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**Algèbre non commutative, groupes quantiques et invariants Septième contact Franco-Belge, Reims, Juin 1995**

J. Alev, G. Cauchon (Éd.)

Séminaires et Congrès

**Coxeter Structure and Finite Group Action**

Anthony JOSEPH

Séminaires et Congrès **2** (1997), 185-219

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**Résumé :**

Let *U*( *g*) be the enveloping algebra of a semi-simple Lie algebra *g*. Very little is known about the nature of . However, if *G* is a finite subgroup of then very general results of Lorenz-Passman and of Montgomery can be used to relate to . As noted by Alev-Polo one may read off the Dynkin diagram of *g* from and they used this to show that *U*( *g*)^{G} could not be again the enveloping algebra of a semi-simple Lie algebra unless *G* is trivial. Again let *U* be the minimal primitive quotient of *U*( *g*) admitting the trivial representation of *g*. A theorem of Polo asserts that if *U*^{G} is isomorphic to a similarly defined quotient of *U*( *g*'): *g*' semi-simple, then . However in this case one cannot say that *G* is trivial.

The main content of this paper is the possible generalization of Polo's theorem to other minimal primitive quotients. A very significant technical difficulty arises from the Goldie rank of the almost minimal primitive quotients being >1. Even under relatively strong hypotheses (regularity and integrality of the central character) one is only able to say that the Coxeter diagrams of *g* and *g*' coincide. The main thrust of the proofs is a systematic use of the Lorenz-Passman-Montgomery theory and the known very detailed description of . Unfortunately there is a severe lack of good examples. During this work some purely ring theoretic results involving Goldie rank comparisons and skew-field extensions are presented. A new inequality for Gelfand-Kirillov dimension is obtained and this leads to an interesting question involving a possible application of the intersection theorem.

**Class. math. :** 20G05, 14L30, 20C30

Publié avec le concours de : Centre National de la Recherche Scientifique