Levi components of parabolic subalgebras of finitary Lie algebras

TL;DR

This paper characterizes Levi components of parabolic subalgebras in infinite-dimensional finitary Lie algebras, providing classifications, estimates, and examples highlighting differences from finite-dimensional cases.

Contribution

It offers a detailed characterization of Levi components and associated parabolic subalgebras in infinite-dimensional finitary Lie algebras, extending finite-dimensional theory.

Findings

Characterization of Levi components in infinite-dimensional setting

Conditions for parabolic subalgebras with given Levi components

Estimate on the number of self-normalizing parabolic subalgebras

Abstract

We characterize locally semisimple subalgebras $\l$ of $\sl_\infty$, $\so_\infty$, and $\sp_\infty$ which are Levi components of parabolic subalgebras. Given $\l$, we characterize the parabolic subalgebras $\p$ such that $\l$ 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.