Abstract
Orbits of the quantum dressing transformation forSU q (N) acting on its solvable dual are introduced. The case is considered when the corresponding classical orbits coincide with Grassmann manifolds. Quantization of the Poisson bracket on a Zariski open subset of the Grassmann manifold yields a *-algebra generated by the quantum coordinate functions. The commutation relations are written in a compact form with the help of theR-matrix. Finite-dimensional irreducible representations ofU h \((\mathfrak{s}\mathfrak{l}(N,\mathbb{C}))\) are derived from the *-algebra structure.
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References
Drinfeld, V.G.: Quantum groups. In: Proc. ICM Berkeley 1986, Providence, RI: AMS 1987, p. 798
Jimbo, M.: Aq-difference analogue ofU(g) and the Yang-Baxter equation. Lett. Math. Phys.10, 63 (1985)
Rosso, M.: Finite dimensional representations fo the quantum analog of the enveloping algebra of a complex simple Lie algebra. Commun. Math. Phys.117, 581 (1988)
Lusztig, G.: Quantum deformations of certain simple modules over enveloping algebras. Adv. Math.70, 237 (1988)
Parshall, B., Wang, J.: Quantum linear groups. Providence, RI: AMS 1991
Lakshmibai, V., Reshetikhin, N.: Quantum deformations of flag and Schubert schemes. C.R. Acad. Sci. Paris313, Série I, 121 (1991)
Levendorskii, S., Soibelman, Ya.: Algebras of functions on cmpact quantum groups, Schubert cells and quantum tori. Commun. Math. Phys.139, 141 (1991)
Semenov-Tian-Shansky, M.A.: Dressing transformations and Poisson Lie group actions. Publ. RIMS, Kyoto University21, 1237 (1985)
Babelon, O., Bernard, D.: Dressing transformations and the origin of the quantum group symmetries. Preprint SPhT-91-016 (1991)
Lu, J.H., Weinstein, A.: Poisson Lie groups, dressing transformations and Bruhat decompositions. J. Diff. Geom.31, 501 (1990)
Reshetikhin, N.Yu., Semenov-Tian-Shansky, M.A.: QuantumR-matrices and factorization problems. J. Geom. Phys.5, 533 (1987)
Podles, P., Woronowicz, S.L.: Quantum deformation of Lorentz group. Commun. Math. Phys.130, 381 (1990)
Reshetikhin, N.Yu., Takhtajan, L.A., Faddeev, L.D.: Quantization of Lie groups and Lie algebras. Algebra i analiz1, 178 (1989) (in Russian)
Jurčo, B., Šťovíček, P.: Quantum dressing orbits on compact groups. Commun. Math. Phys.152, 97 (1993)
Podles, P.: Quantum spheres. Lett. Math. Phys.14, 193 (1987)
Soibelman, Y.: On quantum flag manifolds. RIMS-780 (1991)
Taf, E., Towber, J.: Quantum deformation of flag schemes and Grassmann schemes I. Aq-deformation of the shape-algebra forGL(n). J. Algebra142, No. 1 (1991)
Šťovíček, P.: Quantum line bundles onS 2 and the method of orbits forSU q (2). J. Math. Phys.34, 1606 (1993)
Woronowicz, S.L.: Compact matrix pseudogroups. Commun. Math. Phys.111, 613 (1987)
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Communicated by N. Yu. Reshetikhin
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Šťovíček, P. Quantum Grassmann manifolds. Commun.Math. Phys. 158, 135–153 (1993). https://doi.org/10.1007/BF02097235
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DOI: https://doi.org/10.1007/BF02097235