Locally finite Lie algebras with root decomposition

摘要

Let K be an algebraically closed field of characteristic zero, $frak {g}$ be a countably dimensional locally finite Lie algebra over K, and $frak {h} subset frak {g}$ be a (a priori non-abelian) locally nilpotent subalgebra of $frak {g}$ which coincides with its zero Fitting component. We classify all such pairs $(frak {g}, frak {h})$ under the assumptions that the locally solvable radical of $frak {g}$ equals zero and that $frak {g}$ admits a root decomposition with respect to $frak {h}$. More precisely, we prove that $frak {g}$ is the union of reductive subalgebras $frak {g}_n$ such that the intersections $frak {g}_n cap frak {h}$ are nested Cartan subalgebras of $frak {g}_n$ with compatible root decompositions. This implies that $frak {g}$ is root-reductive and that $frak {h}$ is abelian. Root-reductive locally finite Lie algebras are classified in [6]. The result of the present note is a more general version of the main classification theorem in [9] and is at the same time a new criterion for a locally finite Lie algebra to be root-reductive. Finally we give an explicit example of an abelian selfnormalizing subalgebra $frak {h}$ of $frak {g} = frak {sl}(infty)$ with respect to which $frak {g}$ does not admit a root decomposition.