Coxeter Structure and Finite Group Action

Let U( ) be the enveloping algebra of a semi-simple Lie algebra . Very little is known about the nature of AutU( ). However, if G is a finite subgroup of AutU( ) then very general results of Lorenz-Passman and of Montgomery can be used to relate SpecU( ) to SpecU( ). As noted by Alev-Polo one may read off the Dynkin diagram of from SpecU( ) and they used this to show that U( ) 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( ) admitting the trivial representation of . A theorem of Polo asserts that if U is isomorphic to a similarly defined quotient of U( ) : ′ 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 and ′ coincide. The main thrust of the proofs is a systematic use of the Lorenz-Passman-Montgomery theory and the known very detailed description of PrimU( ). 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. Résumé Soit U( ) l’algèbre enveloppante d’une algèbre de Lie semi-simple . On sait très peu de choses sur AutU( ). Néanmoins, si G désigne un sous-groupe fini de AutU( ), alors des résultats généraux de Lorenz-Passman et Montgomery relient SpecU( ) à SpecU( ). Alev et Polo ont observé qu’on peut lire le diagramme de Dynkin de sur SpecU( ) et ils en ont déduit que U( ) ne peut être isomorphe à l’algèbre enveloppante d’une algèbre de Lie que si G est trivial. Soit U le quotient primitif minimal de U( ) admettant la représentation triviale de . D’après un théorème de Polo, si U est isomorphe à un quotient AMS 1980 Mathematics Subject Classification (1985 Revision): 17B35 ∗Laboratoire de Mathématiques Fondamentales, Université de Pierre et Marie Curie, 4 Place Jussieu, Paris 75252 Cedex 05, France, and The Donald Frey Professorial Chair, Department of Theoretical Mathematics, The Weizmann Institute of Science, Rehovot 76100, Israel. Société Mathématique de France