thanks

]]>(1.) The above proof, in which a representation of a subspace is inductively extended to the whole Lie algebra, indeed seems to be the ‘standard’ proof for Ado’s theorem. A slightly different approach, by Y. Neretin (“A CONSTRUCTION OF FINITE-DIMENSIONAL FAITHFUL

REPRESENTATION OF LIE ALGEBRA,” http://www.ams.org/mathscinet-getitem?mr=1982443), can be found in the literature.

The objects and techniques used in his proof are almost exactly the same but with the following difference: Neretin first embeds the (complex, finite-dimensional) Lie algebra into a Lie algebra that is the semidirect product of a reductive Lie algebra and a nilpotent ideal such that the acts completely reducibly on the . It then suffices to find a (finite-dimensional) faithful representation of . This is done exactly as in the proof above: by letting act on its universal enveloping algebra through derivations (p) and left multiplications (h), and then considering the quotient , where is an invariant ideal that is spanned by all elements that are long enough, in some sense.

The embedding theorem follows from what Neretin calls “elementary expansions,” based on the Jordan-Chevalley decomposition for derivations. I suppose this idea is also implicitly contained in the standard proof.

Neretin’s proof is short and elegant, but it also assumes theorem 7, corollary 8, theorem 10, and so on.

(2.) Concerning “Remark 2″: There have been attempts to determine (bounds for) the minimal degree of a faithful representation of a given Lie algebra in terms of other natural invariants associated to . Some classes admit polynomial bounds, but the general bounds appear to be exponential in the dimension.

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