Serial subalgebras of finitary Lie algebras

Leinen, Felix; Puglisi, Orazio

doi:10.1090/S0002-9939-00-05496-4

Serial subalgebras of finitary Lie algebras
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by Felix Leinen and Orazio Puglisi PDF

Proc. Amer. Math. Soc. 129 (2001), 45-51 Request permission

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.

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