Advertisement

Tracing cyclic homology pairings under twisting of graded algebras

  • Sayan Chakraborty
  • Makoto YamashitaEmail author
Article
  • 16 Downloads

Abstract

We give a description of cyclic cohomology and its pairing with K-groups for 2-cocycle deformation of algebras graded over discrete groups. The proof relies on a realization of monodromy for the Gauss–Manin connection on periodic cyclic cohomology in terms of the cup product action of group cohomology.

Keywords

Graded algebra Cyclic homology Cocycle twisting K-theory 

Mathematics Subject Classification

19D55 46L80 16A58 

Notes

Acknowledgements

The authors would like to thank Wolfgang Lück and Ryszard Nest for fruitful comments. They also thank the following programs/Grants for their support which enabled collaboration for this paper: Simons - Foundation Grant 346300 and the Polish Government MNiSW 2015–2019 matching fund; The Isaac Newton Institute for Mathematical Sciences for the programme Operator algebras: subfactors and their applications. This work was supported by: EPSRC Grant Number EP/K032208/1 and DFG (SFB 878). M.Y. thanks the operator algebra group at University of Münster for their hospitality during his stay, and Yasu Kawahigashi for financial support (JSPS KAKENHI Grant Number 15H02056).

References

  1. 1.
    André, M.: Le \(d_{2}\) de la suite spectrale en cohomologie des groupes. C. R. Acad. Sci. Paris (French) 260, 2669–2671 (1965)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Behrstock, J., Druţu, C.: Divergence, thick groups, and short conjugators. Ill. J. Math. 58(4), 939–980 (2014)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bonic, R.A.: Symmetry in group algebras of discrete groups. Pac. J. Math. 11, 73–94 (1961)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Brown, K.S.: Cohomology of Groups, Graduate Texts in Mathematics, vol. 87. Springer, New York (1994). ISBN=0-387-90688-6 (Corrected reprint of the 1982 original) Google Scholar
  5. 5.
    Buck, J., Walters, S.: Connes–Chern characters of hexic and cubic modules. J. Oper. Theory 57(1), 35–65 (2007)MathSciNetzbMATHGoogle Scholar
  6. 6.
    Burghelea, D.: The cyclic homology of the group rings. Comment. Math. Helv. 60(3), 354–365 (1985).  https://doi.org/10.1007/BF02567420 MathSciNetzbMATHGoogle Scholar
  7. 7.
    Chen, X., Wei, S.: Spectral invariant subalgebras of reduced crossed product \(\text{ C }^*\)-algebras. J. Funct. Anal. 197(1), 228–246 (2003).  https://doi.org/10.1016/S0022-1236(02)00031-9 MathSciNetzbMATHGoogle Scholar
  8. 8.
    Connes, A.: Noncommutative differential geometry. Inst. Hautes Études Sci. Publ. Math. 62, 257–360 (1985)MathSciNetGoogle Scholar
  9. 9.
    Connolly, F., Koźniewski, T.: Rigidity and crystallographic groups. I. Invent. Math. 99(1), 25–48 (1990).  https://doi.org/10.1007/BF01234410 ADSMathSciNetzbMATHGoogle Scholar
  10. 10.
    Cuntz, J.: Cyclic Theory, Bivariant \(K\)-Theory and the Bivariant Chern–Connes Character, Cyclic Homology in Non-commutative Geometry, pp. 1–72. Springer, Berlin (2004). (Operator Algebras and Non-commutative Geometry, II) Google Scholar
  11. 11.
    Davis, J.F., Lück, W.: The topological K-theory of certain crystallographic groups. J. Noncommut. Geom. 7(2), 373–431 (2013).  https://doi.org/10.4171/JNCG/121 MathSciNetzbMATHGoogle Scholar
  12. 12.
    Echterhoff, S., Lück, W., Phillips, N.C., Walters, S.: The structure of crossed products of irrational rotation algebras by finite subgroups of \(\text{ SL }_2(\mathbb{Z})\). J. Reine Angew. Math. 639, 173–221 (2010).  https://doi.org/10.1515/CRELLE.2010.015. arXiv:math/0609784
  13. 13.
    Elliott, G.A.: On the \(K\)-Theory of the \(C^{\ast } \)-Algebra Generated by a Projective Representation of a Torsion-free Discrete Abelian Group, Operator Algebras and Group Representations, vol. I (Neptun, 1980), pp. 157–184. Pitman, Boston (1984)Google Scholar
  14. 14.
    Getzler, E.: Cartan Homotopy Formulas and the Gauss–Manin Connection in Cyclic Homology, Quantum Deformations of Algebras and Their Representations (Ramat-Gan, 1991/1992; Rehovot, 1991/1992), pp. 65–78. Bar-Ilan University, Ramat Gan (1993)Google Scholar
  15. 15.
    Goerss, P.G., Jardine, J.F.: Simplicial Homotopy Theory, Progress in Mathematics, vol. 174. Birkhäuser Verlag, Basel (1999).  https://doi.org/10.1007/978-3-0348-8707-6. ISBN=3-7643-6064-X
  16. 16.
    Higson, N., Kasparov, G.: \(E\)-theory and \(KK\)-theory for groups which act properly and isometrically on Hilbert space. Invent. Math. 144(1), 23–74 (2001).  https://doi.org/10.1007/s002220000118 ADSMathSciNetzbMATHGoogle Scholar
  17. 17.
    Jenkins, J.W.: Symmetry and nonsymmetry in the group algebras of discrete groups. Pac. J. Math. 32, 131–145 (1970)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Ji, R.: A module structure on cyclic cohomology of group graded algebras. K Theory 7(4), 369–399 (1993).  https://doi.org/10.1007/BF00962054 MathSciNetzbMATHGoogle Scholar
  19. 19.
    Ji, R., Ogle, C., Ramsey, B.: Relatively hyperbolic groups, rapid decay algebras and a generalization of the Bass conjecture. J. Noncommut. Geom. 4(1), 83–124 (2010).  https://doi.org/10.4171/JNCG/50. (With an appendix by Ogle)
  20. 20.
    Ji, R., Schweitzer, L.B.: Spectral invariance of smooth crossed products, and rapid decay locally compact groups. K Theory 10(3), 283–305 (1996).  https://doi.org/10.1007/BF00538186 MathSciNetzbMATHGoogle Scholar
  21. 21.
    Jolissaint, P.: Rapidly decreasing functions in reduced \(\text{ C }^*\)-algebras of groups. Trans. Am. Math. Soc. 317(1), 167–196 (1990).  https://doi.org/10.2307/2001458 MathSciNetzbMATHGoogle Scholar
  22. 22.
    Karoubi, M.: Homologie cyclique des groupes et des algèbres. C. R. Acad. Sci. Paris Sér. I Math. 297(7), 381–384 (1983)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Lafforgue, V.: \(K\)-théorie bivariante pour les algèbres de Banach et conjecture de Baum-Connes. Invent. Math. 149(1), 1–95 (2002).  https://doi.org/10.1007/s002220200213 ADSMathSciNetzbMATHGoogle Scholar
  24. 24.
    Lafforgue, V.: A proof of property (RD) for cocompact lattices of \(\text{ SL } (3,{\bf R})\) and \(\text{ SL } (3,{\bf C})\). J. Lie Theory 10(2), 255–267 (2000)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Lafforgue, V.: La conjecture de Baum–Connes à coefficients pour les groupes hyperboliques. J. Noncommut. Geom. 6(1), 1–197 (2012).  https://doi.org/10.4171/JNCG/89. arXiv:1201.4653 [math.OA]
  26. 26.
    Langer, M., Lück, W.: Topological \(K\)-theory of the group \(\text{ C }^*\)-algebra of a semi-direct product \(\mathbb{Z}^n\rtimes \mathbb{Z}/m\) for a free conjugation action. J. Topol. Anal. 4(2), 121–172 (2012).  https://doi.org/10.1142/S1793525312500082 MathSciNetzbMATHGoogle Scholar
  27. 27.
    Loday, J.-L.: Cyclic Homology, Second, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 301. Springer, Berlin (1998). ISBN=3-540-63074-0 (Appendix E by María O. Ronco, Chapter 13 by the author in collaboration with Teimuraz Pirashvili) Google Scholar
  28. 28.
    Loday, J.-L., Quillen, D.: Cyclic homology and the Lie algebra homology of matrices. Comment. Math. Helv. 59(4), 569–591 (1984)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Lück, W., Stamm, R.: Computations of \(K\)- and \(L\)-theory of cocompact planar groups. K Theory 21(3), 249–292 (2000).  https://doi.org/10.1023/A:1026539221644 MathSciNetzbMATHGoogle Scholar
  30. 30.
    Lysënok, I.G.: Some algorithmic properties of hyperbolic groups. Izv. Akad. Nauk SSSR Ser. Mat. 53(4), 814–832 (1989). (912. translation in Math. USSR-Izv. 35 (1990), no. 1, 145–163) MathSciNetzbMATHGoogle Scholar
  31. 31.
    Mathai, V.: Heat kernels and the range of the trace on completions of twisted group algebras. The ubiquitous heat kernel. Contemp. Math. 398, 321–345 (2006). (Am. Math. Soc., Providence, RI. With an appendix by Indira Chatterji) ADSzbMATHGoogle Scholar
  32. 32.
    Mitjagin, B.S.: Approximate dimension and bases in nuclear spaces. Uspehi Mat. Nauk 16(4), 63–132 (1961)MathSciNetGoogle Scholar
  33. 33.
    Mitiagin, B., Rolewicz, S., Żelazko, W.: Entire functions in \(B_{0}\)-algebras. Studia Math. 21, 291–306 (1961/1962)Google Scholar
  34. 34.
    Packer, J.A., Raeburn, I.: Twisted crossed products of \(\text{ C }^*\)-algebras. Math. Proc. Camb. Philos. Soc. 106(2), 293–311 (1989)MathSciNetzbMATHGoogle Scholar
  35. 35.
    Paravicini, W.: The spectral radius in \(\cal{C}_0(X)\)-Banach algebras. J. Noncommut. Geom. 7(1), 135–147 (2013).  https://doi.org/10.4171/JNCG/111 MathSciNetzbMATHGoogle Scholar
  36. 36.
    Phillips, N.C.: \(K\)-theory for Fréchet algebras. Int. J. Math. 2(1), 77–129 (1991).  https://doi.org/10.1142/S0129167X91000077 zbMATHGoogle Scholar
  37. 37.
    Pimsner, M., Voiculescu, D.: Imbedding the irrational rotation \(C^{\ast } \)-algebra into an AF-algebra. J. Oper. Theory 4(2), 201–210 (1980)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Puschnigg, M.: New holomorphically closed subalgebras of \(\text{ C }^*\)-algebras of hyperbolic groups. Geom. Funct. Anal. 20(1), 243–259 (2010).  https://doi.org/10.1007/s00039-010-0062-y MathSciNetzbMATHGoogle Scholar
  39. 39.
    Quddus, S.: Cohomology of \(\cal{A}_\theta ^{\rm alg} \rtimes \mathbb{Z}_2\) and its Chern-Connes pairing. J. Noncommut. Geom. 11(3), 827–843 (2017)MathSciNetzbMATHGoogle Scholar
  40. 40.
    Quddus, S.: Hochschild and cyclic homology of the crossed product of algebraic irrational rotational algebra by finite subgroups of \(SL(2,\mathbb{Z})\). J. Algebra 447, 322–366 (2016).  https://doi.org/10.1016/j.jalgebra.2015.08.019 MathSciNetzbMATHGoogle Scholar
  41. 41.
    Rieffel, M.A.: Irrational rotation C\(^*\)-algebras, short communication. In: Presented at International Congress of Mathematicians, Helsinki (1978)Google Scholar
  42. 42.
    Rinehart, G.S.: Differential forms on general commutative algebras. Trans. Am. Math. Soc. 108, 195–222 (1963)MathSciNetzbMATHGoogle Scholar
  43. 43.
    Sale, A.W.: The Length of Conjugators in Solvable Groups and Lattices of Semisimple Lie Groups. Ph.D. thesis. University of Oxford (2012)Google Scholar
  44. 44.
    Sale, A.: Conjugacy length in group extensions. Commun. Algebra 44(2), 873–897 (2016).  https://doi.org/10.1080/00927872.2014.990021 MathSciNetzbMATHGoogle Scholar
  45. 45.
    Schweitzer, L.B.: Dense \(m\)-convex Fréchet subalgebras of operator algebra crossed products by Lie groups. Int. J. Math. 4(4), 601–673 (1993).  https://doi.org/10.1142/S0129167X93000315 zbMATHGoogle Scholar
  46. 46.
    Schweitzer, L.B.: Spectral invariance of dense subalgebras of operator algebras. Int. J. Math. 4(2), 289–317 (1993).  https://doi.org/10.1142/S0129167X93000157 MathSciNetzbMATHGoogle Scholar
  47. 47.
    Trèves, F.: Topological Vector Spaces, Distributions and Kernels. Academic Press, New York (1967)zbMATHGoogle Scholar
  48. 48.
    Tsygan, B.L.: Homology of matrix Lie algebras over rings and the Hochschild homology. Uspekhi Mat. Nauk 38(2), 217–218 (1983). (Translation in Russ. Math. Survey 38(2) (1983), 198–199) MathSciNetzbMATHGoogle Scholar
  49. 49.
    Tsygan, B.: Cyclic Homology, Cyclic Homology in Non-commutative Geometry, Encyclopaedia of Mathematical Sciences, vol. 121, pp. 73–113. Springer, Berlin (2004). (Operator Algebras and Non-commutative Geometry, II) Google Scholar
  50. 50.
    Valette, A.: Introduction to the Baum–Connes Conjecture, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, Basel (2002). ISBN=3-7643-6706-7, From notes taken by Indira Chatterji, With an appendix by Guido MislinGoogle Scholar
  51. 51.
    Walters, S.G.: Projective modules over the non-commutative sphere. J. Lond. Math. Soc. (2) 51(3), 589–602 (1995).  https://doi.org/10.1112/jlms/51.3.589 MathSciNetzbMATHGoogle Scholar
  52. 52.
    Walters, S.G.: Chern characters of Fourier modules. Can. J. Math. 52(3), 633–672 (2000).  https://doi.org/10.4153/CJM-2000-028-9 MathSciNetzbMATHGoogle Scholar
  53. 53.
    Yamashita, M.: Deformation of Algebras Associated to Group Cocycles (preprint, 2011). arXiv:1107.2512 [math.OA]
  54. 54.
    Yamashita, M.: Monodromy of Gauss-Manin connection for deformation by group cocycles. J. Noncommut. Geom. 11(4), 1237–1265 (2017)MathSciNetzbMATHGoogle Scholar
  55. 55.
    Yashinski, A: Periodic Cyclic Homology and Smooth Deformations. Ph.D. thesis. The Pennsylvania State University (2013)Google Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Mathematisches InstitutWestfälische Wilhelms-Universität MünsterMünsterGermany
  2. 2.Department of MathematicsOchanomizu UniversityBunkyoJapan
  3. 3.Department of MathematicsUniversity of OsloOsloNorway
  4. 4.Stat-Math unit, Indian Statistical InstituteKolkataIndia

Personalised recommendations