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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The Dixmier-Moeglin equivalence
in quantum coordinate rings
and quantized Weyl algebras


Authors: K. R. Goodearl and E. S. Letzter
Journal: Trans. Amer. Math. Soc. 352 (2000), 1381-1403
MSC (1991): Primary 16S36, 16P40, 81R50
DOI: https://doi.org/10.1090/S0002-9947-99-02345-4
Published electronically: October 15, 1999
MathSciNet review: 1615971
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Abstract: We study prime and primitive ideals in a unified setting applicable to quantizations (at nonroots of unity) of $n\times n$ matrices, of Weyl algebras, and of Euclidean and symplectic spaces. The framework for this analysis is based upon certain iterated skew polynomial algebras $A$ over infinite fields $k$ of arbitrary characteristic. Our main result is the verification, for $A$, of a characterization of primitivity established by Dixmier and Moeglin for complex enveloping algebras. Namely, we show that a prime ideal $P$ of $A$ is primitive if and only if the center of the Goldie quotient ring of $A/P$ is algebraic over $k$, if and only if $P$ is a locally closed point - with respect to the Jacobson topology - in the prime spectrum of $A$. These equivalences are established with the aid of a suitable group $\mathcal{H} $ acting as automorphisms of $A$. The prime spectrum of $A$ is then partitioned into finitely many ``$\mathcal{H} $-strata'' (two prime ideals lie in the same $\mathcal{H} $-stratum if the intersections of their $\mathcal{H} $-orbits coincide), and we show that a prime ideal $P$ of $A$ is primitive exactly when $P$ is maximal within its $\mathcal{H} $-stratum. This approach relies on a theorem of Moeglin-Rentschler (recently extended to positive characteristic by Vonessen), which provides conditions under which $\mathcal{H} $ acts transitively on the set of rational ideals within each $\mathcal{H} $-stratum. In addition, we give detailed descriptions of the strata that can occur in the prime spectrum of $A$. For quantum coordinate rings of semisimple Lie groups, results analogous to those obtained in this paper already follow from work of Joseph and Hodges-Levasseur-Toro. For quantum affine spaces, analogous results have been obtained in previous work of the authors.


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  • 1. J. Alev and F. Dumas, Sur les corps de fractions de certaines algèbres de Weyl quantiques, J. Algebra 170 (1994), 229-265. MR 96c:16033
  • 2. M. Artin, W. Schelter, and J. Tate, Quantum deformations of $\text{GL}_{n}$, Communic. Pure Appl. Math. 44 (1991), 879-895. MR 92i:17014
  • 3. A. Borel, Linear Algebraic Groups, Second Enlarged Edition, Springer-Verlag, New York, 1991. MR 92d:20001
  • 4. K. A. Brown and K. R. Goodearl, Prime spectra of quantum semisimple groups, Trans. Amer. Math. Soc. 348 (1996), 2465-2502. MR 96i:17007
  • 5. G. Cauchon, Quotients premiers de $O_{q}({\mathfrak{m}}_{n}(k))$, J. Algebra 180 (1996), 530-545. MR 97e:16078
  • 6. C. De Concini, V. Kac, and C. Procesi, Some remarkable degenerations of quantum groups, Comm. Math. Phys. 157 (1993), 405-427. MR 94i:17019
  • 7. C. De Concini and V. Lyubashenko, Quantum function algebras at roots of $1$, Advances in Math. 108 (1994), 205-262. MR 95m:17014
  • 8. J. Dixmier, Idéaux primitifs dans les algèbres enveloppantes, J. Algebra 48 (1977), 96-112. MR 56:5673
  • 9. -, Enveloping Algebras, The 1996 printing of the 1977 English translation, Amer. Math. Soc., Providence, 1996. MR 97c:17010
  • 10. A. W. Goldie and G. O. Michler, Ore extensions and polycyclic group rings, J. London Math. Soc. (2) 9 (1974), 337-345. MR 50:9968
  • 11. K. R. Goodearl, Uniform ranks of prime factors of skew polynomial rings, in Ring Theory, Proc. Biennial Ohio State - Denison Conf. 1992 (S. K. Jain and S. T. Rizvi, eds.), World Scientific, Singapore, 1993, pp. 182-199. MR 96h:16031
  • 12. K. R. Goodearl and E. S. Letzter, Prime ideals in skew and q-skew polynomial rings, Mem. Amer. Math. Soc. 521 (1994). MR 94j:16051
  • 13. -, Prime and primitive spectra of multiparameter quantum affine spaces, in Trends in Ring Theory. Proc. Miskolc Conf. 1996 (V. Dlab and L. Márki, eds.), Canad. Math. Soc. Conf. Proc. Series 22 (1998), 39-58. CMP 98:07
  • 14. K. R. Goodearl and R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, Cambridge Univ. Press, Cambridge, 1989. MR 91c:16008
  • 15. T. J. Hodges and T. Levasseur, Primitive ideals of ${\mathbf{C}}_{q}[SL(3)]$, Commun. Math. Phys. 156 (1993), 581-605. MR 94k:17023
  • 16. -, Primitive ideals of ${\mathbf{C}}_{q}[SL(n)]$, J. Algebra 168 (1994), 455-468. MR 95i:16038
  • 17. T. J. Hodges, T. Levasseur, and M. Toro, Algebraic structure of multi-parameter quantum groups, Advances in Math. 126 (1997), 52-92. MR 98e:17022
  • 18. J. E. Humphreys, Linear Algebraic Groups, Springer-Verlag, New York, 1975/1981. MR 53:633
  • 19. R. S. Irving, Noetherian algebras and the Nullstellensatz, in Séminaire d'Algèbre Paul Dubreil 31ème année (Paris 1977-78) (M.-P. Malliavin, ed.), Lecture Notes in Math. 740, Springer-Verlag, Berlin, 1979, pp. 80-87. MR 81c:16018
  • 20. R. S. Irving and L. W. Small, On the characterization of primitive ideals in enveloping algebras, Math. Z. 173 (1980), 217-221. MR 82j:17015
  • 21. J. C. Jantzen, Representations of Algebraic Groups, Pure and Applied Math. Series 131, Academic Press, Boston-Orlando, 1987. MR 89c:20001
  • 22. D. A. Jordan, A simple localization of the quantized Weyl algebra, J. Algebra 174 (1995), 267-281. MR 96m:16035
  • 23. A. Joseph, Idéaux premiers et primitifs de l'algèbre des fonctions sur un groupe quantique, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), 1139-1142. MR 94c:17029
  • 24. -, On the prime and primitive spectra of the algebra of functions on a quantum group, J. Algebra 169 (1994), 441-511. MR 96b:17015
  • 25. -, Quantum Groups and Their Primitive Ideals, Ergebnisse der Math. (3) 29, Springer-Verlag, Berlin, 1995. MR 96d:17015
  • 26. G. Maltsiniotis, Calcul différentiel quantique, Groupe de travail, Université Paris VII, 1992.
  • 27. Yu. I. Manin, Multiparametric quantum deformation of the general linear supergroup, Comm. Math. Phys. 123 (1989), 163-175. MR 90h:17018
  • 28. J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Wiley-Interscience, Chichester-New York, 1987. MR 89j:16023
  • 29. C. Moeglin, Idéaux primitifs des algèbres enveloppantes, J. Math. Pures Appl. 59 (1980), 265-336. MR 83i:17008
  • 30. C. Moeglin and R. Rentschler, Orbites d'un groupe algébrique dans l'espace des idéaux rationnels d'une algèbre enveloppante, Bull. Soc. Math. France 109 (1981), 403-426. MR 83i:17009
  • 31. -, Idéaux G-rationnels, Rang de Goldie, unpublished manuscript, 1986.
  • 32. I. M. Musson, Ring theoretic properties of the coordinate rings of quantum symplectic and Euclidean space, in Ring Theory, Proc. Biennial Ohio State-Denison Conf., 1992 (S. K. Jain and S. T. Rizvi, eds.), World Scientific, Singapore, 1993, pp. 248-258. MR 96e:16052
  • 33. C. N[??]ast[??]asescu and F. Van Oystaeyen, Graded Ring Theory, North-Holland, Amsterdam, 1982. MR 84i:16002
  • 34. D. G. Northcott, Affine Sets and Affine Groups, London Math. Soc. Lecture Note Series 39, Cambridge Univ. Press, Cambridge, 1980. MR 82c:14002
  • 35. S.-Q. Oh, Primitive ideals of the coordinate ring of quantum symplectic space, J. Algebra 174 (1995), 531-552. MR 97c:17026
  • 36. -, Primitive ideals in the coordinate ring of quantum Euclidean space, Bull. Austral. Math. Soc. 58 (1998), 57-73. CMP 98:16
  • 37. B. Parshall and J.-p. Wang, Quantum linear groups, Mem. Amer. Math. Soc. 439 (1991). MR 91g:16028
  • 38. D. Quillen, On the endomorphism ring of a simple module over an enveloping algebra, Proc. Amer. Math. Soc. 21 (1969), 171-172. MR 39:252
  • 39. Z. Reichstein and N. Vonessen, Torus actions on rings, J. Algebra 170 (1994), 781-804. MR 95k:16050
  • 40. R. Rentschler, Primitive ideals in enveloping algebras (general case), in Noetherian Rings and their Applications (Oberwolfach, 1983) (L. W. Small, ed.), Math. Surveys Monographs 24, Amer. Math. Soc., Providence, 1987, pp. 37-57. MR 89d:17014
  • 41. N. Yu. Reshetikhin, Multiparameter quantum groups and twisted quasitriangular Hopf algebras, Lett. Math. Phys. 20 (1990), 331-335. MR 91k:17012
  • 42. N. Yu. Reshetikhin, L. A. Takhtadzhyan, and L. D. Faddeev, Quantization of Lie groups and Lie algebras, Leningrad Math. J. 1 (1990), 193-225. MR 97j:17039
  • 43. L. H. Rowen, Ring Theory, Volumes I and II, Academic Press, Boston, 1988. MR 89h:16001; MR 89h:16002
  • 44. A. Sudbery, Consistent multiparameter quantisation of GL($n$), J. Phys. A 23 (1990), L697-L704. MR 91m:17022
  • 45. N. Vonessen, Actions of algebraic groups on the spectrum of rational ideals, J. Algebra 182 (1996), 383-400. MR 97c:16044
  • 46. -, Actions of algebraic groups on the spectrum of rational ideals, II, J. Algebra 208 (1998), 216-261. CMP 99:02
  • 47. S. Yammine, Les théorèmes de Cohen-Seidenberg en algèbre non commutative, in Séminaire d'Algèbre Paul Dubreil 31ème année (Paris 1977-78) (M.-P. Malliavin, ed.), Lecture Notes in Math. 740, Springer-Verlag, Berlin, 1979, pp. 120-169. MR 81i:16004

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Additional Information

K. R. Goodearl
Affiliation: Department of Mathematics, University of California, Santa Barbara, California 93106
Email: goodearl@math.ucsb.edu

E. S. Letzter
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email: letzter@math.tamu.edu

DOI: https://doi.org/10.1090/S0002-9947-99-02345-4
Received by editor(s): August 16, 1997
Published electronically: October 15, 1999
Additional Notes: The research of the first author was partially supported by NSF grant DMS-9622876, and the research of the second author was partially supported by NSF grant DMS-9623579.
Article copyright: © Copyright 1999 American Mathematical Society

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