### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 9 (2013), 051, 15 pages      arXiv:1111.6328      https://doi.org/10.3842/SIGMA.2013.051

### Twisted Cyclic Cohomology and Modular Fredholm Modules

Adam Rennie a, Andrzej Sitarz b, c and Makoto Yamashita d
a) School of Mathematics and Applied Statistics, University of Wollongong, Wollongong NSW 2522, Australia
b) Institute of Mathematics of the Polish Academy of Sciences, ul. Sniadeckich 8, Warszawa, 00-950 Poland
c) Institute of Physics, Jagiellonian University, ul. Reymonta 4, 30-059 Kraków, Poland
d) Department of Mathematics, Ochanomizu University, Otsuka 2-1-1, Tokyo, Japan

Received January 24, 2013, in final form July 22, 2013; Published online July 30, 2013

Abstract
Connes and Cuntz showed in [Comm. Math. Phys. 114 (1988), 515-526] that suitable cyclic cocycles can be represented as Chern characters of finitely summable semifinite Fredholm modules. We show an analogous result in twisted cyclic cohomology using Chern characters of modular Fredholm modules. We present examples of modular Fredholm modules arising from Podleś spheres and from SUq(2).

Key words: twisted cyclic cohomology; spectral triple; modular theory; KMS weight.

pdf (409 kb)   tex (23 kb)

References

1. Carey A.L., Gayral V., Rennie A., Sukochev F.A., Index theory for locally compact noncommutative geometries, Mem. Amer. Math. Soc., to appear, arXiv:1107.0805.
2. Carey A.L., Phillips J., Spectral flow in Fredholm modules, eta invariants and the JLO cocycle, K-Theory 31 (2004), 135-194, math.KT/0308161.
3. Carey A.L., Phillips J., Rennie A., Sukochev F.A., The local index formula in semifinite von Neumann algebras. II. The even case, Adv. Math. 202 (2006), 517-554, math.OA/0411021.
4. Connes A., Noncommutative geometry, Academic Press Inc., San Diego, CA, 1994.
5. Connes A., Cuntz J., Quasi homomorphismes, cohomologie cyclique et positivité, Comm. Math. Phys. 114 (1988), 515-526.
6. Connes A., Moscovici H., Type III and spectral triples, in Traces in Number Theory, Geometry and Quantum Fields, Aspects Math., Vol. E38, Friedr. Vieweg, Wiesbaden, 2008, 57-71, math.OA/0609703.
7. Dąbrowski L., Landi G., Sitarz A., van Suijlekom W., Várilly J.C., The Dirac operator on SUq(2), Comm. Math. Phys. 259 (2005), 729-759, math.OA/0411609.
8. Fathizadeh F., Khalkhali M., The algebra of formal twisted pseudodifferential symbols and a noncommutative residue, Lett. Math. Phys. 94 (2010), 41-61, arXiv:0810.0484.
9. Gracia-Bondía J.M., Várilly J.C., Figueroa H., Elements of noncommutative geometry, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Boston Inc., Boston, MA, 2001.
10. Hadfield T., Twisted cyclic homology of all Podleś quantum spheres, J. Geom. Phys. 57 (2007), 339-351, math.QA/0405243.
11. Hadfield T., Krähmer U., Twisted homology of quantum SL(2) - Part II, J. K-Theory 6 (2010), 69-98, arXiv:0711.4102.
12. Kaad J., Senior R., A twisted spectral triple for quantum SU(2), J. Geom. Phys. 62 (2012), 731-739, arXiv:1109.2326.
13. Krähmer U., Rennie A., Senior R., A residue formula for the fundamental Hochschild 3-cocycle for SUq(2), J. Lie Theory 22 (2012), 557-585, arXiv:1105.5366.
14. Masuda T., Nakagami Y., Watanabe J., Noncommutative differential geometry on the quantum SU(2). I. An algebraic viewpoint, K-Theory 4 (1990), 157-180.
15. Neshveyev S., Tuset L., A local index formula for the quantum sphere, Comm. Math. Phys. 254 (2005), 323-341, math.QA/0309275.
16. Neshveyev S., Tuset L., Hopf algebra equivariant cyclic cohomology, K-theory and index formulas, K-Theory 31 (2004), 357-378, math.KT/0304001.
17. Rennie A., Senior R., The resolvent cocycle in twisted cyclic cohomology and a local index formula for the Podleś sphere, J. Noncommut. Geom., to appear, arXiv:1111.5862.
18. Sheu A.J.L., Quantization of the Poisson SU(2) and its Poisson homogeneous space - the 2-sphere, Comm. Math. Phys. 135 (1991), 217-232.
19. Takesaki M., Theory of operator algebras. II, Encyclopaedia of Mathematical Sciences, Vol. 125, Springer-Verlag, Berlin, 2003.
20. van Suijlekom W., Dąbrowski L., Landi G., Sitarz A., Várilly J.C., The local index formula for SUq(2), K-Theory 35 (2005), 375-394, math.OA/0501287.
21. Wagner E., On the noncommutative spin geometry of the standard Podleś sphere and index computations, J. Geom. Phys. 59 (2009), 998-1016, arXiv:0707.3403.