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Serial subalgebras of finitary Lie algebras
Authors:Felix Leinen   Orazio Puglisi
Affiliation:Fachbereich 17 -- Mathematik, Johannes Gutenberg--Universität Mainz, D--55099 Mainz, Germany ; Dipartimento di Matematica, Università degli Studi di Trento, I--38050 Povo (Trento), Italy
Abstract:

A Lie subalgebra $L$ of ${mathfrak{gl}_{{mathbb{K}}}(V)}$ is said to be finitary if it consists of elements of finite rank. We show that, if $L$ acts irreducibly on $V$, and if $V$ is infinite-dimensional, then every non-trivial ascendant Lie subalgebra of $L$ acts irreducibly on $V$ too. When $operatorname{Char} mathbb{K}neq 2$, it follows that the locally solvable radical of such $L$ is trivial. In general, locally solvable finitary Lie algebras over fields of characteristic $neq 2$ are hyperabelian.

Keywords:Lie algebra   finitary endomorphism   serial subalgebra   locally solvable radical   Hirsch-Plotkin radical
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