Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

The semicenter of an enveloping algebra is factorial


Authors: Lieven Le Bruyn and Alfons I. Ooms
Journal: Proc. Amer. Math. Soc. 93 (1985), 397-400
MSC: Primary 17B35
DOI: https://doi.org/10.1090/S0002-9939-1985-0773989-0
MathSciNet review: 773989
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ L$ be a finite-dimensional Lie algebra over a field $ k$ of characteristic zero, and $ U(L)$ its universal enveloping algebra. We show that the semicenter of $ U(L)$ is a UFD. More generally, the same result holds when $ k$ is replaced by any factorial ring $ R$ of characteristic zero.


References [Enhancements On Off] (What's this?)

  • [1] D. D. Anderson and D. F. Anderson, Divisibility properties of graded domains, Canad. J. Math. 34 (1982), 196-215. MR 650859 (84e:13026)
  • [2] -, Divisorial ideals and invertible ideals in a graded integral domain, J. Algebra (to appear). MR 661873 (83i:13001)
  • [3] D. F. Anderson, Graded Krull domains, Comm. Algebra 7(1) (1979), 79-106. MR 514866 (80c:13015)
  • [4] A. Bouvier, Anneaux de Krull gradués, Université Claude-Bernard, Lyon I, 1981.
  • [5] M. Chamarie, Maximal orders applied to enveloping algebras, Proc. Conf. on Ring Theory (Antwerp, 1980), Lecture Notes in Math., vol. 825. Springer-Verlag, Berlin and New York, 1980, pp. 19-27. MR 590782 (82c:17007)
  • [6] L. Delvaux, E. Nauwelaerts and A. I. Ooms, On the semi-center of a universal enveloping algebra, J. Algebra (to appear). MR 792958 (86k:17007)
  • [7] J. Dixmier, Enveloping algebras, North-Holland Mathematical Library. Vol. 14, North-Holland, Amsterdam, 1977. MR 0498740 (58:16803b)
  • [8] V. Ginzburg, On the ideals of $ U(\mathfrak{g})$ (to appear).
  • [9] M. P. Malliavin, Ultra produit d'algèbres de Lie, Lecture Notes in Math., vol. 924, Springer-Verlag, Berlin and New York, 1982, pp. 157-166. MR 662257 (84k:17007)
  • [10] G. Maury and D. Reynaud, Algèbres enveloppantes de $ R$-algèbres de Lie sur certains anneaux $ R$, Comm. Algebra 11(7), (1983), 753-769. MR 694600 (84e:17011)
  • [11] C. Moeglin, Factorialité dans le algèbres enveloppantes, C. R. Acad. Sci. Paris Ser. A 282 (1976), 1269-1272. MR 0419544 (54:7565)
  • [12] -, Idéux bilatères des algèbres enveloppantes, Bull. Soc. Math. France 108 (1980), 143-186.
  • [13] D. Reynaud, Algèbres enveloppantes de $ R$-algèbres de Lie sur certains anneaux $ R$, Thèse de troisième cycle, Université Claude Bernard, Lyon I, 1982.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 17B35

Retrieve articles in all journals with MSC: 17B35


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1985-0773989-0
Keywords: Finite-dimensional Lie algebra, universal enveloping algebra, semicenter, factorial ring
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society