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The Geometry of Characters of Hopf Algebras

  • Geir Bogfjellmo
  • Alexander SchmedingEmail author
Conference paper
Part of the Abel Symposia book series (ABEL, volume 13)

Abstract

Character groups of Hopf algebras appear in a variety of mathematical contexts. For example, they arise in non-commutative geometry, renormalisation of quantum field theory, numerical analysis and the theory of regularity structures for stochastic partial differential equations. A Hopf algebra is a structure that is simultaneously a (unital, associative) algebra, and a (counital, coassociative) coalgebra that is also equipped with an antiautomorphism known as the antipode, satisfying a certain property. In the contexts of these applications, the Hopf algebras often encode combinatorial structures and serve as a bookkeeping device. Several species of “series expansions” can then be described as algebra morphisms from a Hopf algebra to a commutative algebra. Examples include ordinary Taylor series, B-series, arising in the study of ordinary differential equations, Fliess series, arising from control theory and rough paths, arising in the theory of stochastic ordinary equations and partial differential equations. These ideas are the fundamental link connecting Hopf algebras and their character groups to the topics of the Abelsymposium 2016 on “Computation and Combinatorics in Dynamics, Stochastics and Control”. In this note we will explain some of these connections, review constructions for Lie group and topological structures for character groups and provide some new results for character groups.

Notes

Acknowledgements

This research was partially supported by the European Unions Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 691070 and by the Knut and Alice Wallenberg Foundation grant agreement KAW 2014.0354. We are indebted to K.-H. Neeb and R. Dahmen for discussions which led to Lemma 10. Further, we would like to thank L. Zambotti and Y. Bruned for explaining their results on character groups in the renormalisation of SPDEs. Finally, we thank K.H. Hofmann for encouraging and useful comments and apologize to him for leaving out [28] at first.

References

  1. 1.
    Alzaareer, H., Schmeding, A.: Differentiable mappings on products with different degrees of differentiability in the two factors. Expo. Math. 33(2), 184–222 (2015). https://doi.org/10.1016/j.exmath.2014.07.002 MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bastiani, A.: Applications différentiables et variétés différentiables de dimension infinie. J. Anal. Math. 13, 1–114 (1964)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Beattie, M.: A survey of Hopf algebras of low dimension. Acta Appl. Math. 108(1), 19–31 (2009). https://doi.org/10.1007/s10440-008-9367-3 MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bertram, W., Glöckner, H., Neeb, K.H.: Differential calculus over general base fields and rings. Expo. Math. 22(3), 213–282 (2004). https://doi.org/10.1016/S0723-0869(04)80006-9 MathSciNetCrossRefGoogle Scholar
  5. 5.
    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
  6. 6.
    Bogfjellmo, G., Dahmen, R., Schmeding, A.: Overview of (pro-)Lie group structures on Hopf algebra character groups. In: Ebrahimi-Fard, K., Barbero Linan, M. (eds.) Discrete Mechanics, Geometric Integration and Lie-Butcher Series. Springer Proceedings in Mathematics and Statistics, vol. 267, pp. 287–314. Springer, Cham (2018)Google Scholar
  7. 7.
    Bogfjellmo, G., Schmeding, A.: The tame Butcher group. J. Lie Theor. 26, 1107–1144 (2016)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Bogfjellmo, G., Schmeding, A.: The Lie group structure of the Butcher group. Found. Comput. Math. 17(1), 127–159 (2017). https://doi.org/10.1007/s10208-015-9285-5 MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bourbaki, N.: Lie groups and Lie algebras. Chapters 1–3. Elements of Mathematics (Berlin). Springer, Berlin (1998). Translated from the French, Reprint of the 1989 English translationGoogle Scholar
  10. 10.
    Brouder, C.: Trees, renormalization and differential equations. BIT Num. Anal. 44, 425–438 (2004)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bruned, Y., Hairer, M., Zambotti, L.: Algebraic Renormalisation of Regularity Structures (2016). http://arxiv.org/abs/1610.08468v1
  12. 12.
    Butcher, J.C.: An algebraic theory of integration methods. Math. Comput. 26, 79–106 (1972)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Cartier, P.: A primer of Hopf algebras. In: Frontiers in Number Theory, Physics, and Geometry, vol. II, pp. 537–615. Springer, Berlin (2007)Google Scholar
  14. 14.
    Connes, A., Marcolli, M.: Noncommutative Geometry, Quantum Fields and Motives, American Mathematical Society Colloquium Publications, vol. 55. American Mathematical Society/Hindustan Book Agency, Providence/New Delhi (2008)Google Scholar
  15. 15.
    Floret, K.: Lokalkonvexe Sequenzen mit kompakten Abbildungen. J. Reine Angew. Math. 247, 155–195 (1971)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Glöckner, H.: Algebras whose groups of units are Lie groups. Stud. Math. 153(2), 147–177 (2002). http://dx.doi.org/10.4064/sm153-2-4 MathSciNetCrossRefGoogle Scholar
  17. 17.
    Glöckner, H.: Infinite-dimensional Lie groups without completeness restrictions. In: Geometry and Analysis on Finite- and Infinite-Dimensional Lie Groups (Bȩdlewo, 2000), Banach Center Publication, vol. 55, pp. 43–59. Institute of Mathematics of the Polish Academy of Sciences, Warsaw (2002). https://doi.org/10.4064/bc55-0-3 MathSciNetCrossRefGoogle Scholar
  18. 18.
    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). http://dx.doi.org/10.1215/kjm/1250281028 MathSciNetCrossRefGoogle Scholar
  19. 19.
    Glöckner, H.: Measurable Regularity Properties of Infinite-Dimensional Lie Groups (2015). http://arxiv.org/abs/1601.02568v1
  20. 20.
    Glöckner, H.: Regularity Properties of Infinite-Dimensional Lie Groups, and Semiregularity (2015). http://arxiv.org/abs/1208.0715v3
  21. 21.
    Glöckner, H., Neeb, K.H.: When unit groups of continuous inverse algebras are regular Lie groups. Stud. Math. 211(2), 95–109 (2012). http://dx.doi.org/10.4064/sm211-2-1 MathSciNetCrossRefGoogle Scholar
  22. 22.
    Glöckner, H., Neeb, K.H.: Infinite-dimensional Lie Groups. General Theory and Main Examples (2018). UnpublishedGoogle Scholar
  23. 23.
    Gracia-Bondí a, J.M., Várilly, J.C., Figueroa, H.: Elements of Noncommutative Geometry. Birkhäuser Advanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks]. Birkhäuser Boston, Inc., Boston (2001) https://doi.org/10.1007/978-1-4612-0005-5 CrossRefGoogle Scholar
  24. 24.
    Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Springer Series in Computational Mathematics, vol. 31. Springer, New York (2006)Google Scholar
  25. 25.
    Hairer, M.: A theory of regularity structures. Invent. Math. 198(2), 269–504 (2014). http://dx.doi.org/10.1007/s00222-014-0505-4 MathSciNetCrossRefGoogle Scholar
  26. 26.
    Hewitt, E., Ross, K.A.: Abstract Harmonic Analysis, vol. II: Structure and Analysis for Compact Groups. Analysis on Locally Compact Abelian Groups. Die Grundlehren der mathematischen Wissenschaften, Band 152. Springer, New York/Berlin (1970)Google Scholar
  27. 27.
    Hofmann, K.H., Morris, S.A.: The Lie theory of connected pro-Lie groups. EMS Tracts in Mathematics, vol. 2. EMS, Zürich (2007). https://doi.org/10.4171/032
  28. 28.
    Hofmann, K.H., Morris, S.A.: The Structure of Compact Groups. De Gruyter Studies in Mathematics, vol. 25. De Gruyter, Berlin (2013). https://doi.org/10.1515/9783110296792. A primer for the student—a handbook for the expert, Third edition, revised and augmented
  29. 29.
    Hofmann, K.H., Morris, S.A.: Pro-Lie groups: A survey with open problems. Axioms 4, 294–312 (2015).  https://doi.org/10.3390/axioms4030294 CrossRefGoogle Scholar
  30. 30.
    Jarchow, H.: Locally Convex Spaces. B.G. Teubner, Stuttgart (1981). Mathematische Leitfäden. [Mathematical Textbooks]Google Scholar
  31. 31.
    Keller, H.: Differential Calculus in Locally Convex Spaces. Lecture Notes in Mathematics 417. Springer, Berlin (1974)Google Scholar
  32. 32.
    Kock, J.: Perturbative renormalisation for not-quite-connected bialgebras. Lett. Math. Phys. 105(10), 1413–1425 (2015). https://doi.org/10.1007/s11005-015-0785-7 MathSciNetCrossRefGoogle Scholar
  33. 33.
    König, W.: The Parabolic Anderson Model. Pathways in Mathematics. Birkhäuser/Springer, Cham (2016). https://doi.org/10.1007/978-3-319-33596-4. Random walk in random potentialCrossRefGoogle Scholar
  34. 34.
    Kriegl, A., Michor, P.W.: The Convenient Setting of Global Analysis. Mathematical Surveys and Monographs, vol. 53. AMS, Providence (1997)Google Scholar
  35. 35.
    Majid, S.: Foundations of Quantum Group Theory. Cambridge University Press, Cambridge (1995).  https://doi.org/10.1017/CBO9780511613104
  36. 36.
    Mallios, A.: Topological Algebras. Selected Topics. North-Holland Mathematics Studies, vol. 124. North-Holland, Amsterdam (1986). Notas de Matemática [Mathematical Notes], 109CrossRefGoogle Scholar
  37. 37.
    Manchon, D.: Hopf algebras in renormalisation. In: Handbook of Algebra, vol. 5, pp. 365–427. Elsevier/North-Holland, Amsterdam (2008). https://doi.org/10.1016/S1570-7954(07)05007-3 CrossRefGoogle Scholar
  38. 38.
    McLachlan, R.I., Modin, K., Munthe-Kaas, H., Verdier, O.: B-series methods are exactly the affine equivariant methods. Numer. Math. 133(3), 599–622 (2016). http://dx.doi.org/10.1007/s00211-015-0753-2 MathSciNetCrossRefGoogle Scholar
  39. 39.
    Michaelis, W.: Coassociative coalgebras. In: Handbook of Algebra, vol. 3, pp. 587–788. North-Holland, Amsterdam (2003). http://dx.doi.org/10.1016/S1570-7954(03)80072-4 Google Scholar
  40. 40.
    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
  41. 41.
    Milnor, J.W., Moore, J.C.: On the structure of Hopf algebras. Ann. Math. (2) 81, 211–264 (1965). http://dx.doi.org/10.2307/1970615 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). http://dx.doi.org/10.1016/j.na.2015.10.013 MathSciNetCrossRefGoogle Scholar
  43. 43.
    Neeb, K.H.: Towards a Lie theory of locally convex groups. Japan J. Math. 1(2), 291–468 (2006). https://doi.org/10.1007/s11537-006-0606-y MathSciNetCrossRefGoogle Scholar
  44. 44.
    Schaefer, H.H.: Topological Vector Spaces. Springer, New York/Berlin (1971). Third printing corrected, Graduate Texts in Mathematics, vol. 3CrossRefGoogle Scholar
  45. 45.
    Swan, R.G.: Topological examples of projective modules. Trans. Am. Math. Soc. 230, 201–234 (1977). http://dx.doi.org/10.2307/1997717 MathSciNetCrossRefGoogle Scholar
  46. 46.
    Sweedler, M.E.: Hopf Algebras. Mathematics Lecture Note Series. W. A. Benjamin, Inc., New York (1969)zbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Matematiska vetenskaperChalmers tekniska högskola och Göteborgs universitetGöteborgSweden
  2. 2.Institutt for matematiske fagNTNU TrondheimTrondheimNorway

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