Isomorphism problems and groups of automorphisms for generalized Weyl algebras
HTML articles powered by AMS MathViewer
- by V. V. Bavula and D. A. Jordan PDF
- Trans. Amer. Math. Soc. 353 (2001), 769-794 Request permission
Abstract:
We present solutions to isomorphism problems for various generalized Weyl algebras, including deformations of type-A Kleinian singularities and the algebras similar to $U(\mathfrak {sl}_2)$ introduced by S. P. Smith. For the former, we generalize results of Dixmier on the first Weyl algebra and the minimal primitive factors of $U(\mathfrak {sl}_2)$ by finding sets of generators for the group of automorphisms.References
- S. A. Amitsur, Commutative linear differential operators, Pacific J. Math. 8 (1958), 1–10. MR 95305, DOI 10.2140/pjm.1958.8.1
- M. Artin, J. Tate, and M. Van den Bergh, Modules over regular algebras of dimension $3$, Invent. Math. 106 (1991), no. 2, 335–388. MR 1128218, DOI 10.1007/BF01243916
- V. V. Bavula, Finite-dimensionality of $\textrm {Ext}^n$ and $\textrm {Tor}_n$ of simple modules over a class of algebras, Funktsional. Anal. i Prilozhen. 25 (1991), no. 3, 80–82 (Russian); English transl., Funct. Anal. Appl. 25 (1991), no. 3, 229–230 (1992). MR 1139880, DOI 10.1007/BF01085496
- V. V. Bavula, Generalized Weyl algebras and their representations, Algebra i Analiz 4 (1992), no. 1, 75–97 (Russian); English transl., St. Petersburg Math. J. 4 (1993), no. 1, 71–92. MR 1171955
- V. V. Bavula, Description of two-sided ideals in a class of noncommutative rings. I, Ukraïn. Mat. Zh. 45 (1993), no. 2, 209–220 (Russian, with Russian and Ukrainian summaries); English transl., Ukrainian Math. J. 45 (1993), no. 2, 223–234. MR 1232403, DOI 10.1007/BF01060977
- V. V. Bavula, Description of two-sided ideals in a class of noncommutative rings. II, Ukraïn. Mat. Zh. 45 (1993), no. 3, 307–312 (Russian, with Russian and Ukrainian summaries); English transl., Ukrainian Math. J. 45 (1993), no. 3, 329–334. MR 1238673, DOI 10.1007/BF01061007
- V. V. Bavula, Generalized Weyl algebras, kernel and tensor-simple algebras, their simple modules, CMS Conf. Proc. 14 (1993), 83–107.
- Vladimir Bavula, Global dimension of generalized Weyl algebras, Representation theory of algebras (Cocoyoc, 1994) CMS Conf. Proc., vol. 18, Amer. Math. Soc., Providence, RI, 1996, pp. 81–107. MR 1388045
- Vladimir Bavula, Tensor homological minimal algebras, global dimension of the tensor product of algebras and of generalized Weyl algebras, Bull. Sci. Math. 120 (1996), no. 3, 293–335. MR 1399845
- Richard H. Capps, Orbit depths of affine Kac-Moody algebras, J. Phys. A 23 (1990), no. 11, 1851–1860. MR 1063419, DOI 10.1088/0305-4470/23/11/013
- Jacques Dixmier, Sur les algèbres de Weyl, Bull. Soc. Math. France 96 (1968), 209–242 (French). MR 242897, DOI 10.24033/bsmf.1667
- Jacques Dixmier, Quotients simples de l’algèbre enveloppante de ${\mathfrak {s}}{\mathfrak {l}}_{2}$, J. Algebra 24 (1973), 551–564 (French). MR 310031, DOI 10.1016/0021-8693(73)90127-0
- D. B. Fairlie, Quantum deformations of $\textrm {SU}(2)$, J. Phys. A 23 (1990), no. 5, L183–L187. MR 1048740, DOI 10.1088/0305-4470/23/5/001
- Timothy J. Hodges, Noncommutative deformations of type-$A$ Kleinian singularities, J. Algebra 161 (1993), no. 2, 271–290. MR 1247356, DOI 10.1006/jabr.1993.1219
- David A. Jordan, Iterated skew polynomial rings and quantum groups, J. Algebra 156 (1993), no. 1, 194–218. MR 1213792, DOI 10.1006/jabr.1993.1070
- David A. Jordan, Krull and global dimension of certain iterated skew polynomial rings, Abelian groups and noncommutative rings, Contemp. Math., vol. 130, Amer. Math. Soc., Providence, RI, 1992, pp. 201–213. MR 1176120, DOI 10.1090/conm/130/1176120
- David A. Jordan, Height one prime ideals of certain iterated skew polynomial rings, Math. Proc. Cambridge Philos. Soc. 114 (1993), no. 3, 407–425. MR 1235988, DOI 10.1017/S0305004100071693
- David A. Jordan, Primitivity in skew Laurent polynomial rings and related rings, Math. Z. 213 (1993), no. 3, 353–371. MR 1227487, DOI 10.1007/BF03025725
- David A. Jordan, Finite-dimensional simple modules over certain iterated skew polynomial rings, J. Pure Appl. Algebra 98 (1995), no. 1, 45–55. MR 1316996, DOI 10.1016/0022-4049(95)90017-9
- David A. Jordan, A simple localization of the quantized Weyl algebra, J. Algebra 174 (1995), no. 1, 267–281. MR 1332871, DOI 10.1006/jabr.1995.1128
- D. A. Jordan, Down-up algebras and ambiskew polynomial rings, J. Algebra 228 (2000), 311–346.
- David A. Jordan and Imogen E. Wells, Invariants for automorphisms of certain iterated skew polynomial rings, Proc. Edinburgh Math. Soc. (2) 39 (1996), no. 3, 461–472. MR 1417689, DOI 10.1017/S0013091500023221
- G. R. Krause and T. H. Lenagan, Growth of algebras and Gel′fand-Kirillov dimension, Research Notes in Mathematics, vol. 116, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 781129
- Lieven Le Bruyn, Two remarks on Witten’s quantum enveloping algebra, Comm. Algebra 22 (1994), no. 3, 865–876. MR 1261010, DOI 10.1080/00927879408824881
- J. C. McConnell and J. C. Robson, Homomorphisms and extensions of modules over certain differential polynomial rings, J. Algebra 26 (1973), 319–342. MR 342566, DOI 10.1016/0021-8693(73)90026-4
- 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
- C. Năstăsescu and F. van Oystaeyen, Graded ring theory, North-Holland Mathematical Library, vol. 28, North-Holland Publishing Co., Amsterdam-New York, 1982. MR 676974
- Alexander L. Rosenberg, Noncommutative algebraic geometry and representations of quantized algebras, Mathematics and its Applications, vol. 330, Kluwer Academic Publishers Group, Dordrecht, 1995. MR 1347919, DOI 10.1007/978-94-015-8430-2
- Louis H. Rowen, Ring theory. Vol. I, Pure and Applied Mathematics, vol. 127, Academic Press, Inc., Boston, MA, 1988. MR 940245
- 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}
- S. P. Smith, A class of algebras similar to the enveloping algebra of $\textrm {sl}(2)$, Trans. Amer. Math. Soc. 322 (1990), no. 1, 285–314. MR 972706, DOI 10.1090/S0002-9947-1990-0972706-5
- J. T. Stafford, Homological properties of the enveloping algebra $U(\textrm {Sl}_{2})$, Math. Proc. Cambridge Philos. Soc. 91 (1982), no. 1, 29–37. MR 633253, DOI 10.1017/S0305004100059089
- Edward Witten, Gauge theories, vertex models, and quantum groups, Nuclear Phys. B 330 (1990), no. 2-3, 285–346. MR 1043385, DOI 10.1016/0550-3213(90)90115-T
- S. L. Woronowicz, Twisted $\textrm {SU}(2)$ group. An example of a noncommutative differential calculus, Publ. Res. Inst. Math. Sci. 23 (1987), no. 1, 117–181. MR 890482, DOI 10.2977/prims/1195176848
- Cosmas Zachos, Elementary paradigms of quantum algebras, Deformation theory and quantum groups with applications to mathematical physics (Amherst, MA, 1990) Contemp. Math., vol. 134, Amer. Math. Soc., Providence, RI, 1992, pp. 351–377. MR 1187297, DOI 10.1090/conm/134/1187297
Additional Information
- V. V. Bavula
- Affiliation: Department of Mathematics and Computer Science, University of Antwerp, U. I. A., B-2610 Wilrijk, Belgium
- Address at time of publication: Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom
- MR Author ID: 293812
- Email: v.bavula@sheffield.ac.uk
- D. A. Jordan
- Affiliation: Department of Pure Mathematics, University of Sheffield, Hicks Building, Sheffield S3 7RH, United Kingdom
- MR Author ID: 199952
- Email: d.a.jordan@sheffield.ac.uk
- Received by editor(s): July 12, 1999
- Published electronically: October 13, 2000
- Additional Notes: This work was done during visits to the University of Sheffield by the first author with the support of the London Mathematical Society
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 353 (2001), 769-794
- MSC (2000): Primary 16S36, 16W20, 16W35
- DOI: https://doi.org/10.1090/S0002-9947-00-02678-7
- MathSciNet review: 1804517