General information
Latest issue
Impact factor

Search papers
Search references

Latest issue
Current issues
Archive issues
What is RSS

Mosc. Math. J.:

Personal entry:
Save password
Forgotten password?

Mosc. Math. J., 2004, Volume 4, Number 1, Pages 67–109 (Mi mmj143)  

This article is cited in 32 scientific papers (total in 32 papers)

Modular Hecke algebras and their Hopf symmetry

A. Connesa, H. Moscovicib

a Collège de France
b Ohio State University

Abstract: We introduce and begin to analyse a class of algebras, associated to congruence subgroups, that extend both the algebra of modular forms of all levels and the ring of classical Hecke operators. At the intuitive level, these are algebras of “polynomial coordinates” for the “transverse space” of lattices modulo the action of the Hecke correspondences. Their underlying symmetry is shown to be encoded by the same Hopf algebra that controls the transverse geometry of codimension 1 foliations. Its action is shown to span the “holomorphic tangent space” of the noncommutative space, and each of its three basic Hopf cyclic cocycles acquires a specific meaning. The Schwarzian 1-cocycle gives an inner derivation implemented by the level 1 Eisenstein series of weight 4. The Hopf cyclic 2-cocycle representing the transverse fundamental class provides a natural extension of the first Rankin–Cohen bracket to the modular Hecke algebras. Lastly, the Hopf cyclic version of the Godbillon–Vey cocycle gives rise to a 1-cocycle on $PSL(2,\mathbb Q)$ with values in Eisenstein series of weigh 2, which, when coupled with the “period” cocycle, yields a representative of the Euler class.

Key words and phrases: Modular forms, Hecke correspondences, transverse geometry, Hopf cyclic homology, Dedekind eta function, Schwarzian cocycle, Euler class of $PSL(2,\mathbb Q)$, Dedekind sums


Full text:
References: PDF file   HTML file

Bibliographic databases:

MSC: 11F32, 11F75, 58B34
Received: May 14, 2003

Citation: A. Connes, H. Moscovici, “Modular Hecke algebras and their Hopf symmetry”, Mosc. Math. J., 4:1 (2004), 67–109

Citation in format AMSBIB
\by A.~Connes, H.~Moscovici
\paper Modular Hecke algebras and their Hopf symmetry
\jour Mosc. Math.~J.
\yr 2004
\vol 4
\issue 1
\pages 67--109

Linking options:

    SHARE: FaceBook Twitter Livejournal

    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. A. Connes, H. Moscovici, “Rankin-Cohen brackets and the Hopf algebra of transverse geometry”, Mosc. Math. J., 4:1 (2004), 111–130  mathnet  mathscinet  zmath
    2. Bonneau P., Gerstenhaber M., Giaquinto A., Sternheimer D., “Quantum groups and deformation quantization: explicit approaches and implicit aspects”, J. Math. Phys., 45:10 (2004), 3703–3741  crossref  mathscinet  zmath  adsnasa  isi  scopus
    3. Figueroa H., Gracia-Bondía J.M., “Combinatorial Hopf algebras in quantum field theory. I”, Rev. Math. Phys., 17:8 (2005), 881–976  crossref  mathscinet  zmath  adsnasa  isi  scopus
    4. Fox D.J.F., “Projectively invariant star products”, IMRP Int. Math. Res. Pap., 2005, no. 9, 461–510  crossref  mathscinet  zmath  isi
    5. Sternheimer D., “Quantization is deformation”, Noncommutative Geometry and Representation Theory in Mathematical Physics, Contemporary Mathematics Series, 391, 2005, 331–352  crossref  mathscinet  zmath  isi
    6. Connes A., Marcolli M., “$\mathbb Q$-lattices: quantum statistical mechanics and Galois theory”, J. Geom. Phys., 56:1 (2006), 2–23  crossref  mathscinet  zmath  adsnasa  isi  scopus
    7. El Gradechi A.M., “The Lie theory of the Rankin-Cohen brackets and allied bi-differential operators”, Adv. Math., 207:2 (2006), 484–531  crossref  mathscinet  zmath  isi  scopus
    8. Ihara K., Kaneko M., Zagier D., “Derivation and double shuffle relations for multiple zeta values”, Compos. Math., 142:2 (2006), 307–338  crossref  mathscinet  zmath  isi  scopus
    9. Connes A., Marcolli M., “From physics to number theory via noncommutative geometry”, Frontiers in Number Theory, Physics and Geometry I - ON RANDOM MATRICES, ZETA FUNCTIONS, AND DYNAMICAL SYSTEMS, 2006, 269–349  crossref  mathscinet  isi
    10. Connes A., “On the foundations of noncommutative geometry”, Unity of Mathematics - IN HONOR OF THE NINETIETH BIRTHDAY OF I.M. GELFAND, Progress in Mathematics, 244, 2006, 173–204  crossref  mathscinet  zmath  isi  scopus
    11. Laca M., Larsen N.S., Neshveyev S., “Phase transition in the Connes-Marcolli $\mathrm{GL}_2$-system”, J. Noncommut. Geom., 1:4 (2007), 397–430  crossref  mathscinet  zmath  isi
    12. Hadfield T., Majid S., “Bicrossproduct approach to the Connes-Moscovici Hopf algebra”, J. Algebra, 312:1 (2007), 228–256  crossref  mathscinet  zmath  isi  scopus
    13. Bieliavsky P., Tang Xiang, Yao Yijun, “Rankin-Cohen brackets and formal quantization”, Adv. Math., 212:1 (2007), 293–314  crossref  mathscinet  zmath  isi  scopus
    14. Moscovici H., Rangipour B., “Cyclic cohomology of Hopf algebras of transverse symmetries in codimension 1”, Adv. Math., 210:1 (2007), 323–374  crossref  mathscinet  zmath  isi  scopus
    15. Cartier P., “A primer of Hopf algebras”, Frontiers in Number Theory, Physics, and Geometry II: On Conformal Field Theories, Discrete Groups and Renormalization, 2007, 537–615  mathscinet  zmath  isi
    16. Kaneko M., “On an extension of the derivation relation for multiple zeta values”, Conference on L-Functions, 2007, 89–94  mathscinet  zmath  isi
    17. Kordyukov Yu.A., “Noncommutative geometry of foliations”, J. K-Theory, 2:2, Special issue in memory of Yurii Petrovich Solovyev, Part 1 (2008), 219–327  crossref  mathscinet  zmath  isi  scopus
    18. Yu. A. Kordyukov, “Index theory and non-commutative geometry on foliated manifolds”, Russian Math. Surveys, 64:2 (2009), 273–391  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    19. Rangipour B., “Cyclic Cohomology of Corings”, J. K-Theory, 4:1 (2009), 193–207  crossref  mathscinet  zmath  isi  scopus
    20. Tang Xiang, Yao Yi-Jun, “A universal deformation formula for $\mathscr H_1$ without projectivity assumption”, J. Noncommut. Geom., 3:2 (2009), 151–179  crossref  mathscinet  zmath  isi
    21. Kaygun A., “A survey on Hopf-cyclic cohomology and Connes-Moscovici characteristic map”, Noncommutative Geometry and Global Analysis, Contemporary Mathematics, 546, 2011, 171–179  crossref  mathscinet  zmath  isi
    22. Khalkhali M., Pourkia A., “A Super Version of the Connes-Moscovici Hopf Algebra”, Noncommutative Geometry and Global Analysis, Contemporary Mathematics, 546, 2011, 181–198  crossref  mathscinet  zmath  isi
    23. Banerjee A., “Hopf action and Rankin-Cohen brackets on an Archimedean complex”, J Noncommut Geom, 5:3 (2011), 401–421  crossref  mathscinet  zmath  isi  scopus
    24. Rochberg R. Tang X. Yao Y.-j., “A Survey on Rankin-Cohen Deformations”, Perspectives on Noncommutative Geometry, Fields Institute Communications, 61, ed. Khalkhali M. Yu G., Amer Mathematical Soc, 2011, 133–151  mathscinet  zmath  isi
    25. Banerjee A., “Hecke operators on line bundles over modular curves”, J Number Theory, 132:4 (2012), 714–734  crossref  mathscinet  zmath  isi  scopus
    26. Yao Y., “Rankin-Cohen Deformations and Representation Theory”, Chin. Ann. Math. Ser. B, 35:5 (2014), 817–840  crossref  mathscinet  zmath  isi  scopus
    27. Moscovici H., “Geometric Construction of Hopf Cyclic Characteristic Classes”, Adv. Math., 274 (2015), 651–680  crossref  mathscinet  zmath  isi  scopus
    28. Banerjee A., “Action of Hopf on the Twisted Modular Hecke Operator”, J. Noncommutative Geom., 9:4 (2015), 1155–1173  crossref  mathscinet  zmath  isi  scopus
    29. Moscovici H., Rangipour B., “Hopf Algebras and Universal Chern Classes”, J. Noncommutative Geom., 11:1 (2017), 71–109  crossref  mathscinet  zmath  isi  scopus
    30. Rangipour B., Sutlu S., Aliabadi F.Ya., “Hopf-Cyclic Cohomology of the Connes-Moscovici Hopf Algebras With Infinite Dimensional Coefficients”, J. Homotopy Relat. Struct., 13:4 (2018), 927–969  crossref  mathscinet  zmath  isi  scopus
    31. Banerjee A., “Modular Hecke Algebras Over Mobius Categories”, J. Geom. Phys., 131 (2018), 23–40  crossref  mathscinet  zmath  isi  scopus
    32. Banerjee A., “Quasimodular Hecke Algebras and Hopf Actions”, J. Noncommutative Geom., 12:3 (2018), 1040–1079  crossref  mathscinet  isi  scopus
  • Moscow Mathematical Journal
    Number of views:
    This page:199

    Contact us:
     Terms of Use  Registration  Logotypes © Steklov Mathematical Institute RAS, 2020