By Albert A.A.

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Let L be a Lie algebra over K, and let S be a subspace of L. 3 {; € s 1 [;. L 1 c s} . Using this, we define inductively a descending sequence {Di,(S)}p_ >O of subspaces of S, by (1. 6) for convenience, we set Di,(S} = L, for p < 0 . We record the following properties of the derived subspace: Proposition 1. 2.. Let L be a Lie algebra, and let S be a subspace of L. (i) S. The derived subspace DL(S) is a subalgebra of L contained in Moreover, if S is a subalgebra of L, then DL(S) is an ideal of S.

Subalgebra of L. (Z; of L by subalgebras, with for p < 0 , M, LP = Di,(M) Then, the filtration {LP} "O €Hu for p > 0 . endows L with a structure of filtered Lie 27 algebra, that is, if p s; q, then (1. 8) and, for all p, q e. Z:, (1. 9) Furthermore, if L is a linearly compact Lie algebra, and M is an open (resp. closed) subalgebra of L, then each of the subalgebras LP is open (resp. closed) in L, for all p e. Z:. Proof: The first inclusion (1. 8) is obvious. by induction on p + q; the case p + q < 0 is trivial, From the definition of the filtration {Lr}r e.

Let V be a finite-dimensional vector space over K. Recall that, for any vector space W over K, we have defined a natural structure of * S(V)-module on the tensor product W IC\ ~K S(V ) ; if vE V ov for multiplication by v in the module W@K S(V*). G = © Gp' with = S 1 (V), we write Suppose that Gp c W(8)K sP(v'~) ' pEZ is a graded S(V)-submodule of W(8)K S(V,~). For all pEZ, there is a natural mapping such that, for all a E G if p > 0, p and vE V, the mapping 6 is injective. Assume that dim(V) = n; we write Aq(V1' ) for the q-th exterior power of v'~, for q ~ 0.