On normed Lie algebras with sufficiently many subalgebras of codimension I

Abstract

Let H be a finite or infinite dimensional Lie algebra. Barnes [2] and Towers [5] considered the case when H is a finite-dimensional Lie algebra over an arbitrary field, and all maximal subalgebras of H have codimension 1. Barnes, using the cohomology theory of Lie algebras, investigated solvable algebras, and Towers extended Barnes's results to include all Lie algebras. In [4] complex finite-dimensional Lie algebras were considered for the case when all the maximal subalgebras of H are not necessarily of codimension 1 but when S(H)where S(H) is the set of all Lie subalgebras in H of codimension 1. Amayo [1] investigated the finite-dimensional Lie algebras with core-free subalgebras of codimension 1 and also obtained some interesting results about the structure of infinite dimensional Lie algebras with subalgebras of codimension 1.By X we shall denote the class of complex finite or infinite dimensional normed Lie algebras for which (1) holds. In Section 2 the results of Amayo will be applied in order to prove that for every complex normed Lie algebra H and for every subalgebra M e S(H) the largest Lie ideal /(M) of H contained in M has codimension less or equal to 3. Using this result for the case when HeX we shall show that, if S k (H) = {MeS(H):codim/(M) = /c}, for fc = l,2,3, thenis a semisimple ideal in H and R(H) = f) M es 3 (H)I(M) is the radical of H. We shall also prove that H (2) £L(H), so that i?(H) (2) =0. If H is finite-dimensional, then it was proved in [4] that H = L{H) + R{H) and that L{H) = L l + ••• +Ln , where all L t are Lie ideals in H and isomorphic to s/(2, C). If H is infinite dimensional, then this does not necessarily hold any longer. We shall consider an example of a normed Lie algebra H from 3E such that R(H) = 0 but L(H)^H. We shall also show that the property of belonging to X is inherited by all closed subalgebras of H and by all quotient algebras H/Sr s where S is any subset of S(H) and where y s = f) Ms sl(M)-Finally, we shall consider the set T of all ideals /(M) such that codim /(M) = 3 and shall introduce a Jacobson's topology on T.In Section 3 the. structure of solvable algebras from X is investigated. For every R e X we consider a special set £ of functionals on R from R°l } (Rf t) is the polar of R (1) ) and 199

Banach Lie algebras with Lie subalgebras of finite codimension: Their invariant subspaces and Lie ideals

“…Amayo showed in Lemma 2.2(c) of [1] that, if L is a finite-dimensional Lie algebra over field of characteristic 0 and codim(L 0 ) = 1, then L 0 contains a Lie ideal K of L such that dim(L/K) 3. The extension of this result to Banach Lie algebras obtained in [10] is an easy consequence of Corollary 9.1. Proof.…”

“…Recall that a complex Lie algebra L is called a Banach Lie algebra, if it is a Banach space in some norm · that satisfies The next result gives a partial answer to the question raised in [10]. Amayo showed in Lemma 2.2(c) of [1] that, if L is a finite-dimensional Lie algebra over field of characteristic 0 and codim(L 0 ) = 1, then L 0 contains a Lie ideal K of L such that dim(L/K) 3.…”

“…The first author in [10] investigated the structure of Banach Lie algebras L with "sufficiently" many Lie subalgebras of codimension 1 and showed that dim(L/I (L 0 )) 3 for each Lie subalgebra L 0 of codimension 1. The question was also raised as to whether dim(L/I (L 0 )) < ∞ for each closed Lie subalgebra L 0 of L of finite codimension.…”

Growth rates for persistently excited linear systems

“…Using the fact that d ≥ 3 and a theorem of Amayo (see [1] and also [15,Theorem 1.1]), one gets that the only Lie subalgebra of M d (R) of codimension 1 is sl(d, R). Since Tr(BK) = 0 then L K = sl(d, R) and therefore L K must be equal to M d (R).…”

“…Acknowledgements It is a pleasure to acknowledge U. Helmke and P. Kokkonen for pointing out, respectively, the papers [13,21] and [1,15], which led us to Proposition 5.1. We also thank J-P. Gauthier and F. Wirth for several fruitful exchanges.…”