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

SIGMA 16 (2020), 098, 18 pages      arXiv:2003.10305      https://doi.org/10.3842/SIGMA.2020.098
Contribution to the Special Issue on Noncommutative Manifolds and their Symmetries in honour of Giovanni Landi

Twisted Hochschild Homology of Quantum Flag Manifolds and Kähler Forms

Marco Matassa
OsloMet - Oslo Metropolitan University, Oslo, Norway

Received March 31, 2020, in final form September 25, 2020; Published online October 03, 2020

Abstract
We study the twisted Hochschild homology of quantum flag manifolds, the twist being the modular automorphism of the Haar state. We prove that every quantum flag manifold admits a non-trivial class in degree two, with an explicit representative defined in terms of a certain projection. The corresponding classical two-form, via the Hochschild-Kostant-Rosenberg theorem, is identified with a Kähler form on the flag manifold.

Key words: quantum flag manifolds; twisted Hochschild homology; Kähler forms.

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References

  1. Akrami S.E., Majid S., Braided cyclic cocycles and nonassociative geometry, J. Math. Phys. 45 (2004), 3883-3911, arXiv:math.QA/0406005.
  2. Baez J.C., Hochschild homology in a braided tensor category, Trans. Amer. Math. Soc. 344 (1994), 885-906.
  3. Beggs E., Paul Smith S., Non-commutative complex differential geometry, J. Geom. Phys. 72 (2013), 7-33, arXiv:1209.3595.
  4. Borel A., Hirzebruch F., Characteristic classes and homogeneous spaces. I, Amer. J. Math. 80 (1958), 458-538.
  5. Brown K.A., Zhang J.J., Dualising complexes and twisted Hochschild (co)homology for Noetherian Hopf algebras, J. Algebra 320 (2008), 1814-1850, arXiv:math.RA/0603732.
  6. Čap A., Slovák J., Parabolic geometries. I. Background and general theory, Mathematical Surveys and Monographs, Vol. 154, Amer. Math. Soc., Providence, RI, 2009.
  7. Connes A., Noncommutative differential geometry, Inst. Hautes Études Sci. Publ. Math. (1985), 41-144.
  8. Dolgushev V., The Van den Bergh duality and the modular symmetry of a Poisson variety, Selecta Math. (N.S.) 14 (2009), 199-228, arXiv:math.QA/0612288.
  9. Feng P., Tsygan B., Hochschild and cyclic homology of quantum groups, Comm. Math. Phys. 140 (1991), 481-521.
  10. Hadfield T., Twisted cyclic homology of all Podleś quantum spheres, J. Geom. Phys. 57 (2007), 339-351, arXiv:math.QA/0405243.
  11. Hadfield T., Krähmer U., Twisted homology of quantum ${\rm SL}(2)$, $K$-Theory 34 (2005), 327-360, arXiv:math.QA/0405249.
  12. Hadfield T., Krähmer U., Braided homology of quantum groups, J. K-Theory 4 (2009), 299-332, arXiv:math.QA/0701193.
  13. Heckenberger I., Kolb S., De Rham complex for quantized irreducible flag manifolds, J. Algebra 305 (2006), 704-741, arXiv:math.QA/0307402.
  14. Hochschild G., Kostant B., Rosenberg A., Differential forms on regular affine algebras, Trans. Amer. Math. Soc. 102 (1962), 383-408.
  15. Hong J., Kang S.J., Introduction to quantum groups and crystal bases, Graduate Studies in Mathematics, Vol. 42, Amer. Math. Soc., Providence, RI, 2002.
  16. Huybrechts D., Complex geometry: an introduction, Universitext, Springer-Verlag, Berlin, 2005.
  17. Karoubi M., Homologie cyclique et $K$-théorie, Astérisque 149 (1987), 147 pages.
  18. Klimyk A., Schmüdgen K., Quantum groups and their representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997.
  19. Knapp A.W., Lie groups beyond an introduction, Progress in Mathematics, Vol. 140, Birkhäuser Boston, Inc., Boston, MA, 1996.
  20. Kustermans J., Murphy G.J., Tuset L., Differential calculi over quantum groups and twisted cyclic cocycles, J. Geom. Phys. 44 (2003), 570-594, arXiv:math.QA/0110199.
  21. Loday J.-L., Cyclic homology, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Vol. 301, Springer-Verlag, Berlin, 1998.
  22. Matassa M., Kähler structures on quantum irreducible flag manifolds, J. Geom. Phys. 145 (2019), 103477, 16 pages, arXiv:1901.09544.
  23. Matassa M., Twisted Hochschild homology of quantum flag manifolds: 2-cycles from invariant projections, J. Algebra Appl., to appear, arXiv:1703.00725.
  24. Ó Buachalla R., Noncommutative complex structures on quantum homogeneous spaces, J. Geom. Phys. 99 (2016), 154-173, arXiv:1108.2374.
  25. Ó Buachalla R., Noncommutative Kähler structures on quantum homogeneous spaces, Adv. Math. 322 (2017), 892-939, arXiv:1602.08484.
  26. Sepanski M.R., Compact Lie groups, Graduate Texts in Mathematics, Vol. 235, Springer, New York, 2007.
  27. Snow D.M., Homogeneous vector bundles, in Group Actions and Invariant Theory (Montreal, PQ, 1988), CMS Conf. Proc., Vol. 10, Amer. Math. Soc., Providence, RI, 1989, 193-205.
  28. Stokman J.V., Dijkhuizen M.S., Quantized flag manifolds and irreducible $*$-representations, Comm. Math. Phys. 203 (1999), 297-324, arXiv:math.QA/9802086.
  29. van den Bergh M., A relation between Hochschild homology and cohomology for Gorenstein rings, Proc. Amer. Math. Soc. 126 (1998), 1345-1348.
  30. Woronowicz S.L., Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math. Phys. 122 (1989), 125-170.

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