Invariants pour l'action d'un groupe fini sur l'algèbre enveloppante d'une algèbre de Lie semi-simple

Isomorphisms of Finite Invariant for Enveloping Algebras, Semi-simple Case

“…Let , * the projection from U onto UÂAnn(V(*)). As in [10] and [4], we show that for any * in P + , the algebras , * (U H ) and , * (C[H]) are in duality. This duality is a consequence of the following lemma:…”

“…K Remark 1. This theorem due to Kraft and Small, [10], is precised in [4], where the description of the U H -module P i and its annihilator is given from the data of N i . Let us note that the method in [4] can be generalized, by the results of [14], in the case of the action of a semi-simple finite dimensional Hopf algebra H, on an algebra whose finite dimensional representations are completely reducible, and on which H acts in a compatible way.…”

“…Moreover, this action is compatible with the action of U on V(*), cf. [4]. Let , * the projection from U onto UÂAnn(V(*)).…”

“…More precisely, S= * # P + V(*), in which V(*) denotes the (finite dimensional) U-module with highest weight *. In [10], [4], the dimension of a U H -submodule of V(*) is seen as the multiplicity of an irreducible representation of H. With the help of a multiparametered version of a theorem of Howe, [6], settled in 3.1, we obtain the following fact: the quotients of the dimensions of the finite dimensional U H -modules corresponding to an integral regular central character / * , tend, when * tends toward infinity (inside a cone), to the quotients of the dimensions of the irreducible representations of H. So, U H``s ees'' these dimensions and we can recover the algebra C[H]. In the case SL 2 , the classification of the finite subgroups as well as the constant {, [4], associated to each one of these subgroups, enable us to recover the group H.…”