Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

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

 
 

 

Quantum $n$-space as a quotient of classical $n$-space


Authors: K. R. Goodearl and E. S. Letzter
Journal: Trans. Amer. Math. Soc. 352 (2000), 5855-5876
MSC (2000): Primary 16D60, 16P40, 16S36, 16W35; Secondary 20G42, 81R50
DOI: https://doi.org/10.1090/S0002-9947-00-02639-8
Published electronically: August 8, 2000
MathSciNet review: 1781280
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The prime and primitive spectra of $\mathcal{O}_{\mathbf q}(k^{n})$, the multiparameter quantized coordinate ring of affine $n$-space over an algebraically closed field $k$, are shown to be topological quotients of the corresponding classical spectra, $\operatorname{spec} \mathcal{O}(k^{n})$ and $\max \mathcal{O}(k^{n})\approx k^{n}$, provided the multiplicative group generated by the entries of $\mathbf{q}$ avoids $-1$.


References [Enhancements On Off] (What's this?)

  • 1. G. Abrams and J. Haefner, Primeness conditions for group graded rings, in Ring Theory, Proc. Biennial Ohio State - Denison Conf. 1992, World Scientific, Singapore, 1993, pp. 1-19. MR 96f:16050
  • 2. M. Artin, W. Schelter, and J. Tate, Quantum deformations of $GL_{n}$, Communic. Pure Appl. Math. 44 (1991), 879-895. MR 92i:17014
  • 3. K. A. Brown and K. R. Goodearl, Prime spectra of quantum semisimple groups, Trans. Amer. Math. Soc. 348 (1996), 2465-2502. MR 96i:17007
  • 4. C. De Concini, V. Kac, and C. Procesi, Some remarkable degenerations of quantum groups, Comm. Math. Phys. 157 (1993), 405-427. MR 94i:17019
  • 5. 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
  • 6. K. R. Goodearl and E. S. Letzter, 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 Amer. Math. Soc., Providence, RI, 1998, pp. 39-58. MR 99h:16045
  • 7. K. R. Goodearl and R. B. Warfield, Jr., An Introduction to Noncommutative Noetherian Rings, London Math. Soc. Student Texts 16, Cambridge Univ. Press, Cambridge, 1989. MR 91c:16001
  • 8. T. J. Hodges and T. Levasseur, Primitive ideals of ${\mathbf{C}}_{q}[SL(3)]$, Commun. Math. Phys. 156 (1993), 581-605. MR 94k:17023
  • 9. -, Primitive ideals of ${\mathbf{C}}_{q}[SL(n)]$, J. Algebra 168 (1994), 455-468. MR 95i:16038
  • 10. T. J. Hodges, T. Levasseur, and M. Toro, Algebraic structure of multi-parameter quantum groups, Advances in Math. 126 (1997), 52-92. MR 98e:17023
  • 11. A. Joseph, On the prime and primitive spectra of the algebra of functions on a quantum group, J. Algebra 169 (1994), 441-511. MR 96b:17015
  • 12. -, Quantum Groups and Their Primitive Ideals, Ergebnisse der Math. (3) 29, Springer-Verlag, Berlin, 1995. MR 96d:17015
  • 13. J. C. McConnell and J. J. Pettit, Crossed products and multiplicative analogs of Weyl algebras, J. London Math. Soc. (2) 38 (1988), 47-55. MR 90c:16011
  • 14. D. G. Northcott, Affine Sets and Affine Groups, London Math. Soc. Lecture Note Series 39, Cambridge Univ. Press, Cambridge, 1980. MR 82c:14002
  • 15. M. Vancliff, Primitive and Poisson spectra of twists of polynomial rings, Algebras and Representation Theory 2 (1999), 269-285. CMP 2000:02

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 16D60, 16P40, 16S36, 16W35, 20G42, 81R50

Retrieve articles in all journals with MSC (2000): 16D60, 16P40, 16S36, 16W35, 20G42, 81R50


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
Address at time of publication: Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19122
Email: letzter@math.tamu.edu, letzter@math.temple.edu

DOI: https://doi.org/10.1090/S0002-9947-00-02639-8
Received by editor(s): April 22, 1999
Published electronically: August 8, 2000
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 2000 American Mathematical Society

American Mathematical Society