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Krull dimension of factor rings of the enveloping algebra of a semi-simple Lie algebra

Published online by Cambridge University Press:  24 October 2008

S. P. Smith
S. P. Smith
Affiliation:
University of Southern California, Los Angeles, California
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Extract

Let g be a semi-simple complex Lie algebra with enveloping algebra U(g). It is shown that the Krull dimension of U(g) is bounded above by dim g − r, where r is half the minimal dimension of a non-trivial G orbit in g (G is the adjoint group of g).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 93 , Issue 3 , May 1983 , pp. 459 - 466
Copyright
Copyright © Cambridge Philosophical Society 1983

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References

REFERENCES

(1)Bernstein, I. N.The analytic continuation of generalized functions with respect to a parameter. Funktional. Anal. Prilozen 6 (1972), 26–40.Google Scholar
(2)Bourbaki, N.Groupes et algèbres de Lie, chapitres 7–8 (Hermann, Paris, 1975).Google Scholar
(3)Dixmier, J.Enveloping algebras (North-Holland Mathematical Library, 1977).Google Scholar
(4)Duflo, M.Sur la classification des ideaux primitifs dans l'algèbre enveloppante d'une algèbre de Lie semi-simple. Ann. of Math. 105 (1977), 107–120.CrossRefGoogle Scholar
(5)Gabber, O.The integrability of the characteristic variety. Amer. J. Math. 103 (1981), 445–468.CrossRefGoogle Scholar
(6)Gabber, O. Private communication.Google Scholar
(7)Gordon, R. and Robson, J. C.Krull dimension. Amer. Math. Soc. Mem. 133 (1973).Google Scholar
(8)Humphreys, J. E. Introduction to Lie algebras and representation theory. Graduate Texts in Math. no. 9 (Springer-Verlag, New York, 1972).Google Scholar
(9)Joseph, A.A generalisation of Quillen's Lemma and its application to the Weyl algebras. Israel J. Math. 28 (1977), 177–192.Google Scholar
(10)Joseph, A.Minimal realizations and spectrum generating algebras. Comm. Math. Phys. 36 (1974), 325–328.Google Scholar
(11)Joseph, A.The minimal orbit in a simple Lie algebra and its associated maximal ideal. Ann. Scient. Ec. Norm. Sup. 9 (1976), 1–30.CrossRefGoogle Scholar
(12)Joseph, A.Gelfand-Kiiillov dimension for the annihilators of simple quotients of Verma modules. J. Lond. Math. Soc. 18 (1978), 50–60.CrossRefGoogle Scholar
( 13)Joseph, A. Application de la théorie des anneaux aux algèbres enveloppantes. Paris Lecture Notes 1981.Google Scholar
(14)Levasseur, T.Dimension injective de quotients primitifs minimaux de l'algèbre envelop-pante d'une algèbre de Lie semi-simple. C.R.A.S. 292 (1981), 385–387.Google Scholar
(15)Levasseur, T.Sur la dimension de Krull de l'algèbre enveloppante d'une algèbre de Lie semi-simple. Preprint (University of Paris VI, 1982).Google Scholar
(16)Rentschler, R. and Gabriel, P.Sur la dimension des anneaux et ensembles ordonnés. C.R. Acad. Sci. 265 (1967), 712–715.Google Scholar
(17)Roos, J. E.Complements a l'étude des quotients primitifs des algebres enveloppantes des algebres de Lie semi-simples. C.R. Acad. Sci. 276 (1973), 447–450.Google Scholar
(18)Smith, S. P.Krull dimension of the enveloping algebra of sl(2, ). J. Alg. 71 (1981), 189–194.Google Scholar
(19)Smith, S. P. Central localisation and Gelfand-Kirillov dimension. Submitted.Google Scholar
(20)Smith, S. P. Krull dimension and Gelfand-Kirillov dimension of modules over enveloping algebras. Thesis (Leeds, 1981).Google Scholar