2-affine complete algebras need not be affine complete by Aichinger E.

By Aichinger E.

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Example text

Otherwise, it follows easily from part (i) that there exists a non-zero maximal closed ideal J• of L'; the preimage J of J' in L is then a strictly maximal closed ideal of L. Let L be a primitive Lie algebra over K. of L is a faithful, A primitive realization transitive representation >.. of L on F{v':'}, for some finite-dimensional vector space V, such that the isotropy subalgebra of >.. is a primitive subalgebra of L. :. _ unique primitive subalgebra LO; thus we see from Theorem 1. 2 that a primitive realization of such a primitive Lie algebra is uniquely determined, up to the action of an isomorphism of F{(L/LO>*}.

2, we see that a linearly compact Lie algebra L is transitive if and only if there exists a neighborhood 0 of 0 which contains no ideals of L except {o}, for, if S is an open subspace of L contained in {) , then DL(S) is a fundamental subalgebra of L; the converse is obvious. Lemma 1. 1. If (L, LO) is a transitive Lie algebra, then the sequence {Di(L0 )} p_ > 1 forms a fundamental system of neighborhoods of 0 in L. Proof: According to Proposition 1. 2, (iii), the spaces Dt(Lo), for p ;;::: 1, form a descending chain of open subalgebras of L, and, by n Dt(LO) = {o}.

3 that each of the summands grP(L, 6) is finite-dimensional; moreover, for p ~ -2 we have grP(L, 6) = the definition of 6. {o}, by Thus, the space is a finite-dimensional, abelian Lie subalgebra of gr(L, 6). The universal enveloping algebra of V is naturally isomorphic to the algebra S(V) of symmetric tensors on V; therefore, the adjoint representation of V on gr(L, 6) gives rise to a canonical structure of S(V)-module on gr(L, 6). This structure satisfies the relation for all p;;:: 0 and qEZ. For convenience, if U is a vector space over K we write the symmetric algebra S(U) as S(U) G) sP(U) p€Z with sP(U) = {o} for p

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