Algebras and Representation Theory

, Volume 2, Issue 3, pp 269–285 | Cite as

Primitive and Poisson Spectra of Twists of Polynomial Rings

  • Michaela Vancliff


A family of flat deformations of a commutative polynomial ring S on n generators is considered, where each deformation B is a twist of S by a semisimple, linear automorphism σ of ℙn−1, such that a Poisson bracket is induced on S. We show that if the symplectic leaves associated with this Poisson structure are algebraic, then they are the orbits of an algebraic group G determined by the Poisson bracket. In this case, we prove that if σ is 'generic enough', then there is a natural one-to-one correspondence between the primitive ideals of B and the symplectic leaves if and only if σ has a representative in GL(ℂ n ) which belongs to G. As an example, the results are applied to the coordinate ring \(\mathcal{O}_q (M_2 )\) of quantum 2 × 2 matrices which is not a twist of a polynomial ring, although it is a flat deformation of one; if q is not a root of unity, then there is a bijection between the primitive ideals of \(\mathcal{O}_q (M_2 )\) and the symplectic leaves.

twisted homogeneous coordinate ring symplectic leaf Poisson manifold primitive ideal quantum matrices 


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  1. 1.
    Artin, M., Tate, J. and Van den Bergh, M.: Some algebras associated to automorphisms of elliptic curves, in: P. Cartier et al. (eds), The Grothendieck Festschrift 1, Birkhäuser, Basel, 1990, pp. 33-85.Google Scholar
  2. 2.
    Artin, M., Tate, J. and Van den Bergh, M.: Modules over regular algebras of dimension 3, Invent. Math. 106 (1991), 335-388.Google Scholar
  3. 3.
    Brown, K. A. and Goodearl, K. R.: Prime spectra of quantum semisimple groups, Trans. Amer. Math. Soc. 348(6) (1996), 2465-2502.Google Scholar
  4. 4.
    Dixmier, J.: Enveloping Algebras, North-Holland, Amsterdam, 1977.Google Scholar
  5. 5.
    Drinfel'd, V. G.: Quantum groups, in: Proc. Internat. Congr. Math. Berkeley 1, 1986, pp. 798-820.Google Scholar
  6. 6.
    Faddeev, L. D., Reshetikhin, N. Yu. and Takhtadzhyan, L. A.: Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1(1) (1990), 193-225.Google Scholar
  7. 7.
    Hodges, T. J. and Levasseur, T.: Primitive ideals of ℂq[SL(3)], Comm. Math. Phys. 156(3) (1993), 581-605.Google Scholar
  8. 8.
    Hodges, T. J. and Levasseur, T.: Primitive ideals of ℂq[SL(n)], J. Algebra 168(2) (1994), 455-468.Google Scholar
  9. 9.
    Joseph, A.: Idéaux premiers et primitifs de l'algèbre des fonctions sur un groupe quantique, C.R. Acad. Sci. Paris, Sér. I 316 (1993), 1139-1142.Google Scholar
  10. 10.
    Joseph, A.: On the prime and primitive spectra of the algebra of functions on a quantum group, J. Algebra 169 (1994), 441-511.Google Scholar
  11. 11.
    Joseph, A.: Quantum Groups and their Primitive Ideals, Ergeb. Math. Grenzgeb. (3) 29, Springer-Verlag, Berlin, 1995.Google Scholar
  12. 12.
    Kirillov, A. A.: Local Lie algebras, Uspekhi Mat. Nauk 31(4) (1976), 57-76; English transl. in Russian Math. Surveys 31 (1976).Google Scholar
  13. 13.
    Vancliff, M.: Quadratic algebras associated with the union of a quadric and a line in ℙ3, J. Algebra 165(1) (1994), 63-90.Google Scholar
  14. 14.
    Vancliff, M.: The defining relations of quantum n × n matrices, J. London Math. Soc. 52(2) (1995), 255-262.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Michaela Vancliff
    • 1
  1. 1.Department of MathematicsUniversity of Texas at ArlingtonArlingtonU.S.A.

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