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

Kissin, Edward; Shulman, Victor S.; Turovskiı, Yuriı V.

doi:10.1016/j.jfa.2008.10.012

Cited by 4 publications

(2 citation statements)

References 21 publications

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“…One of the main obstacles in transferring this theory to infinite-dimensional Lie algebras is the fact that, in the contrast to associative algebras (see [L]), an infinitedimensional Lie algebra with a maximal Lie subalgebra of finite codimension may have no Lie ideals of finite codimension [A1, A2]. Recently the authors proved in [KST1,KST2] that a Banach Lie algebra with a maximal Lie subalgebra of finite codimension always has a Lie ideal of finite codimension (for Banach Lie algebras with Lie subalgebras of codimension 1 this was proved in [K]). This result provides a powerful tool for the study of infinite-dimensional Frattini-free and Jacobson-free Banach Lie algebras.…”

Section: Introductionmentioning

confidence: 99%

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Topological Radicals and Frattini Theory of Banach Lie Algebras

Kissin

Shulman

Turovskiı

2012

Integr. Equ. Oper. Theory

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In this paper we develop the theory of topological radicals of Banach Lie algebras and apply it to the study of the structure of Banach Lie algebras with sufficiently many closed Lie subalgebras of finite codimensions, that is, the intersection of all these subalgebras is zero. The first part is devoted to the radical theory of Banach Lie algebras; the second develops some technique of construction of preradicals via subspace-multifunctions and analyses the corresponding radicals, and the third part contains the Frattini theory of infinite-dimensional Banach Lie algebras. It is shown that the multifunctions of closed Lie subalgebras of finite codimension (closed Lie ideals of finite codimension, closed maximal Lie subalgebras of finite codimension, closed maximal Lie ideals of finite codimension) produce different preradicals, and that these preradicals generate the same radical, the Frattini radical. The main attention is given to structural properties of Frattini-semisimple Banach Lie algebras and, in particular, to a novel infinite-dimensional phenomenon associated with the strong Frattini preradical introduced in this paper. A new constructive description of Frattini-free Banach Lie algebras is obtained.1991 Mathematics Subject Classification. Primary 46H70, 46H99; Secondary 16N80, 17B65. Lemma 2.2. Let J be a closed linear subspace of a Banach Lie algebra L invariant for all bounded Lie isomorphisms of L. Then J is a characteristic Lie ideal of L. Proof. For each δ ∈ D (L) , exp(tδ) = ∞ i=0 t n δ n n! , for t ∈ R, is a one-parameter group of bounded Lie automorphisms of L: exp(tδ)([a, b]) = [exp(tδ)(a), exp(tδ)(b)], for all a, b ∈ L. Hence exp(tδ)(J) ⊆ J. Since δ(a) = lim t→0 (exp(tδ)(a) − a)/t, for each a ∈ L, J is invariant for δ, so it is a characteristic Lie ideal of L. Clearly, the intersection and the closed linear span of a family of characteristic Lie ideals are characteristic Lie ideals. Lemma 2.3. Let L be a Banach Lie algebra, let J ⊳ ch L and q : L −→ L/J be the quotient map. If I ⊳ ch L/J then q −1 (I) ⊳ ch L. β α=0 be the P J -superposition series of closed Lie ideals of L. Then P β J (L) = P • J (L) = F (L). As P α+1 J(L) = P J P α J (L) , we have from Proposition 8.1 that there is a complete chain C α of closed Lie ideals of P α J (L) such that it is a lower finite-gap chain, s (C α ) = P α J (L) and p (C α ) = P α+1 J

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Section: Introductionmentioning

confidence: 99%

“…The existence of Lie ideals and characteristic Lie ideals of finite codimension was studied in [KST1,KST2]. We will often use the following result obtained in [KST2].…”

Section: Introductionmentioning

confidence: 99%

Topological Radicals and Frattini Theory of Banach Lie Algebras

Kissin

Shulman

Turovskiı

2012

Integr. Equ. Oper. Theory

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Victor Shulman: The Gentle Art of Mathematics

Todorov

Turowska

2013

Operator Theory: Advances and Applications

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On Tractability and Ideal Problem in Non-Associative Operator Algebras

Brešar

Shulman

Turovskii

2010

Integr. Equ. Oper. Theory

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The question of the existence of non-trivial ideals of Lie algebras of compact operators is considered from different points of view. One of the approaches is based on the concept of a tractable Lie algebra, which can be of interest independently of the main theme of the paper. Among other results it is shown that an infinite-dimensional closed Lie or Jordan algebra of compact operators cannot be simple. Several partial answers to Wojtyński's problem on the topological simplicity of Lie algebras of compact quasinilpotent operators are also given.

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

Cited by 4 publications

References 21 publications

Topological Radicals and Frattini Theory of Banach Lie Algebras

Topological Radicals and Frattini Theory of Banach Lie Algebras

Victor Shulman: The Gentle Art of Mathematics

On Tractability and Ideal Problem in Non-Associative Operator Algebras

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