Diagrammes de Dynkin et algèbres enveloppantes d'algèbres de lie semi-simples

Abstract

Let g be a semi-simple complex Lie algebra, U = U( g ) its enveloping algebra, and A a minimal primitive factor of U, with central character χ. Under the assumption that χ is regular and integral, we prove that the Dynkin diagram of g is a Morita invariant of A. Further, a slight refinement implies that the flag variety of g is determined, within all generalized flag varieties, by its ring of differential operators. Then we derive the following consequences. First, if U( g ) ≅ U( g ′), for some Lie algebra g ′, then g ′ ≅ g . Second, any automorphism of U acts on the centre, and on some dense open subset of the primitive spectrum, as a diagram automorphism. We conjecture that this result holds true on the whole primitive spectrum, and give K-theoretic versions of the result and the conjecture. We also improve a key result of [1]. Finally, when χ is only assumed to be regular, we prove, using a result of Soergel, that the Weyl group of g is a Morita invariant of A.