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

Goodearl, K.; Letzter, E.

doi:10.1090/S0002-9947-00-02639-8

Quantum $n$-space as a quotient of classical $n$-space
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by K. R. Goodearl and E. S. Letzter PDF

Trans. Amer. Math. Soc. 352 (2000), 5855-5876 Request permission

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

Gene Abrams and Jeremy Haefner, Primeness conditions for group graded rings, Ring theory (Granville, OH, 1992) World Sci. Publ., River Edge, NJ, 1993, pp. 1–19. MR 1344219
Michael Artin, William Schelter, and John Tate, Quantum deformations of $\textrm {GL}_n$, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 879–895. MR 1127037, DOI 10.1002/cpa.3160440804
K. A. Brown and K. R. Goodearl, Prime spectra of quantum semisimple groups, Trans. Amer. Math. Soc. 348 (1996), no. 6, 2465–2502. MR 1348148, DOI 10.1090/S0002-9947-96-01597-8
C. De Concini, V. G. Kac, and C. Procesi, Some remarkable degenerations of quantum groups, Comm. Math. Phys. 157 (1993), no. 2, 405–427. MR 1244875
Alfred Goldie and Gerhard Michler, Ore extensions and polycyclic group rings, J. London Math. Soc. (2) 9 (1974/75), 337–345. MR 357500, DOI 10.1112/jlms/s2-9.2.337
K. R. Goodearl and E. S. Letzter, Prime and primitive spectra of multiparameter quantum affine spaces, Trends in ring theory (Miskolc, 1996) CMS Conf. Proc., vol. 22, Amer. Math. Soc., Providence, RI, 1998, pp. 39–58. MR 1491917
K. R. Goodearl and R. B. Warfield Jr., An introduction to noncommutative Noetherian rings, London Mathematical Society Student Texts, vol. 16, Cambridge University Press, Cambridge, 1989. MR 1020298
Timothy J. Hodges and Thierry Levasseur, Primitive ideals of $\textbf {C}_q[\textrm {SL}(3)]$, Comm. Math. Phys. 156 (1993), no. 3, 581–605. MR 1240587
Timothy J. Hodges and Thierry Levasseur, Primitive ideals of $\textbf {C}_q[\textrm {SL}(n)]$, J. Algebra 168 (1994), no. 2, 455–468. MR 1292775, DOI 10.1006/jabr.1994.1239
Hans Plesner Jakobsen, Tensoring with small quantized representations, J. Math. Phys. 38 (1997), no. 8, 4323–4335. MR 1459660, DOI 10.1063/1.532096
Anthony Joseph, On the prime and primitive spectra of the algebra of functions on a quantum group, J. Algebra 169 (1994), no. 2, 441–511. MR 1297159, DOI 10.1006/jabr.1994.1294
Anthony Joseph, Quantum groups and their primitive ideals, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 29, Springer-Verlag, Berlin, 1995. MR 1315966, DOI 10.1007/978-3-642-78400-2
J. C. McConnell and J. J. Pettit, Crossed products and multiplicative analogues of Weyl algebras, J. London Math. Soc. (2) 38 (1988), no. 1, 47–55. MR 949080, DOI 10.1112/jlms/s2-38.1.47
D. G. Northcott, Affine sets and affine groups, London Mathematical Society Lecture Note Series, vol. 39, Cambridge University Press, Cambridge-New York, 1980. MR 569353
M. Vancliff, Primitive and Poisson spectra of twists of polynomial rings, Algebras and Representation Theory 2 (1999), 269-285.