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Transactions of the American Mathematical Society
ISSN 1088-6850(e) ISSN 0002-9947(p)

     

On Gelfand-Kirillov Transcendence Degree

Author(s): James J. Zhang
Journal: Trans. Amer. Math. Soc. 348 (1996), 2867-2899.
MSC (1991): Primary 16P90, 12E15, 16K40, 16S80
MathSciNet review: 1370657
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Abstract: We study some basic properties of the Gelfand-Kirillov transcendence degree and compute the transcendence degree of various infinite-dimensional division algebras including quotient division algebras of quantized algebras related to quantum groups, 3-dimensional Artin-Schelter regular algebras and the 4-dimensional Sklyanin algebra.

References:

[AS]
M. Artin and W. Schelter, Graded algebras of global dimension 3, Adv. Math. 66 (1987), 171--216. MR 88k:16003

[AST]
M. Artin, W. Schelter and J. Tate, Quantum deformations of $GL_{n}$, Comm. Pure Appl. Math. 64 (1991), 879--895. MR 92i:17014

[ASta]
M. Artin and J. T. Stafford, Noncommutative graded domains with quadratic growth, Invent. Math. 122 (1995), 231--276. CMP 96:3

[ATV1]
M. Artin, J. Tate and M. van den Bergh, Some algebras associated to automorphisms of elliptic curves, The Grothendieck Festschrift (P. Cartier et al., eds.), Birkhäuser, Boston, 1990, Vol. 1, pp. 33--85. MR 92e:14002

[ATV2]
M. Artin, J. Tate and M. van den Bergh, Modules over regular algebras of dimension 3, Invent Math. 106 (1991), 335--388. MR 93e:16055

[Be]
G. M. Bergman, A note on growth functions of algebras and semigroups, mimeographed notes, University of California, Berkeley, 1978.

[BK]
W. Borho and H. Kraft, Über die Gelfand-Kirillov-Dimension, Math. Annalen 220 (1976), 1--24. MR 54:367

[Co]
P. M. Cohn, Skew field constructions, London Math. Soc. Lec. Notes Series 27, Cambridge University Press, 1977. MR 57:3190

[GK]
I. M. Gelfand and A. A. Kirillov, Sur les corps liés aux algèbres enveloppantes des algèbres de Lie, Publ. Math. IHES 31 (1966), 5--19. MR 34:7731

[GZ]
A. Giaquinto and J. J. Zhang, Quantum Weyl algebras, J. Alg. 176 (1995), 861--881. CMP 96:1

[Ja]
A. V. Jategaonkar, Ore domains and free algebras, Bull. London Math. Soc. 1 (1969), 45--46. MR 39:241

[Ji]
M. Jimbo, A $q$-difference analogue of $U({\mathfrak {g}})$ and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63--69. MR 86k:17008

[JZ]
N. Jing and J. J. Zhang, Quantum Weyl algebras and deformations of $U({\mathfrak {g}})$, Pacific J. Math. (in press).

[Jo]
A. Joseph, Gelfand-Kirillov dimension for algebras associated with the Weyl algebra, Ann. Inst. H. Poincaré Sect. A (N.S.) 17 (1972), 325--336. MR 49:2868

[KL]
G. Krause and T. H. Lenagan, Growth of algebras and Gelfand-Kirillov dimension, Research Notes in Mathematics, Pitman Adv. Publ. Program, vol 116 (1985). MR 86g:16001

[LMO]
A. Leroy, J. Matczuk and J. Okninski, On the Gelfand-Kirillov dimension of normal localizations and twisted polynomial rings, Perspectives in Ring Theory (F. van Oystaeyen and L. Le Bruyn, eds.), Kluwer Academic Publishers, 1988, pp. 205--214. MR 91c:16020

[Le]
T. Levasseur, Some properties of non-commutative regular graded rings, Glasgow J. Math. 34 (1992), 277--300. MR 93k:16045

[Lo]
M. Lorenz, On the transcendence degree of group algebras of nilpotent groups, Glasgow Math. J. 25 (1984), 167--174. MR 86c:16005

[M-L]
L. Makar-Limanov, The skew field of fractions of the Weyl algebra contains a free noncommutative subalgebra, Comm. Algebra 11 (1983), 2003--2006. MR 84j:16012

[Ma]
W. S. Martindale III, On semiprime PI rings, Proc. Amer. Math. Soc. 40 (2) (1973), 365--369. MR 47:6762

[MR]
J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Wiley-Interscience, Chichester, 1987. MR 89j:16023

[NV1]
C. N\v{a}st\v{a}sescu and F. Van Oystaeyen, Graded Ring Theory, North Holland, Amsterdam, 1982. MR 84i:16002

[NV2]
C. N\v{a}st\v{a}sescu and F. Van Oystaeyen, Dimensions of Ring Theory, D. Reidel Publishing Company, 1987. MR 88f:16029

[Pa]
D. S. Passman, The Algebraic Structure of Group Rings, Wiley-Interscience, 1977. MR 81d:16001

[Sc]
A. H. Schofield, Questions on skew fields (F. van Oystaeyen, ed.), Methods in Ring Theory, D. Reidel Publishing Company, 1984, pp. 489--495. MR 86f:16017

[SSW]
L. W. Small, J. T. Stafford and R. B. Warfield, Affine algebras of Gelfand-Kirillov dimension one are PI, Math. Proc. Camb. Phil. Soc. 97 (1985), 407--414. MR 86g:16025

[SSta]
S. P. Smith and J. T. Stafford, Regularity of the four dimensional Sklyanin algebra, Compositio Math. 83 (1992), 259--289. MR 93h:16037

[Su]
A. Sudbery, Consistent multiparameter quantisation of $GL(n)$, J. Phys., A 23 (1990), L697--704. MR 91m:17022

[Z1]
J. J. Zhang, Twisted graded algebras and equivalences of graded categories, Proc. London Math. Soc. (in press).

[Z2]
J. J. Zhang, On lower transcendence degree, in preparation.

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Additional Information:

James J. Zhang
Affiliation: Department of Mathematics, Box 354350, University of Washington, Seattle, Washington 98195
Email: Zhang@math.washington.edu

DOI: 10.1090/S0002-9947-96-01702-3
PII: S 0002-9947(96)01702-3
Keywords: Gelfand-Kirillov transcendence degree, Gelfand-Kirillov dimension, division algebra, noncommutative domain
Received by editor(s): May 9, 1995
Additional Notes: This research was supported in part by the NSF
Copyright of article: Copyright 1996, American Mathematical Society




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