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On Cyclic Cohomology of ×-Hopf algebras

Published online by Cambridge University Press:  02 January 2014

Mohammad Hassanzadeh
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Abstract

In this paper we study the cyclic cohomology of certain ×-Hopf algebras: universal enveloping algebras, quantum algebraic tori, the Connes-Moscovici ×-Hopf algebroids and the Kadison bialgebroids. Introducing their stable anti Yetter-Drinfeld modules and cocyclic modules, we compute their cyclic cohomology. Furthermore, we provide a pairing for the cyclic cohomology of ×-Hopf algebras which generalizes the Connes-Moscovici characteristic map to ×-Hopf algebras. This enables us to transfer the ×-Hopf algebra cyclic cocycles to algebra cyclic cocycles.

Keywords

Cyclic cohomologyHopf algebrasNoncommutative geometry19D5516T0511M55
Type
Research Article
Information
Journal of K-Theory , Volume 13 , Issue 1 , February 2014 , pp. 147 - 170
Copyright
Copyright © ISOPP 2014 

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References

BSz.Böhm, G., and Szlachányi, K., Hopf algebroids with bijective antipodes: axioms, integrals, and duals, J. Algebra 274(2) (2004), 708–750.CrossRefGoogle Scholar
BS.Böhm, G., and Stefan, D., (co)cyclic (co)homology of bialgebroids: An approach via (co)monads, Commun. Math. Phys 282 (2008), 239–286.Google Scholar
BS2.Böhm, G., and Stefan, D., Examples of para-cocyclic objects induced by BD-laws, Algebr. Represent. Theory 12(2–5) (2009), 153–180.CrossRefGoogle Scholar
BW.Brzezinski, T., and Wisbauer, R., Corings and Comodules, L. M. S. Lecture Note Series 309, Cambridge University Press, Cambridge, 2003.Google Scholar
C-Book.Connes, A., Noncommutative geometry, Academic Press, 1994.Google Scholar
CM98.Connes, A., and Moscovici, H., Hopf algebras, cyclic cohomology and the transverse index theorem, Comm. Math. Phys. 198 (1998), 199–246.Google Scholar
CM00.Connes, A., and Moscovici, , Cyclic cohomology and Hopf algebra symmetry, Lett. Phys. 52(1) (2000), 1–28.Google Scholar
CM01.Connes, A., and Moscovici, H., Differential cyclic cohomology and Hopf algebraic structures in transverse geometry, in: Essays on geometry and related topics 1–2, Monogr. Enseign. Math. 38, Enseignement Math, Geneva 2001, pp. 217–255.Google Scholar
CM2004.Connes, A., and Moscovici, H., Rankin-Cohen brackets and the Hopf algebra of transverse geometry, Moscow Mathematical Journal 4(1) (2004), 111–130.Google Scholar
HK1.Hadfield, T., and Krähmer, U., Twisted homology of quantum SL(2), K-Theory 34(4) (2005), 327–360.Google Scholar
HK2.Hadfield, T., and Krähmer, U., Twisted homology of quantum SL(2)-part II, J. K-Theory 6(1) (2010), 69–98.Google Scholar
CR.Crainic, M., Cyclic cohomology of Hopf algebras, J. Pure Appl. Algebra 166(1–2) (2002), 29–66.Google Scholar
HKRS1.Hajac, P.M., Khalkhali, M., Rangipour, B., and Sommerhäuser, Y., Stable anti-Yetter-Drinfeld modules, C. R. Math. Acad. Sci. Paris 338 (2004), 587–590.Google Scholar
HKRS2.Hajac, P.M., Khalkhali, M., Rangipour, B., and Sommerhäuser, Y., Hopf-cyclic homology and cohomology with coefficients, C. R. Math. Acad. Sci. Paris 338 (2004), 667–672.Google Scholar
HR1.Hassanzadeh, M., Rangipour, B., Equivariant Hopf Galois extensions and Hopf cyclic cohomology, To apear in Journal of noncommutative Geometry, 33 pages, (2010).Google Scholar
HR2.Hassanzadeh, M., Rangipour, B., Quantum Groupoids and their Hopf Cyclic Cohomology, http://arxiv.org/abs/1108.0030, 48 pages, (2011).Google Scholar
Kad.Kadison, L., Pseudo-Galois extensions and Hopf algebroids, in: Modules and Comod-ules, (Porto conf. 09 (2006) for Robert Wisbauer) eds. Brzezinski, et alBirkhauser Trends in Math. XII (2008), 247–264.Google Scholar
Kay1.Kaygun, A., A survey on Hopf-cyclic cohomology and Connes-Moscovici characteristic map, Contemp. Math. 546 (2001), 171–179, (Proceedings of Henrifest).Google Scholar
Kay2.Kaygun, A., Products in Hopf-cyclic cohomology, Homology, Homotopy and Applications 10(2) (2008), 115–133.Google Scholar
Kay3.Kaygun, A., The universal Hopf cyclic theory, Journal of Noncommutative Geometry 2(2) (2008), 333–351.CrossRefGoogle Scholar
Kay4.Kaygun, A., Uniqueness of pairings in Hopf-cyclic cohomology, J. K-Theory 6(1) 2010, 1–21.CrossRefGoogle Scholar
Ko.Kowalzig, N., Hopf Algebroids and Their Cyclic Theory, Ph. D. Thesis, 2009.Google Scholar
KK.Kowalzig, N., and Krähmer, U., Cyclic structures in algebraic (co)homology theories, Homology, Homotopy and Applications 13(1) (2011), 297–318.CrossRefGoogle Scholar
KP.Khalkhali, M., and Pourkia, A., Hopf cyclic cohomology in braided monoidal categories, Homology, Homotopy and Applications 12(1) (2010), 111–155.CrossRefGoogle Scholar
KR1.Khalkhali, M., and Rangipour, B., A new cyclic module for Hopf algebras, K-Theory 27 (2002), 111–131.Google Scholar
KR2.Khalkhali, M., and Rangipour, B., Cup products in Hopf-cyclic cohomology, C. R. Math. Acad. Sci. Paris 340(1) (2005), 9–14.CrossRefGoogle Scholar
KR3.Khalkhali, M., and Rangipour, B., Para-Hopf algebroids and their cyclic cohomology, Lett. Math. Phys. 70(3) (2004), 259–272.CrossRefGoogle Scholar
KRT.Kustermans, J., Rognes, J., and Tuset, L., The Connes-Moscovici approach to cyclic cohomology for compact quantum groups, K-Theory 26(2) (2002), 101–137.CrossRefGoogle Scholar
Lu.Lu, J.H., Hopf algebroids and quantum groupoids, Int. J. Math. 7 (1996), 47–70.CrossRefGoogle Scholar
NS.Nikonov, I. M., and Sharygin, G. I., On the Hopf-type cyclic cohomology with coefficients, C.-algebras and elliptic theory, Trends Math., Birkhäuser, Basel, 2006, pp. 203–212.Google Scholar
Mil.Miller, H. R., Some Algebraic Aspects of the Adams-Novikov Spectral Sequence, Ph.D. Thesis, Princeton University, 1974.Google Scholar
PVO.Panaite, F., and Oystaeyen, F. Van, Some bialgebroids constructed by Kadison and Connes-Moscovici are isomorphic, Appl. Categorical Structures 14(5–6) (2006), 627–632.Google Scholar
R1.Rangipour, B., Cup products in Hopf cyclic cohomology via cyclic modules I, Homology Homotopy Appl. 10(2) (2008), 273–286.Google Scholar
Sch98.Schauenburg, P., Bialgebras over noncommutative rings and a structure theorem for Hopfbimodules, Appl. Categorical Structures 6 (1998), 193–222.CrossRefGoogle Scholar
Sch.Schauenburg, P., Duals and doubles of quantum groupoids (×R-Hopf algebras), in: Andruskiewitsch, N., Ferrer-Santos, W.R. and Schneider, H.-J. (eds.) Contemp. Math. 267 (2000), 273–293.Google Scholar