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)



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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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, P.-P. Grivel, B. Kaup, A. Haefliger, B. Malgrange, and F. Ehlers, Algebraic 𝐷-modules, Perspectives in Mathematics, vol. 2, Academic Press, Inc., Boston, MA, 1987. MR 882000
  • 3. P. Jarratt, A numerical method for determining points of inflexion, Nordisk Tidskr. Informations-Behandling (BIT) 8 (1968), 31–35. MR 0226845
  • 4. Frank Grosshans, Observable groups and Hilbert’s fourteenth problem, Amer. J. Math. 95 (1973), 229–253. MR 0325628
  • 5. A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. III, Inst. Hautes Études Sci. Publ. Math. 28 (1966), 255. MR 0217086
  • 6. Thierry Levasseur, Anneaux d’opérateurs différentiels, Lecture Notes in Math., vol. 867, Springer, Berlin-New York, 1981, pp. 157–173 (French). MR 633520
  • 7. Thierry Levasseur, Sur la dimension de Krull de l’algèbre enveloppante d’une algèbre de Lie semi-simple, Paul Dubreil and Marie-Paule Malliavin Algebra Seminar, 34th Year (Paris, 1981) Lecture Notes in Math., vol. 924, Springer, Berlin-New York, 1982, pp. 173–183 (French). MR 662259
  • 8. Thierry Levasseur, La dimension de Krull de 𝑈(𝑠𝑙(3)), J. Algebra 102 (1986), no. 1, 39–59 (French, with English summary). MR 853230, 10.1016/0021-8693(86)90127-4
  • 9. 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
  • 10. S. P. Smith, Krull dimension of the enveloping algebra of 𝑠𝑙(2,𝐶), J. Algebra 71 (1981), no. 1, 189–194. MR 627433, 10.1016/0021-8693(81)90114-9
  • 11. È. B. Vinberg and V. L. Popov, A certain class of quasihomogeneous affine varieties, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972), 749–764 (Russian). MR 0313260

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 16Sxx, 17Bxx

Retrieve articles in all journals with MSC (2000): 16Sxx, 17Bxx

Additional Information

Thierry Levasseur
Affiliation: Département de Mathématiques, Université de Brest, 29285 Brest cedex, France

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