Quantum Dimension and Quantum Projective Spaces

Marco Matassa

Symmetry, integrability and geometry: methods and applications, Tome 10 (2014)

Résumé

We show that the family of spectral triples for quantum projective spaces introduced by D'Andrea and Da̧browski, which have spectral dimension equal to zero, can be reconsidered as modular spectral triples by taking into account the action of the element $K_{2\rho}$ or its inverse. The spectral dimension computed in this sense coincides with the dimension of the classical projective spaces. The connection with the well known notion of quantum dimension of quantum group theory is pointed out.

Détail

Keywords: quantum projective spaces; quantum dimension; modular spectral triples.

Bibliographie

[1] Brown K. A., Zhang J. J., “Dualising complexes and twisted {H}ochschild (co)homology for {N}oetherian {H}opf algebras”, J. Algebra, 320 (2008), 1814–1850, arXiv: math.RA/0603732 | DOI | MR | Zbl

[2] Carey A. L., Phillips J., Rennie A., “Twisted cyclic theory and an index theory for the gauge invariant {KMS} state on the {C}untz algebra {$O_n$}”, J. $K$-Theory, 6 (2010), 339–380, arXiv: 0801.4605 | DOI | MR | Zbl

[3] Carey A. L., Rennie A., Sedaev A., Sukochev F., “The {D}ixmier trace and asymptotics of zeta functions”, J. Funct. Anal., 249 (2007), 253–283, arXiv: math.OA/0611629 | DOI | MR | Zbl

[4] Carey A. L., Rennie A., Tong K., “Spectral flow invariants and twisted cyclic theory for the {H}aar state on {${\rm SU}_q(2)$}”, J. Geom. Phys., 59 (2009), 1431–1452, arXiv: 0802.0317 | DOI | MR | Zbl

[5] Connes A., Noncommutative geometry, Academic Press, Inc., San Diego, CA, 1994 | MR | Zbl

[6] D'Andrea F., D{a̧}browski L., “Dirac operators on quantum projective spaces”, Comm. Math. Phys., 295 (2010), 731–790, arXiv: 0901.4735 | DOI | MR | Zbl

[7] D'Andrea F., D{a̧}browski L., Landi G., “The noncommutative geometry of the quantum projective plane”, Rev. Math. Phys., 20 (2008), 979–1006, arXiv: 0712.3401 | DOI | MR | Zbl

[8] D{a̧}browski L., Sitarz A., “Dirac operator on the standard {P}odleś quantum sphere”, Noncommutative Geometry and Quantum Groups ({W}arsaw, 2001), Banach Center Publ., 61, eds. P. M. Hajac, W. Pusz, Polish Acad. Sci., Warsaw, 2003, 49–58, arXiv: math.QA/0209048 | DOI

[9] Fuchs J., Affine {L}ie algebras and quantum groups. An introduction, with applications in conformal field theory, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 1995 | MR | Zbl

[10] Gracia-Bondía J. M., Várilly J. C., Figueroa H., Elements of noncommutative geometry, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Boston Inc., Boston, MA, 2001 | MR | Zbl

[11] Hadfield T., “Twisted cyclic homology of all {P}odleś quantum spheres”, J. Geom. Phys., 57 (2007), 339–351, arXiv: math.QA/0405243 | DOI | MR | Zbl

[12] Hadfield T., Krähmer U., “Twisted homology of quantum {${\rm SL}(2)$}”, $K$-Theory, 34 (2005), 327–360, arXiv: math.QA/0405249 | DOI | MR | Zbl

[13] Heckenberger I., Kolb S., “The locally finite part of the dual coalgebra of quantized irreducible flag manifolds”, Proc. London Math. Soc., 89 (2004), 457–484, arXiv: math.QA/0301244 | DOI | MR | Zbl

[14] Kaad J., On modular semifinite index theory, arXiv: 1111.6546

[15] Kaad J., Senior R., “A twisted spectral triple for quantum {${\rm SU}(2)$}”, J. Geom. Phys., 62 (2012), 731–739, arXiv: 1109.2326 | DOI | MR | Zbl

[16] Klimyk A., Schmüdgen K., Quantum groups and their representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997 | DOI | MR | Zbl

[17] Krähmer U., “Dirac operators on quantum flag manifolds”, Lett. Math. Phys., 67 (2004), 49–59, arXiv: math.QA/0305071 | DOI | MR

[18] Krähmer U., “On the {H}ochschild (co)homology of quantum homogeneous spaces”, Israel J. Math., 189 (2012), 237–266, arXiv: 0806.0267 | DOI | MR

[19] Krähmer U., Tucker-Simmons M., On the Dolbeault–Dirac operator of quantized symmetric spaces, arXiv: 1307.7106

[20] Krähmer U., Wagner E., “A residue formula for the fundamental {H}ochschild class on the {P}odleś sphere”, J. $K$-Theory, 12 (2013), 257–271, arXiv: 1008.1830 | DOI | MR

[21] Matassa M., Non-commutative integration, zeta functions and the Haar state for ${\rm SU}_{q}(2)$, arXiv: 1310.7477

[22] Neshveyev S., Tuset L., “A local index formula for the quantum sphere”, Comm. Math. Phys., 254 (2005), 323–341, arXiv: math.QA/0309275 | DOI | MR | Zbl