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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Krull dimension of the enveloping algebra of a semisimple Lie algebra


Author: Thierry Levasseur
Journal: Proc. Amer. Math. Soc. 130 (2002), 3519-3523
MSC (2000): Primary 16Sxx, 17Bxx
Published electronically: May 15, 2002
MathSciNet review: 1918828
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Abstract: Let $\mathfrak{g}$ be a complex semisimple Lie algebra and $U(\mathfrak{g})$ be its enveloping algebra. We deduce from the work of R. Bezrukavnikov, A. Braverman and L. Positselskii that the Krull-Gabriel-Rentschler dimension of $U(\mathfrak{g})$ is equal to the dimension of a Borel subalgebra of $\mathfrak{g}$.


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

  • 1. R. Bezrukavnikov, A. Braverman and L. Positselskii, Gluing of abelian categories and differential operators on the basic affine space, preprint, 13 pages, arXiv:math.RT/0104114.
  • 2. A. Borel et al., Algebraic $D$-modules, Academic Press, London-New York, 1987. MR 89g:32014
  • 3. P. Gabriel et R. Rentschler, Sur la dimension des anneaux et ensembles ordonnés, C. R. Acad. Sci. Paris Sér. I, 265 (1967), 712-715. MR 37:2431
  • 4. F. Grosshans, Observable groups and Hilbert's fourteenth problem, Amer. J. Math., 95 (1979), 229-253. MR 48:3975
  • 5. A. Grothendieck, Éléments de géométrie algébrique, Chapitre IV, Publications de l'IHES, 32 (1967). MR 36:178
  • 6. T. Levasseur, Anneaux d'opérateurs différentiels, in Séminaire P. Dubreil et M.-P. Malliavin (1980), Lecture Notes in Math. 867, 1981, Springer-Verlag, Berlin, 157-173. MR 84j:32009
  • 7. -, Sur la dimension de Krull de l'algèbre enveloppante d'une algèbre de Lie semi-simple, in Séminaire P. Dubreil et M.-P. Malliavin (1981), Lecture Notes in Math. 924, 1982, Springer-Verlag, Berlin, 173-183. MR 84j:17011
  • 8. -, La dimension de Krull de $U({sl}(3))$, J. Algebra , 102 (1986), 39-59. MR 87m:17019
  • 9. J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, John Wiley & Sons, Chichester, 1987. MR 89j:16023
  • 10. S. P. Smith, Krull dimension of the enveloping algebra of $sl(2,{\mathbb{C} })$, J. Algebra , 71 (1981), 89-94. MR 82m:17005
  • 11. E. B. Vinberg and V. L. Popov, On a class of quasi-homogeneous affine varieties, Math. USSR Izvestija, 6 (1972), 743-758. (Russian original, MR 47:1815)

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Additional Information

Thierry Levasseur
Affiliation: Département de Mathématiques, Université de Brest, 29285 Brest cedex, France
Email: Thierry.Levasseur@univ-brest.fr

DOI: http://dx.doi.org/10.1090/S0002-9939-02-06507-3
PII: S 0002-9939(02)06507-3
Keywords: Krull dimension, semisimple Lie algebra, enveloping algebra, differential operators
Received by editor(s): July 30, 2001
Published electronically: May 15, 2002
Communicated by: Lance W. Small
Article copyright: © Copyright 2002 American Mathematical Society