Abstract
If g is a semisimple Lie algebra in characteristic zero then the universal enveloping algebra U(g) has two important properties. First, as shown by H. Weyl, every nite dimensional representation of U(g) is completely reducible (this property characterizes nite dimensional semisimple Lie algebras in characteristic zero). Second, as shown by Harish-Chandra, there are enough nite dimensional representations, in the sense that the intersection of the kernels of all the nite dimensional representations of U(g) is zero. H.P. Kraft and L.W. Small [1] call an algebra A over a eld k an FCR-algebra (FCR coming from \Finite dimensional modules are Completely Reducible) if A satis es the two properties below:
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