Abstract
Let G be the complex general linear group and \( \mathfrak{g} \) its Lie algebra equipped with a factorizable Lie bialgebra structure; let Uħ(\( \mathfrak{g} \)) be the corresponding quantum group. We construct explicit Uħ(\( \mathfrak{g} \))-equivariant quantization of Poisson orbit bundles O λ → O μ in \( \mathfrak{g} \)*.
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References
A. A. Belavin and V. G. Drinfeld, On solutions of YangBaxter equation, Functional Analysis and Applications 16 (1982), 1–29. English translation: Functional Analysis and Applications 32 (1985), 254-255.
J. Donin, Double quantization on coadjoint representations of simple Lie groups and its orbits, preprint MPIM1999-103, September 1999.
J. Donin, Quantum g-manifolds, Contemporary Mathematics 315 (2002), 47–60.
J. Donin and A. Mudrov, Explicit quantization on coadjoint orbits of gl(n,C), Letters in Mathematical Physics 62 (2002), 17–32.
J. Donin and A. Mudrov, Method of quantum characters in equivariant quantization, Communications in Mathematical Physics 234 (2003), 533–555.
J. Donin and A. Mudrov, Quantum coadjoint orbits of GL(n) and generalized Verma modules, Letters in Mathematical Physics 67 (2004), 167–184.
J. Donin and V. Ostapenko, Equivariant quantization on quotients of simple Lie groups by reductive subgroups, Czechoslovak Journal of Physics 52 (2002), 1213–1218.
J. Donin and S. Shnider, Quantum symmetric spaces, Journal of Pure and Applied Algebra 100 (1995), 103–115.
J. Donin, D. Gurevich and S. Shnider, Double quantization on some orbits in the coadjoint representations of simple Lie groups, Communications in Mathematical Physics 204 (1999), 39–60, math.QA/9807159.
V. G. Drinfeld, Quantum groups, in Proceedings of International Congress of Mathematicians, Berkley 1986, vol. 1, American Mathematical Society, Providence, RI, 1986, pp. 798–820.
V. G. Drinfeld, Almost cocommutative Hopf algebras, Algebra i Analys, 1 (1989), 321–342. English translation: Leningrad Journal of Mathematics 1 (1990), 321-342.
P. Etingoff and D. Kazhdan, Quantization of Lie bialgebras, Selecta Mathematicae 2 (1996), 1–41.
D. I. Gurevich and P. A. Saponov, Geometry of non-commutative orbits related to Hecke symmetries, Contemporary Mathematics 433, “Quantum Groups”, Proceedings of a conference in memory of Joseph Donin, 2007, pp. 209-250.
E. Karolinskii, A classification of Poisson homogeneous spaces of complex reductive PoissonLiegroups, in Poissongeometry, P. Urbanski and J. Grabowski, eds., vol. 51, Banach Center, Warsaw, 2000, pp. 103–108.
P. P. Kulish and A. I. Mudrov, Dynamical reflection equation, Contemporary Mathematics Contemporary Mathematics, “Quantum Groups”, Proceedings of a conference in memory of Joseph Donin, 2007, pp. 281–309.
A. Mudrov, Quantum conjugacy classes of simple matrix groups, Communications in Mathematical Physics 272 (2007), 635–660.
M. Semenov-Tian-Shansky, Poisson-Lie groups, quantum duality principle, and the quantum double, Contemporary Mathematics 175 (1994), 219–248.
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Mudrov, A., Ostapenko, V. Quantization of orbit bundles in \( \mathfrak{g}\mathfrak{l}_n^* (\mathbb{C}) \) . Isr. J. Math. 172, 399–423 (2009). https://doi.org/10.1007/s11856-009-0080-3
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DOI: https://doi.org/10.1007/s11856-009-0080-3