Levi components of parabolic subalgebras of finitary Lie algebras

Abstract

We characterize locally semisimple subalgebras $ł$ of $\sl_\infty$, $\so_\infty$, and $\sp_\infty$ which are Levi components of parabolic subalgebras. Given $ł$, we characterize the parabolic subalgebras $\p$ such that $ł$ is a Levi component of $\p$. When the set of such self-normalizing parabolic subalgebras $\p$ is finite, we prove an estimate on its cardinality. We consider various examples which highlight the differences from the case of parabolic subalgebras of finite-dimensional simple Lie algebras.