Abstract
Cyclic cohomology has been recently adapted to the treatment of Hopf symmetry in noncommutative geometry. The resulting theory of characteristic classes for Hopf algebras and their actions on algebras allows the expansion of the range of applications of cyclic cohomology. It is the goal of this Letter to illustrate these recent developments, with special emphasis on the application to transverse index theory, and point towards future directions. In particular, we highlight the remarkable accord between our framework for cyclic cohomology of Hopf algebras on the one hand and both the algebraic as well as the analytic theory of quantum groups on the other, manifest in the construction of the modular square.
Similar content being viewed by others
References
Adams, J. F.: On the cobar construction, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 409–412.
Beals, R. and Greiner, P.: Calculus on Heisenberg Manifolds, Ann. of Math. Stud. 119, Princeton Univ. Press, Princeton, N.J., 1988.
Burghelea, R.: The cyclic homology of the group rings, Comment.Math. Helv. 60 (1985), 354–365.
Cartier, P.: Cohomologie des coalgèbres (Exposé 5), In: Séminaire Sophus Lie, 1955–56, Facult des Sciences de Paris, 1957.
Connes, A.: Spectral sequence and homology of currents for operator algebras, Math. Forschungsinst. Oberwolfach Tagungsber. 41/81; Funktionalanalysis und C*-Algebren, 27-9/3-10, 1981.
Connes, A.: C* algèbres et géométrie differentielle, C. R. Acad. Sci. Paris, Sé r. A-B 290 (1980), 599–604.
A. Connes: Noncommutative differential geometry. Part I: The Chern character in K-homology, Preprint IHES (M/82/53), 1982; Part II: de Rham homology and noncommutative algebra, Preprint IHES (M/83/19), 1983.
Connes, A.: Noncommutative differential geometry, Inst. Hautes Etudes Sci. Publ. Math. 62 (1985), 257–360.
Connes, A.: Cohomologie cyclique et foncteur Ext n, C.R. Acad. Sci. Paris, Sér. I Math 296 (1983), 953–958.
Connes, A.: Cyclic cohomology and the transverse fundamental class of a foliation, In: H. Araki and E. G. Effros (eds), Geometric Methods in Operator Algebras (Kyoto, dy1983), Pitman Res. Notes in Math. 123, Longman Sci. Tech., Harlow, 1986, pp. 52–144.
Connes, A.: Noncommutative Geometry, Academic Press, San Diego, 1994.
Connes, A. and Moscovici, H.: Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology 29 (1990), 345–388.
Connes, A. and Moscovici, H.: The local index formula in noncommutative geometry, GAFA 5 (1995), 174–243.
Connes, A. and Moscovici, H.: Hopf algebras, cyclic cohomology and the transverse index theorem, Comm. Math. Phys. 198 (1998), 199–246.
Connes, A. and Moscovici, H.: Cyclic cohomology and Hopf algebras, Lett. Math. Phys. 48 (1999), 97–108.
Crainic, M.: Cyclic cohomology of Hopf algebras and a noncommutative Chern–Weil theory, preprint QA/9812113.
Drinfeld, V. G.: Quantum groups, In: Proc. Internat. Congr. Math. (Berkeley, 1986), Amer. Math. Soc., Providence, RI, 1987, pp. 798–820.
Gelfand, I. M. and Fuchs, D. B.: Cohomology of the Lie algebra of formal vector fields, Izv. Akad. Nauk SSSR 34 (1970), 322–337.
Godbillon, G.: Cohomologies d'algè bres de Lie de champs de vecteurs formels, In: Sé minaire Bourbaki (dy1971/1972), Exposé No.421, Lecture Notes in Math. 383, Springer, Berlin, 1974, pp. 69–87.
Hilsum, M. and Skandalis, G.: Morphismes K-orienté s d'espaces de feuilles et fonctorialité en thé orie de Kasparov, Ann. Sci. Ecole Norm. Sup. (4) 20 (1987), 325–390.
Kassel, C.: Quantum Groups, Springer-Verlag, New York, 1995.
Kustermans, J. and Vaes, S.: A simple definition for locally compact quantum groups, C.R. Acad. Sci. Paris, Sèr. I Math 328 (1999), 871–876.
Loday, J.L.: Cyclic Homology, Springer-Verlag, Berlin, 1992, 1998.
Reshetikhin, N. Yu. and Turaev, V. G.: Ribbon graphs and their invariants derived from quantum groups, Comm. Math. Phys. 127 (1990), 1–26.
Sweedler, M.E.: Hopf Algebras, Benjamin, New York, 1969.
Tsygan, B. L.: Homology of matrix Lie algebras over rings and the Hochschild homology, Uspekhi Math. Nauk. 38 (1983), 217–218.
Van Daele, A.: An algebraic framework for group duality, Adv. Math. 140 (1998), 323–366.
Woronowicz, S.L.: Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613–665.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Connes, A., Moscovici, H. Cyclic Cohomology and Hopf Algebra Symmetry. Letters in Mathematical Physics 52, 1–28 (2000). https://doi.org/10.1023/A:1007698216597
Issue Date:
DOI: https://doi.org/10.1023/A:1007698216597