Prime factor algebras of the coordinate ring of quantum matrices

Goodearl, K. R.; Letzter, E. S.

doi:10.1090/S0002-9939-1994-1211579-1

Prime factor algebras of the coordinate ring of quantum matrices
HTML articles powered by AMS MathViewer

by K. R. Goodearl and E. S. Letzter PDF

Proc. Amer. Math. Soc. 121 (1994), 1017-1025 Request permission

Abstract:

It is proved that every prime factor algebra of the coordinate ring ${\mathcal {O}_q}({M_n}(k))$ of quantum $n \times n$ matrices over a field k is an integral domain (albeit not necessarily commutative) when q is not a root of unity. The same conclusion follows for the quantum groups ${\mathcal {O}_q}({\text {SL}_n}(k))$ and ${\mathcal {O}_q}({\text {GL}_n}(k))$. The proof uses a q-analog of Sigurdsson’s theorem bounding the Goldie ranks of prime factors of differential operator rings; this q-analog in turn is based on results from the authors’ recent work on q-skew polynomial rings.

References

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
Allen D. Bell, Goldie dimension of prime factors of polynomial and skew polynomial rings, J. London Math. Soc. (2) 29 (1984), no. 3, 418–424. MR 754928, DOI 10.1112/jlms/s2-29.3.418
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, Prime ideals in skew polynomial rings and quantized Weyl algebras, J. Algebra 150 (1992), no. 2, 324–377. MR 1176901, DOI 10.1016/S0021-8693(05)80036-5
K. R. Goodearl, Uniform ranks of prime factors of skew polynomial rings, Ring theory (Granville, OH, 1992) World Sci. Publ., River Edge, NJ, 1993, pp. 182–199. MR 1344230
K. R. Goodearl and E. S. Letzter, Prime ideals in skew and $q$-skew polynomial rings, Mem. Amer. Math. Soc. 109 (1994), no. 521, vi+106. MR 1197519, DOI 10.1090/memo/0521
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
Sei-Qwon Oh, Finite-dimensional simple modules over the coordinate ring of quantum matrices, Bull. London Math. Soc. 25 (1993), no. 5, 427–430. MR 1233404, DOI 10.1112/blms/25.5.427
A. V. Jategaonkar, Localization in Noetherian rings, London Mathematical Society Lecture Note Series, vol. 98, Cambridge University Press, Cambridge, 1986. MR 839644, DOI 10.1017/CBO9780511661938
Edward S. Letzter, Finite correspondence of spectra in Noetherian ring extensions, Proc. Amer. Math. Soc. 116 (1992), no. 3, 645–652. MR 1098402, DOI 10.1090/S0002-9939-1992-1098402-1
T. Levasseur and J. T. Stafford, The quantum coordinate ring of the special linear group, J. Pure Appl. Algebra 86 (1993), no. 2, 181–186. MR 1215645, DOI 10.1016/0022-4049(93)90102-Y
Yu. I. Manin, Multiparametric quantum deformation of the general linear supergroup, Comm. Math. Phys. 123 (1989), no. 1, 163–175. MR 1002037
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
J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Pure and Applied Mathematics (New York), John Wiley & Sons, Ltd., Chichester, 1987. With the cooperation of L. W. Small; A Wiley-Interscience Publication. MR 934572
Brian Parshall and Jian Pan Wang, Quantum linear groups, Mem. Amer. Math. Soc. 89 (1991), no. 439, vi+157. MR 1048073, DOI 10.1090/memo/0439
J. E. Roseblade, Prime ideals in group rings of polycyclic groups, Proc. London Math. Soc. (3) 36 (1978), no. 3, 385–447. MR 491797, DOI 10.1112/plms/s3-36.3.385
Gunnar Sigurdsson, Differential operator rings whose prime factors have bounded Goldie dimension, Arch. Math. (Basel) 42 (1984), no. 4, 348–353. MR 753356, DOI 10.1007/BF01246126
S. P. Smith, Quantum groups: an introduction and survey for ring theorists, Noncommutative rings (Berkeley, CA, 1989) Math. Sci. Res. Inst. Publ., vol. 24, Springer, New York, 1992, pp. 131–178. MR 1230220, DOI 10.1007/978-1-4613-9736-6_{6}

Consistent multiparameter quantisation of

23