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Bibliography BIBLIOGRAPHY 1. E. Abe, Hopf Algebras, Cambridge Univ. Press, Cambridge, 1980. 2. 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. 3. J. Alev and F. Dumas, Sur Ie corps de fractions de certaines algebres quantiques, J. Algebra 110 (1994), 229-265. 4. J. Alev, A. 1. Ooms, and M. Van den Bergh, A class of counterexamples to the Gel'fand­ Kirillov conjecture, Trans. Amer. Math. Soc. 348 (1996), 1709-1716. 5. A. S. Amitsur, Algebras over infinite fields, Proc. Amer. Math. Soc. 1 (1956), 35-48. 6. H. H. Andersen, J. C. Jantzen, and W. Soergel, Representations of quantum groups at a p-th root of unity and of semisimple groups in characteristic p: independence of p, Asterisque 220 (1994), 1-321. 1. H. H. Andersen, P. Polo and K. Wen, Representations of quantum algebras, Invent. Math. 104 (1991), 1-59; Addendum, Invent. Math. 120 (1995), 409-410. 8. M. Artin, W. Schelter, and J. Tate, Quantum deformations of GLn , Comm. Pure Appl. Math. 44 (1991), 879-895. 9. M. Auslander and O. Goldman, Maximal orders, Trans. Amer. Math. Soc. 91 (1960),1-24. 10. H. Bass, Algebraic K-Theory, Benjamin, New York, 1968. 11. A. Bell and R. Farnsteiner, On the theory of Frobenius extensions and its application to Lie superalgebras, Trans. Amer. Math. Soc. 335 (1993), 407-424. 12. A. D. Bell and G. Sigurdsson, Catenarity and Gel'fand-Kirillov dimension in Ore exten­ sions, J. Algebra 121 (1989), 409-425. 13. G. M. Bergman, The diamond lemma for ring theory, Advances in Math. 29 (1978), 178- 218. 14. S. M. Bhatwadekar, On the global dimension of some filtered algebras, J. London Math. Soc. (2) 13 (1976), 239-248. 15. J.-E. Bjork, The Auslander condition on Noetherian rings, in Sem. d'Algebre P. Dubreil et M.-P. Malliavin 1987-88 (M.-P. Malliavin, ed.), Lecture Notes in Math. 1404, Springer­ Verlag, Berlin, 1989, pp. 137-173. 16. W. Borho, P. Gabriel, and R. Rentschler, Primideale in Einhiillenden aufiosbarer Lie­ Algebren, Lecture Notes in Math. 357, Springer-Verlag, Berlin, 1973. 11. W. Borho and J. C. Jantzen, Uber primitive Ideale in der Einhiillenden einer halbeinfachen Lie-Algebra, Invent. Math. 39 (1977), 1-53. 18. K. A. Brown, Representation theory of Noetherian Hopf algebras satisfying a polynomial identity, in Trends in the Representation Theory of Finite Dimensional Algebras (Seattle 1997) (E. L. Green and B. Huisgen-Zimmermann, Eds.), Contemp. Math. 229 (1998), 49-79. 19. K. A. Brown and K. R. Goodearl, Prime spectra of quantum semisimple groups, Trans. Amer. Math. Soc. 348 (1996), 2465-2502. 20. ___ , A Hilbert basis theorem for quantum groups, Bull. London Math. Soc. 29 (1997), 150-158. 21. ___ , Homological aspects of noetherian PI Hopf algebras and irreducible modules of maximal dimension, J. Algebra 198 (1997), 240-265. 332 BIBLIOGRAPHY 22. K. A. Brown and 1. Gordon, The ramification of centres: Lie algebras in positive charac­ teristic and quantized enveloping algebras, Math. Zeit. 238 (2001), 733-779. 23. ___ , The ramifications of the centres: quantised function algebras at roots of unity, Proc. London Math. Soc. (to appear). 24. K. A. Brown, 1. Gordon, and J. T. Stafford, O€(G) is a free module over O[G], Preprint (2000); available at http://arXiv.org/abs/math . QA/0007179. 25. K. A. Brown and C. R. Hajarnavis, Homologically homogeneous rings, Trans. Amer. Math. Soc. 281 (1984), 197-208. 26. J. L. Bueso, J. G6mez-Torrecillas, and F. J. Lobillo, Re-filtering and exactness of the Gelfand-Kirillov dimension (to appear). 27. P. Caldero, Etude des q-commutations dans l'algebre Uq(n+), J. Algebra 178 (1995), 444- 457. 28. ___ , On the Gelfand-Kirillov conjecture for quantum algebras, Proc. AMS 128 (2000), 943-95l. 29. A. Cannas da Silva and A. Weinstein, Geometric Models for Noncommutative Algebras, Berkeley Math. Lecture Notes 10, Amer. Math. Soc., Providence, 1999. 30. N. Cantarini, The quantized enveloping algebra Uq (sl( n)) at the roots of unity, Communic. Math. Phys. 211 (2000), 207-230. 31. G. Cauchon, Quotients premiers de Oq(ffin(k)), J. Algebra 180 (1996), 530-545. 32. ___ , Effacement des derivations. Spectres premiers et primitifs des algebres quantiques (to appear). 33. ___ , Spectre premier de Oq(Mn(k)). Image canonique et separation normale (to ap­ pear). 34. V. Chari and A. Pressley, A Guide to Quantum Groups, Cambridge Univ. Press, Cam­ bridge, 1994. 35. N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry, Birkhiiuser, Boston, 1997. 36. G. Cliff, The division ring of quotients of the coordinate ring of the quantum general linear group, J. London Math. Soc. (2) 51 (1995),503-513. 37. L. Conlon, Differentiable Manifolds: A First Course, Birkhiiuser, Boston, 1993. 38. L. J. Corwin, 1. M. Gel'fand, and R. Goodman, Quadratic algebras and skew-fields, in Representation Theory and Analysis on Homogeneous Spaces (New Brunswick, NJ 1993), Contemp. Math. 177 (1994), 217-225. 39. L. Dabrowski, C. Reina, and A. Zampa, A[SLq(2)] at roots of unity is a free module over A[SL(2)], math.QA/0004092 (2000). 40. C. De Concini and V. G. Kac, Representations of quantum groups at roots of 1, in Opera­ tor Algebras, Unitary Representations, Enveloping Algebras, and Invariant Theory (Paris 1989), Birkhiiuser, Boston, 1990, pp. 471-506. 41. ___ , Representations of quantum groups at roots of 1: reduction to the exceptional case, in Infinite Analysis, Parts A,B (Kyoto, 1991), World Scientific, River Edge, N.J., 1992, pp. 141-149. 42. C. De Concini, V. G. Kac, and C. Procesi, Quantum coadjoint action, J. Amer. Math. Soc. 5 (1992), 151-189. 43. C. De Concini and V. Lyubashenko, Quantum function algebra at roots of 1, Advances in Math. 108 (1994), 205-262. 44. C. De Concini and C. Procesi, Quantum groups, in D-Modules, Representation Theory, and Quantum Groups (Venezia, June 1992) (G. Zampieri, and A. D'Agnolo, eds.), Lecture Notes in Math. 1565, Springer-Verlag, Berlin, 1993, pp. 31-140. BIBLIOGRAPHY 333 45. Quantum Schubert cells and representations at roots of 1, in Algebraic Groups and Lie Groups (G.l. Lehrer, ed.), Austral. Math. Soc. Lecture Series 9, Cambridge Univ. Press, Cambridge, 1997, pp. 127-160. 46. E. E. Demidov, Modules over a quantum Weyl algebra, Moscow Univ. Math. Bull. 48 (1993), 49-51. 47. ___ , Some aspects of the theory of quantum groups, Russian Math. Surveys 48:6 (1993), 41-79. 48. J. Dixmier, Representations irreductibles des algebres de Lie resolubles, J. Math. Pures Appl. (9) 45 (1966), 1-66. 49. ___ , Ideaux primitifs dans les algebres enveloppantes, J. Algebra 48 (1977), 96-112. 50. V. G. Drinfel'd, Hopf algebras and the quantum Yang-Baxter equation, Soviet Math. Dok­ lady 32 (1985), 254-258. 51. ___ , Quantum groups, in Proc. Internat. Congress of Mathematicians (Berkeley 1986), I, Amer. Math. Soc., Providence, 1987, pp. 798-820. 52. ___ , Quantum groups, J. Soviet Math. 41:2 (1988), 898-915. 53. E. K. Ekstrom, The Auslander condition on graded and filtered Noetherian rings, in Sem. d'Algebre P. Dubreil et M.-P. Malliavin 1987-88 (M.-P. Malliavin, ed.), Lecture Notes in Math. 1404, Springer-Verlag, Berlin, 1989, pp. 220-245. 54. L. D. Faddeev, N. Yu. Reshetikhin, and L. A. Takhtajan, Quantization of Lie groups and Lie algebras, in Algebraic Analysis, Vol. I (M. Kashiwara and T. Kawai, eds.), Academic Press, Boston, 1988, pp. 129-139. 55. R. Farnsteiner and H. Strade, Shapiro's lemma and its consequences in the cohomology theory of modular Lie algebras, Math. Zeitschr. 206 (1991), 153-168. 56. F. Fauquant-Millet, Quantification de la localisation de Dixmier de U(Sln+l(C)), J. Al­ gebra 218 (1999), 93-116. 57. O. Gabber, Equidimensionalite de la variete caracteristique, Expose de O. Gabber redige par T. Levasseur, Universite de Paris VI (1982). 58. P. Gabriel, Representations des algebres de Lie resolubles (d'apres J. Dixmier), in Semi­ naire Bourbaki 1968/69, Lecture Notes in Math. 179, Springer-Verlag, Berlin, 1971, pp. 1- 22. 59. 1. M. Gel'fand and A. A. Kirillov, Fields associated with enveloping algebras of Lie algebras, Soviet Math. Doklady 7 (1966), 407-409. 60. ___ , Sur les corps Lies aux algebres enveloppantes des algebres de Lie, PubI. Math. 1.H.E.S. 31 (1966), 5-19. 61. ___ , The structure of the quotient field of the enveloping algebra of a semisimple Lie algebra, Soviet Math. Doklady 9 (1968), 669-671. 62. ___ , The structure of the Lie field connected with a split semisimple Lie algebra, Func. Anal. Applic. 3 (1969), 6-21. 63. M. Gerstenhaber, A. Giaquinto, and S. D. Schack, Quantum symmetry, in Quantum Groups (Leningrad 1990), Lecture Notes in Math. 1510, Springer-Verlag, Berlin, 1992, pp.9-46. 64. A. Giaquinto, Quantization of tensor representations and deformations of matrix bialge­ bras, J. Pure Appl. Algebra 79 (1992), 169-190. 65. J. Gomez Torrecillas, Gelfand-Kirillov dimension of multi-filtered algebras, Proc. Edin­ burgh Math. Soc. (2) 42 (1999), 155-168. 66. J. Gomez Torrecillas and F. J. Lobillo, Global homological dimension of multifiltered rings and quantized enveloping algebras, J. Algebra 225 (2000), 522-533. 67. K. R. Goodearl, Global dimension of differential operator rings. II, Trans. Amer. Math. Soc. 209 (1975), 65-85. 334 BIBLIOGRAPHY 68. ___ , Prime ideals in skew polynomial rings and quantized Weyl algebras, J. Algebra 150 (1992), 324-377. 69. ___ , 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.
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