Abstract
We construct equivariant vector bundles over quantum projective spaces using parabolic Verma modules over the quantum general linear group. Using an alternative realization of the quantized coordinate ring of the projective space as a subalgebra in the algebra of functions on the quantum group, we reformulate quantum vector bundles in terms of quantum symmetric pairs. We thus prove the complete reducibility of modules over the corresponding coideal stabilizer subalgebras, via the quantum Frobenius reciprocity.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
F. Bayen, M. Flato, C. Fronsdal, A. Lichnerowicz, and D. Sternheimer, “Deformation theory and quantization: I. Deformations of symplectic structures,” Ann. Phys., 111, 61–110 (1978).
J. Donin and A. Mudrov, “Dynamical Yang–Baxter equation and quantum vector bundles,” Commun. Math. Phys., 254, 719–760 (2005).
A. Mudrov, “Quantum conjugacy classes of simple matrix groups,” Commun. Math. Phys., 272, 635–660 (2007).
T. Ashton and A. Mudrov, “Representations of quantum conjugacy classes of orthosymplectic groups,” J. Math. Sci., 213, 637–650 (2016).
A. Mudrov, “Contravariant form on tensor product of highest weight modules,” arXiv:1709.08394v6 [math.QA] (2017).
M. Semenov–Tian–Shansky, “Poisson–Lie groups, quantum duality principle, and the quantum double,” in: Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups (Contemp. Math., Vol. 175, P. J. Sally Jr., M. Flato, J. Lepowsky, N. Reshetikhin, and G. J. Zuckerman, eds.), Amer. Math. Soc., Providence, R. I. (1994), pp. 219–248.
J.–P. Serre, “Faisceaux algébriques cohérents,” Ann. Math., 61, 197–278 (1955).
R. Swan, “Vector bundles and projective modules,” Trans. Amer. Math. Soc., 105, 264–277 (1962).
G. Letzter, “Symmetric pairs for quantized enveloping algebras,” J. Algebra, 220, 729–767 (1999).
S. Kolb, “Quantum symmetric Kac–Moody pairs,” Adv. Math., 267, 395–469 (2014).
J. Donin and A. Mudrov, “Method of quantum characters in equivariant quantization,” Commun. Math. Phys., 234, 533–555 (2003); arXiv:math/0204298v3 (2002).
V. G. Drinfeld, “Quantum groups,” J. Soviet Math., 41, 898–915 (1988).
V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge Univ. Press, Cambridge (1994).
P. Etingof and A. Varchenko, “Dynamical Weyl groups and applications,” Adv. Math., 167, 74–127 (2002).
A. I. Molev, “Gelfand–Tsetlin bases for classical Lie algebras,” in: Handbook of Algebra (M. Hazewinkel, eds.), Vol. 4, Elsevier, Amsterdam (2006), pp. 109–170.
S. Khoroshkin and O. Ogievetsky, “Mickelsson algebras and Zhelobenko operators,” J. Algebra, 319, 2113–2165 (2008).
E. Karolinsky, A. Stolin, and V. Tarasov, “Equivariant quantization of Poisson homogeneous spaces and Kostant’s problem,” J. Algebra, 409, 362–381 (2014).
N. Yu. Reshetikhin, L. A. Takhtadzhyan, and L. D. Faddeev, “Quantization of Lie groups and Lie algebras,” Leningrad Math. J., 1, 193–225 (1990).
A. Mudrov, “Characters of the Uq(sl(n))–reflection equation algebra,” Lett. Math. Phys., 60, 283–291 (2002).
P. P. Kulish and E. K. Sklyanin, “Algebraic structure related to the reflection equation,” J. Phys. A, 25, 5963–5975 (1992); arXiv:hep–th/9209054v1 (1992).
Author information
Authors and Affiliations
Corresponding author
Additional information
Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, Vol. 198, No. 2, pp. 326–340, February, 2019.
Rights and permissions
About this article
Cite this article
Mudrov, A.I. Equivariant Vector Bundles Over Quantum Projective Spaces. Theor Math Phys 198, 284–295 (2019). https://doi.org/10.1134/S0040577919020090
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577919020090