)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Glöckner, H.: Direct limits of infinite-dimensional Lie groups. In: Developments and trends in infinite-dimensional Lie theory, Progress in Mathematics, vol. 288, pp. 243–280. Birkhäuser Boston, Inc., Boston, MA (2011)Google Scholar
  39. 39.
    Bonet, J., Pérez Carreras, P.: Barrelled locally convex spaces. In: North-Holland Mathematics Studies, vol. 131. North-Holland Publishing Co., Amsterdam. Notas de Matemática [Mathematical Notes], 113 (1987)Google Scholar
  40. 40.
    Grosse-Erdmann, K.-G.: The locally convex topology on the space of meromorphic functions. J. Aust. Math. Soc. Ser. A 59(3), 287–303 (1995)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Arens, R.: Linear topological division algebras. Bull. Am. Math. Soc. 53, 623–630 (1947)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Murua, A., Sanz-Serna, J.M.: Computing normal forms and formal invariants of dynamical systems by means of word series. Nonlinear Anal. 138, 326–345 (2016)MathSciNetCrossRefGoogle Scholar
  43. 43.
    van Suijlekom, W.D.: Renormalization of gauge fields: a Hopf algebra approach. Comm. Math. Phys. 276(3), 773–798 (2007)MathSciNetCrossRefGoogle Scholar
  44. 44.
    van Suijlekom, W.D.: The structure of renormalization Hopf algebras for gauge theories. I. representing Feynman graphs on BV-algebras. Comm. Math. Phys. 290(1), 291–319 (2009)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Chartier, P., Murua, A., Sanz-Serna, J.M.: Higher-order averaging, formal series and numerical integration II: the quasi-periodic case. Found. Comput. Math. 12(4), 471–508 (2012)MathSciNetCrossRefGoogle Scholar
  46. 46.
    Murua, A.: The Hopf algebra of rooted trees, free Lie algebras, and Lie series. Found. Comput. Math. 6(4), 387–426 (2006)MathSciNetCrossRefGoogle Scholar
  47. 47.
    Hairer, E., Lubich, C.: The life-span of backward error analysis for numerical integrators. Numerische Math. 76(4), 441–462 (1997)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Dahmen, R.: Analytic mappings between LB-spaces and applications in infinite-dimensional Lie theory. Math. Z. 266(1), 115–140 (2010)MathSciNetCrossRefGoogle Scholar
  49. 49.
    Bogfjellmo, G., Schmeding, A.: The tame Butcher group. J. Lie Theor. 26(4), 1107–1144 (2016). cf. arXiv:1509.03452v3
  50. 50.
    Munthe-Kaas, H.Z., Lundervold, A.: On post-Lie algebras, Lie-Butcher series and moving frames. Found. Comput. Math. 13(4), 583–613 (2013)MathSciNetCrossRefGoogle Scholar
  51. 51.
    Chartier, P., Murua, A., Sanz-Serna, J.M.: Higher-order averaging, formal series and numerical integration III: error bounds. Found. Comput. Math. 15(2), 591–612 (2015)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Gray, W.S., Duffaut, L.A.: Espinosa, and K. Ebrahimi-Fard. Faà di Bruno Hopf algebra of the output feedback group for multivariable Fliess operators. Syst. Control Lett. 74, 64–73 (2014)CrossRefGoogle Scholar
  53. 53.
    Ebrahimi-Fard, K., Gray, W.S.: Center problem, Abel equation and the Faà di Bruno Hopf algebra for output feedback. Int. Math. Res. Not. IMRN 17, 5415–5450 (2017)Google Scholar
  54. 54.
    Dahmen, R., Schmeding, A.: Lie groups of controlled characters of combinatorial Hopf algebras (2018). arXiv:1609.02044v4
  55. 55.
    Butcher, J.C.: An algebraic theory of integration methods. Math. Comp. 26, 79–106 (1972)MathSciNetCrossRefGoogle Scholar
  56. 56.
    Butcher, J.C.: Numerical methods for ordinary differential equations, 2nd edn. Wiley, Chichester (2008)CrossRefGoogle Scholar
  57. 57.
    Hairer, E., Lubich, C., Wanner, G.: Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations In: Springer Series in Computational Mathematics vol. 31, 2nd edn. Springer-Verlag, Berlin (2006)Google Scholar
  58. 58.
    Casas, F., Murua, A.: An efficient algorithm for computing the Baker-Campbell-Hausdorff series and some of its applications. J. Math. Phys. 50(3), 033513, 23 (2009)MathSciNetCrossRefGoogle Scholar
  59. 59.
    Glöckner, H.: Lie group structures on quotient groups and universal complexifications for infinite-dimensional Lie groups. J. Funct. Anal. 194(2), 347–409 (2002)MathSciNetCrossRefGoogle Scholar
  60. 60.
    Milnor, J.: On infinite-dimensional Lie groups, 1982. unpublished preprintGoogle Scholar
  61. 61.
    Hamilton, R.S.: The inverse function theorem of Nash and Moser. Bull. Amer. Math. Soc. (N.S.), 7(1), 65–222 (1982)MathSciNetCrossRefGoogle Scholar
  62. 62.
    Glöckner, H.: Regularity properties of infinite-dimensional Lie groups, and semiregularity (2015). arXiv:1208.0715v3
  63. 63.
    Hofmann, K.H., Morris, S.A.: Pro-Lie groups: a survey with open problems. Axioms 4(3), 294–312 (2015)CrossRefGoogle Scholar
  64. 64.
    Hofmann, K.H., Morris, S.A.: The structure of compact groups. In: De Gruyter Studies in Mathematics, vol. 25, 3rd edn. De Gruyter, Berlin (2013)Google Scholar
  65. 65.
    Yamabe, H.: On the conjecture of Iwasawa and Gleason. Ann. Math. 2(58), 48–54 (1953)MathSciNetCrossRefGoogle Scholar
  66. 66.
    Michaelis, W.: Coassociative coalgebras. In: Handbook of Algebra, vol. 3, pp. 587–788. North-Holland, Amsterdam (2003)Google Scholar
  67. 67.
    Hofmann, K.H., Neeb, K.-H.: Pro-Lie groups which are infinite-dimensional Lie groups. Math. Proc. Cambridge Philos. Soc. 146(2), 351–378 (2009)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Chalmers Technical University & Gothenburg UniversityGothenburgSweden
  2. 2.TU DarmstadtDarmstadtGermany
  3. 3.Department of Mathematical SciencesNorwegian University of Science and Technology—NTNUTrondheimNorway

Personalised recommendations