Journal of Mathematical Sciences

, Volume 142, Issue 2, pp 1969–1976 | Cite as

Representations of quantum orders

  • A. N. Panov


We study finite-dimensional algebras that appear as fibers of quantum orders over a given point of variety of center. We present a formula for the number of irreducible representations and check it for the algebra of twisted polynomials, the quantum Weyl algebra, and the algebra of regular functions on a quantum group.


Irreducible Representation Hopf Algebra Quantum Group Poisson Algebra Coordinate Ring 
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  1. 1.
    K. A. Brown and I. Gordon, “Poisson orders, symplection reflection algebras and representation theory,” J. Reine Angew. Math., 559, 193–216 (2003).MATHMathSciNetGoogle Scholar
  2. 2.
    G. Cauchon, “Effacement des dérivations et spèctres premiers des algèbres quantiques,” J. Algebra, 260, No. 2, 476–518 (2003).MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    C. De Concini and V. G. Kac, “Representations of quantum groups at roots of 1,” in: Operator algebras, unitary representations, enveloping algebras, and invariant theory, Proc. Colloq. in Honour of J. Dixmier, Paris 1989, Progress in Math., Vol. 92, Birkhäuser (1990), pp. 471–506.Google Scholar
  4. 4.
    C. De Concini, V. G. Kac, and C. Procesi, “Quantum coadjoint action,” J. Amer. Math. Soc., 5, 151–189 (1992).MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    C. De Concini and V. Lyubashenko, “Quantum function algebra at roots of 1,” Adv. Math., 108, 205–262 (1994).MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    C. De Concini and C. Procesi, “Quantum groups,” in: G. Zampieri et al., ed., D-modules, representation theory, and quantum groups. Lectures given at the 2nd session of the Centro Internazionale Matematico Estivo (C.I.M.E.) held in Venezia, Italy, June 12–20, 1992, Lect. Notes Math., Vol. 1565, Springer, Berlin (1993), pp. 31–140.Google Scholar
  7. 7.
    C. De Concini and C. Procesi, “Quantum Schubert cells and representations at roots of 1,” in: G. I. Lehrer, ed., Algebraic Groups and Lie Groups, Australian Math. Soc. Lect. Series, Vol. 9, Cambridge University Press, Cambridge (1997).Google Scholar
  8. 8.
    K. R. Goodearl, “Prime spectra of quantized coordinate rings,” in: F. Van Oystaeyen et al., ed., Interactions between ring theory and representations of algebras. Proceedings of the conference, Murcia, Spain, Lect. Notes Pure Appl. Math., Vol. 210, Marcel Dekker, New York (2000), pp. 205–237.Google Scholar
  9. 9.
    K. L. Horton, “The prime and primitive spectra of multilinear quantum symplectic and Euclidean spaces,” Comm. Algebra, 31, No. 10, 4713–4743 (2003).MATHMathSciNetGoogle Scholar
  10. 10.
    H. Jakobsen and H. Zhang, “Quantized Heisenberg spaces,” Algebras Represent. Theory, 2, No. 2, 151–174 (2000).CrossRefMathSciNetGoogle Scholar
  11. 11.
    A. Joseph, Quantum Groups and Their Primitive Ideals, Springer, Berlin (1995).MATHGoogle Scholar
  12. 12.
    M. V. Karasev and V. P. Maslov, Nonlinear Poisson Brackets. Geometry and Quantization [in Russian], Nauka, Moscow (1991).MATHGoogle Scholar
  13. 13.
    J. C. McConnel and J. C. Robson, Noncommutative Noetherian Rings, Wileys-Interscience, New York (1987).Google Scholar
  14. 14.
    A. N. Panov, “Fields of fractions of quantum solvable algebras,” J. Algebra, 236, 110–121 (2001).MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    A. N. Panov, “Quantum solvable algebras. Ideals and representations at roots of 1,” Transform. Groups, 7, No. 4, 379–402 (2002).MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    A. N. Panov, “Irreducible representations of quantum solvable algebras at roots of 1,” Algebra Analiz, 15, No. 4, 229–259 (2003).Google Scholar
  17. 17.
    Oh Sei-Qwon, “Primitive ideals of the coordinate ring of quantum symplectic space,” J. Algebra, 174, 531–552 (1995).MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    H. Zhang, “The irreducible representations of the coordinate ring of the quantum matrix space,” Algebra Colloq., 9, No. 4, 383–392 (2002).MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2007

Authors and Affiliations

  • A. N. Panov
    • 1
  1. 1.Mathematical DepartmentSamara State UniversitySamaraRussia

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