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

SIGMA 18 (2022), 070, 21 pages      arXiv:2202.09842      https://doi.org/10.3842/SIGMA.2022.070

Bimodule Connections for Relative Line Modules over the Irreducible Quantum Flag Manifolds

Alessandro Carotenuto and Réamonn Ó Buachalla
Mathematical Institute of Charles University, Sokolovská 83, Prague, Czech Republic

Received April 14, 2022, in final form September 18, 2022; Published online October 01, 2022

Abstract
It was recently shown (by the second author and Díaz García, Krutov, Somberg, and Strung) that every relative line module over an irreducible quantum flag manifold $\mathcal{O}_q(G/L_S)$ admits a unique $\mathcal{O}_q(G)$-covariant connection with respect to the Heckenberger-Kolb differential calculus $\Omega^1_q(G/L_S)$. In this paper we show that these connections are bimodule connections with an invertible associated bimodule map. This is proved by applying general results of Beggs and Majid, on principal connections for quantum principal bundles, to the quantum principal bundle presentation of the Heckenberger-Kolb calculi recently constructed by the authors and Díaz García. Explicit presentations of the associated bimodule maps are given first in terms of generalised quantum determinants, then in terms of the FRT presentation of the algebra $\mathcal{O}_q(G)$, and finally in terms of Takeuchi's categorical equivalence for relative Hopf modules.

Key words: quantum groups; noncommutative geometry; bimodule connections; quantum principal bundles; quantum flag manifolds; complex geometry.

pdf (478 kb)   tex (27 kb)  

References

  1. Aschieri P., Schenkel A., Noncommutative connections on bimodules and Drinfeld twist deformation, Adv. Theor. Math. Phys. 18 (2014), 513-612, arXiv:1210.0241.
  2. Beggs E., Majid S., Spectral triples from bimodule connections and Chern connections, J. Noncommut. Geom. 11 (2017), 669-701, arXiv:1508.04808.
  3. Beggs E., Majid S., Quantum Riemannian geometry, Grundlehren Math. Wiss., Vol. 355, Springer, Cham, 2020.
  4. Bhowmick J., Goswami D., Landi G., On the Koszul formula in noncommutative geometry, Rev. Math. Phys. 32 (2020), 2050032, 33 pages, arXiv:1910.09306.
  5. Brzeziński T., Janelidze G., Maszczyk T., Galois structures, in Lecture Notes on Noncommutative Geometry and Quantum Groups, Editor P.M. Hajac, 2008, 707-711, Notes by P. Witkowski, available at http://www.mimuw.edu.pl/ pwit/toknotes/toknotes.pdf.
  6. Brzeziński T., Majid S., Quantum group gauge theory on quantum spaces, Comm. Math. Phys. 157 (1993), 591-638, arXiv:hep-th/9208007.
  7. Carotenuto A., Díaz García F., Ó Buachalla R., A Borel-Weil theorem for the irreducible quantum flag manifolds, Int. Math. Res. Not., to appear, arXiv:2112.03305.
  8. Carotenuto A., Mrozinski C., Ó Buachalla R., A Borel-Weil theorem for the quantum Grassmannians, arXiv:1611.07969.
  9. Carotenuto A., Ó Buachalla R., Principal pairs of quantum homogeneous spaces, arXiv:2111.11284.
  10. Díaz García F., Krutov A., Ó Buachalla R., Somberg P., Strung K.R., Positive line bundles over the irreducible quantum flag manifolds, arXiv:1912.08802.
  11. Díaz García F., Krutov A., Ó Buachalla R., Somberg P., Strung K.R., Holomorphic relative Hopf modules over the irreducible quantum flag manifolds, Lett. Math. Phys. 111 (2021), 10, 24 pages, arXiv:2005.09652.
  12. Dubois-Violette M., Lectures on graded differential algebras and noncommutative geometry, in Noncommutative Differential Geometry and its Applications to Physics (Shonan, 1999), Math. Phys. Stud., Vol. 23, Kluwer Acad. Publ., Dordrecht, 2001, 245-306, arXiv:math.QA/9912017.
  13. Dubois-Violette M., Madore J., Masson T., Mourad J., Linear connections on the quantum plane, Lett. Math. Phys. 35 (1995), 351-358, arXiv:hep-th/9410199.
  14. Dubois-Violette M., Masson T., On the first-order operators in bimodules, Lett. Math. Phys. 37 (1996), 467-474, arXiv:q-alg/9507028.
  15. Dubois-Violette M., Michor P.W., Connections on central bimodules in noncommutative differential geometry, J. Geom. Phys. 20 (1996), 218-232, arXiv:q-alg/9503020.
  16. Ghobadi A., Hopf algebroids, bimodule connections and noncommutative geometry, arXiv:2001.08673.
  17. 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.
  18. Heckenberger I., Kolb S., De Rham complex for quantized irreducible flag manifolds, J. Algebra 305 (2006), 704-741, arXiv:math.QA/0307402.
  19. Humphreys J.E., Introduction to Lie algebras and representation theory, Grad. Texts in Math., Vol. 9, Springer-Verlag, New York - Berlin, 1972.
  20. Khalkhali M., Landi G., van Suijlekom W.D., Holomorphic structures on the quantum projective line, Int. Math. Res. Not. 2011 (2011), 851-884, arXiv:0907.0154.
  21. Khalkhali M., Moatadelro A., The homogeneous coordinate ring of the quantum projective plane, J. Geom. Phys. 61 (2011), 276-289, arXiv:1007.3255.
  22. Klimyk A., Schmüdgen K., Quantum groups and their representations, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1997.
  23. Madore J., An introduction to noncommutative differential geometry and its physical applications, 2nd ed., London Math. Soc. Lecture Note Ser., Vol. 257, Cambridge University Press, Cambridge, 1999.
  24. Madore J., Masson T., Mourad J., Linear connections on matrix geometries, Classical Quantum Gravity 12 (1995), 1429-1440, arXiv:hep-th/9411127.
  25. Majid S., Noncommutative Riemannian and spin geometry of the standard $q$-sphere, Comm. Math. Phys. 256 (2005), 255-285, arXiv:math.QA/0307351.
  26. Matassa M., Fubini-Study metrics and Levi-Civita connections on quantum projective spaces, Adv. Math. 393 (2021), 108101, 56 pages, arXiv:2010.03291.
  27. Năstăsescu C., Van Oystaeyen F., Methods of graded rings, Lecture Notes in Math., Vol. 1836, Springer-Verlag, Berlin, 2004.
  28. Ó Buachalla R., Quantum bundle description of quantum projective spaces, Comm. Math. Phys. 316 (2012), 345-373, arXiv:1105.1768.
  29. Ó Buachalla R., Noncommutative complex structures on quantum homogeneous spaces, J. Geom. Phys. 99 (2016), 154-173, arXiv:1108.2374.
  30. Reshetikhin N.Yu., Takhtadzhyan L.A., Faddeev L.D., Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1 (1990), 193-225.
  31. Stokman J.V., The quantum orbit method for generalized flag manifolds, Math. Res. Lett. 10 (2003), 469-481, arXiv:math.QA/0206245.
  32. Takeuchi M., Relative Hopf modules - equivalences and freeness criteria, J. Algebra 60 (1979), 452-471.

Previous article  Next article  Contents of Volume 18 (2022)