Inventiones mathematicae

, Volume 56, Issue 3, pp 191–213 | Cite as

Kostant's problem, Goldie rank and the Gelfand-Kirillov conjecture

  • A. Joseph
Article

Abstract

Let g be a complex semisimple Lie algebra andU(g) its enveloping algebra. GivenM a simpleU(g) module, letL(M, M) denote the subspace of ad g finite elements of Hom(M, M). Kostant has asked if the natural homomorphism ofU(g) intoL(M, M) is surjective. Here the question is analysed for simple modules with a highest weight vector. This has a negative answer if g admits roots of different length ([7], 6.5). Here general conditions are obtained under whichU(g)/AnnM andL(M, M) have the same ring of fractions—in particular this is shown to always hold if g has only typeA n factors. Combined with [21], this provides a method for determining the Goldie ranks for the primitive quotients ofU(g). Their precise form is given in typeA n (Cartan notation) for which the generalized Gelfand-Kirillov conjecture for primitive quotients is also established.

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Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • A. Joseph
    • 1
  1. 1.Centre de MathématiquesOrsayFrance

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