Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-08T13:21:16.607Z Has data issue: false hasContentIssue false
  • English
  • Français

Twisted Homology of Quantum SL(2) - Part II

Published online by Cambridge University Press:  02 October 2009

Tom Hadfield  and
Ulrich Krähmer
Tom Hadfield
Affiliation:
Gloucester Research, Whittington House, 19-30 Alfred Place, London WC1E 7EA, UK; [email protected]
Ulrich Krähmer
Affiliation:
University of Glasgow, Department of Mathematics, University Gardens, Glasgow G12 8QW, UK; [email protected]
Get access

Abstract

We complete the calculation of the twisted cyclic homology of the quantised coordinate ring = ℂq [SL(2)] of SL(2) that we began in [14]. In particular, a nontrivial cyclic 3-cocycle is constructed which also has a nontrivial class in Hochschild cohomology and thus should be viewed as a noncommutative geometry analogue of a volume form.

Keywords

Hochschild homologyCyclic homologyquantum groups
Type
Research Article
Information
Journal of K-Theory , Volume 6 , Issue 1 , August 2010 , pp. 69 - 98
Copyright
Copyright © ISOPP 2009

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Apel, J., Uwe, K., FELIX – an assistant for algebraists, see http://felix.hgb-leipzig.deGoogle Scholar
2.Brown, K., Zhang, J.: Dualising complexes and twisted Hochschild (co)homology for Noetherian Hopf algebras, J. Algebra 320, no. 5, 1814–1850 (2008).CrossRefGoogle Scholar
3.Cartan, H., Eilenberg, S., Homological algebra, Princeton University Press (1956).Google Scholar
4.Connes, A., Cohomologie cyclique et foncteurs Extn, C. R. Acad. Sci. Paris I Math. 296, no. 23, 953–958 (1983).Google Scholar
5.Connes, A., Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math. 62, 257–360 (1985).Google Scholar
6.Connes, A., Cyclic cohomology, quantum group symmetries and the local index formula for SUq(2), J. Inst. Math. Jussieu 3 no. 1, 17–68 (2004).Google Scholar
7.Connes, A., Marcolli, M., Noncommutative Geometry, Quantum Fields and Motives, AMS (2008).Google Scholar
8.Connes, A., Moscovici, H., Hopf algebras, cyclic cohomology and the transverse index theorem, Comm. Math. Phys. 198, no. 1, 199–246 (1998).Google Scholar
9.Dabrowski, L., Landi, G., Sitarz, A., van Suijlekom, W., Varilly, J., The local index formula for SU q(2), K-Theory 35, no. 3-4, 375–394 (2005).Google Scholar
10.Etingof, P., Dolgushev, V., Hochschild cohomology of quantized symplectic orbifolds and the Chen-Ruan cohomology, Int. Math. Res. Not. 27 1657–1688 (2005).Google Scholar
11.Farinati, M., Hochschild duality, localization, and smash products, J. Algebra 284, no. 1, 415–434 (2005).Google Scholar
12.Feng, P., Tsygan, B., Hochschild and cyclic homology of quantum groups, Comm. Math. Phys. 140, no. 3, 481–521 (1991).CrossRefGoogle Scholar
13.Getzler, E., Jones, J., The cyclic homology of crossed product algebras, J. Reine Angew. Math., 445, 161–174 (1993).Google Scholar
14.Hadfield, T., Krähmer, U., Twisted homology of quantum SL(2), K-theory 34, no. 4, 327–360 (2005)Google Scholar
15.Hadfield, T., Krähmer, U., On the Hochschild homology of quantum SL(N), Comptes Rendus Acad. Sci. Paris, Ser. I 343, 9–13 (2006)CrossRefGoogle Scholar
16.Hadfield, T., Krähmer, U., Braided homology of quantum groups, to appear in J. K-theory.Google Scholar
17.Hajac, P., Khalkhali, M., Rangipour, B., Sommerhäuser, Y., Hopf-cyclic homology and cohomology with coefficients, C. R. Math. Acad. Sci. Paris 338, no. 9, 667–672 (2004).CrossRefGoogle Scholar
18.Heckenberger, I., Classification of left-covariant differential calculi on the quantum group SLq(2), J. Algebra 237, 203–237 (2001).Google Scholar
19.Joseph, A., Quantum groups and their primitive ideals, Springer (1995).CrossRefGoogle Scholar
20.Klimyk, A., Schmüdgen, K., Quantum groups and their representations, Springer (1997).Google Scholar
21.Kowalzig, N., Krähmer, U., Duality and products in algebraic (co)homology theories, arXiv:0812.4312 (2008).Google Scholar
22.Krähmer, U., Poincaré duality in Hochschild (co)homology, in Caenepeel, S., Van Oystayen, F. (eds.): New techniques in Hopf algebras and graded ring theory, 117–126 (2007).Google Scholar
23.Krähmer, U., twisted2.flx, http://www.maths.gla.ac.uk/~ukraehmer/twisted2.flxGoogle Scholar
24.Kustermans, J., Murphy, G., Tuset, L., Differential calculi over quantum groups and twisted cyclic cocycles, J. Geom. Phys. 44, no. 4, 570–594 (2003).Google Scholar
25.Launois, S., Lenagan, T., The first Hochschild cohomology group of quantum matrices and the quantum special linear group, J. Noncomm. Geom. 1, no. 3, 281–309 (2007).Google Scholar
26.Masuda, T., Nakagami, Y., Watanabe, J., Noncommutative differential geometry on the quantum SU(2), Part I, K-theory 4, no. 2, 157–180 (1990).CrossRefGoogle Scholar
27.Loday, J.-L., Cyclic homology, Springer-Verlag (1998).Google Scholar
28.Neshveyev, S., Tuset, L., A local index formula for the quantum sphere, Comm. Math. Phys. 254, no. 2, 323–341 (2005).Google Scholar
29.Nest, R., Tsygan, B., On the cohomology ring of an algebra, Progr. Math. 172, 337–370 (1999).Google Scholar
30.Schmüdgen, K., Wagner, E., Dirac operator and a twisted cyclic cocycle on the standard Podle's quantum sphere, J. Reine Angew. Math. 574, 219–235 (2004).Google Scholar
31.Tsygan, B., Homology of matrix Lie algebras over rings and the Hochschild homology, Uspekhi Mat. Nauk 38, no. 2 (230), 217–218 (1983).Google Scholar
32.Van den Bergh, M., A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Amer. Math. Soc. 126, no. 5, 1345–1348 (1998). Erratum: Proc. Amer. Math. Soc. 130, no. 9, 2809–2810 (electronic) (2002).CrossRefGoogle Scholar
33.Weibel, C., An introduction to homological algebra, Cambridge Univ. Press (1995).Google Scholar