Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-12T19:49:20.259Z Has data issue: false hasContentIssue false
  • English
  • Français

A residue formula for the fundamental Hochschild class on the Podleś sphere

Published online by Cambridge University Press:  07 August 2013

Ulrich Krähmer  and
Elmar Wagner
Get access

Abstract

The fundamental Hochschild cohomology class of the standard Podleś quantum sphere is expressed in terms of the spectral triple of Dąabrowski and Sitarz by means of a residue formula.

Keywords

quantum sphereHochschild cohomologyfundamental classresidue formulasPrimary 58J42Secondary 16E4058B32
Type
Research Article
Information
Journal of K-Theory , Volume 12 , Issue 2 , October 2013 , pp. 257 - 271
Copyright
Copyright © ISOPP 2013 

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.Cartan, Henri, Eilenberg, Samuel, Homological algebra. Princeton University Press 1956.Google Scholar
2.Chakraborty, Partha Sarathi, Pal, Arupkumar, On equivariant Dirac operators for SUq (2). Proc. Indian Acad. Sci. Math. Sci. 116(4) (2006), 531–541.Google Scholar
3.Cherednik, Ivan, On q-analogues of Riemann's zeta function. Selecta Math. (N.S.) 7 (4) (2001), 447–491.Google Scholar
4.Connes, Alain, Moscovici, Henri, Type III and spectral triples. Traces in number theory, geometry and quantum fields, 57–71, Aspects Math., E38, Friedr. Vieweg, Wiesbaden, 2008.Google Scholar
5.Connes, Alain, Noncommutative geometry, Academic Press 1994.Google Scholar
6.Connes, Alain, Moscovici, Henri, The local index formula in noncommutative geometry. Geom. Funct. Anal. 5(2) (1995), 174–243.Google Scholar
7.D'Andrea, Francesco, Dabrowski, Ludwik, Landi, Giovanni, The noncommutative geometry of the quantum projective plane. Rev. Math. Phys. 20(8) (2008), 979–1006.Google Scholar
8.Dabrowski, Ludwik, The local index formula for quantum SU(2). Traces in number theory, geometry and quantum fields, 99–110, Aspects Math., E38, Friedr. Vieweg, Wiesbaden, 2008.Google Scholar
9.Dąbrowski, Ludwik, D'Andrea, Francesco, Landi, Giovanni, Wagner, Elmar, Dirac operators on all Podleś quantum spheres. J. Noncommut. Geom. 1(2) (2007), 213–239.Google Scholar
10.Dabrowski, Ludwik, Sitarz, Andrzej, Dirac operator on the standard Podleś quantum sphere. Noncommutative geometry and quantum groups (Warsaw, 2001), 49–58, Banach Center Publ. 61, Polish Acad. Sci., Warsaw, 2003.Google Scholar
11.Decker, Wolfram, Greuel, Gert-Martin, Pfister, Gerhard, Schonemann, Hans, Singular - A computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2010).Google Scholar
12.Hadfield, Tom, Twisted cyclic homology of all Podleś quantum spheres. J. Geom. Phys. 57(2) (2007), 339–351.Google Scholar
13.Higson, Nigel, Meromorphic continuation of zeta functions associated to elliptic operators. Operator algebras, quantization, and noncommutative geometry, 129–142, Contemp. Math. 365, Amer. Math. Soc., Providence, RI, 2004.CrossRefGoogle Scholar
14.Klimyk, Anatoli, Schmüdgen, Konrad, Quantum groups and their representations. Texts and Monographs in Physics. Springer-Verlag, Berlin, 1997.Google Scholar
15.Krähmer, Ulrich, Dirac operators on quantum flag manifolds. Lett. Math. Phys. 67 (1) (2004), 49–59.Google Scholar
16.Krähmer, Ulrich, On the Hochschild (co)homology of Quantum Homogeneous Spaces. Israel J. Math. 189 (2012), 237–266.Google Scholar
17.Krähmer, Ulrich, Poincaré duality in Hochschild (co)homology, New techniques in Hopf algebras and graded ring theory, K. Vlaam. Acad. Belgie Wet. Kunsten (KVAB), Brussels, 2007, pp. 117–125.Google Scholar
18.Krähmer, Ulrich, The Hochschild cohomology ring of the standard Podleś quantum sphere. Arab. J. Sci. Eng. Sect. C Theme Issues 33(2) (2008), 325–335.Google Scholar
19.http://www.maths.gla.ac.uk/~ljukraehmer/podneu.sinGoogle Scholar
20.Loday, Jean-Louis, Cyclic homology, second ed., Grundlehren der Mathematischen Wissenschaften 301, Springer-Verlag, Berlin, 1998.Google Scholar
21.Masuda, Tetsuya, Nakagami, Yoshiomi, Watanabe, Junsei, Noncommutative differential geometry on the quantum two sphere of Podleś I. An algebraic viewpoint. K-Theory 5(2) (1991), 151–175.Google Scholar
22.Neshveyev, Sergey, Tuset, Lars, A local index formula for the quantum sphere. Comm. Math. Phys. 254(2) (2005), 323–341.Google Scholar
23.Nest, Ryszard, Voigt, Christian, Equivariant Poincaré duality for quantum group actions. J. Funct. Anal. 258(5) (2010), 1466–1503.Google Scholar
24.Podleś, Piotr, Quantum spheres. Lett. Math. Phys. 14(3) (1987), 193–202.CrossRefGoogle Scholar
25.Schmüdgen, Konrad, Wagner, Elmar, Dirac operator and a twisted cyclic cocycle on the standard Podleś quantum sphere. J. Reine Angew. Math. 574 (2004), 219–235.Google Scholar
26.Sitarz, Andrzej, Equivariant spectral triples. Noncommutative geometry and quantum groups (Warsaw, 2001), 231–263, Banach Center Publ. 61, Polish Acad. Sci., Warsaw, 2003.Google Scholar
27.Ueno, Kimio, Nishizawa, Michitomo, Quantum groups and zeta-functions. Quantum groups (Karpacz, 1994), 115–126, PWN, Warsaw, 1995.Google Scholar
28.den Bergh, Michel Van, A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Amer. Math. Soc. 126(5) (1998), 1345–1348, Erratum: Proc. Amer. Math. Soc. 130 (9), 2809–2810 (electronic) (2002).Google Scholar
29.Wagner, Elmar, On the noncommutative spin geometry of the standard Podleś sphere and index computations. J. Geom. Phys. 59(7) (2009), 998–1016.Google Scholar