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Quantum duality principle for quantum Grassmannians

  • Rita Fioresi
  • Fabio Gavarini

Abstract

The quantum duality principle (QDP) for homogeneous spaces gives four recipes to obtain, from a quantum homogeneous space, a dual one, in the sense of Poisson duality. One of these recipes fails (for lack of the initial ingredient) when the homogeneous space we start from is not a quasi-affine variety. In this work we solve this problem for the quantum Grassmannian, a key example of quantum projective homogeneous space, providing a suitable analogue of the QDP recipe.

Keywords

Hopf Algebra Homogeneous Space Quantum Group Quantum Deformation Grassmann Variety 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    [1] V. Chari, A. Pressley, Quantum Groups, Cambridge Univ. Press, Cambridge (1994).MATHGoogle Scholar
  2. [2]
    [2] N. Ciccoli, R. Fioresi, F. Gavarini, Quantum Duality Principle for Projective Homogeneous Spaces, J. Noncommut. Geom. 2 (2008), 449–496.MATHCrossRefMathSciNetGoogle Scholar
  3. [3]
    [3] N. Ciccoli, F. Gavarini, A quantum duality principle for coisotropic subgroups and Poisson quotients, Adv. Math. 199 (2006), 104–135.MATHCrossRefMathSciNetGoogle Scholar
  4. [4]
    [4] C. De Concini, D. Eisenbud, C. Procesi, Young Diagrams and Determinantal Varieties, Invent. Math. 56 (1980), 129–165.CrossRefMathSciNetGoogle Scholar
  5. [5]
    [5] V. G. Drinfeld, Quantum groups, Proc. Intern. Congress of Math. (Berkeley, 1986) (1987), 798–820.Google Scholar
  6. [6]
    [6] R. Fioresi, Quantum deformation of the Grassmannian manifold, J. Algebra 214 (1999), 418–447.MATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    [7] R. Fioresi, A deformation of the big cell inside the Grassmannian manifold G(r, n), Rev. Math. Phys. 11 (1999), 25–40.MATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    [8] F. Gavarini, Quantum function algebras as quantum enveloping algebras, Comm. Algebra 26 (1998), 1795–1818.MATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    [9] F. Gavarini, The quantum duality principle, Ann. Inst. Fourier (Grenoble) 52 (2002), 809–834.MATHMathSciNetGoogle Scholar
  10. [10]
    [10] F. Gavarini, The global quantum duality principle: theory, examples, and applications, 120 pages, see http://arxiv.org/abs/math.QA/0303019 (2003).
  11. [11]
    [11] F. Gavarini, The global quantum duality principle, J. Reine Angew. Math. 612 (2007), 17–33.MATHMathSciNetGoogle Scholar
  12. [12]
    [12] V. Lakshmibai, N. Reshetikhin, Quantum flag and Schubert schemes, Contemp. Math. 134, Amer. Math. Soc., Providence, RI (1992), 145–181.MathSciNetGoogle Scholar
  13. [13]
    [13] E. Taft, J. Towber, Quantum deformation of flag schemes and Grassmann schemes, I. A q-deformation of the shape-algebra for GL(n), J. Algebra 142 (1991), 1–36.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Vieweg+Teubner Verlag | Springer Fachmedien Wiesbaden GmbH 2011

Authors and Affiliations

  • Rita Fioresi
    • 1
  • Fabio Gavarini
    • 2
  1. 1.Dipartimento di MatematicaUniversitá di BolognaBolognaItaly
  2. 2.Dipartimento di MatematicaUniversitá di Roma “Tor Vergata”RomaItaly

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