Parabolic and Levi subalgebras of finitary Lie algebras

TL;DR

This paper characterizes parabolic and Levi subalgebras of certain infinite-dimensional Lie algebras using stabilizers of generalized flags, introducing trace conditions and extending classical concepts to the finitary setting.

Contribution

It provides a comprehensive description of parabolic subalgebras in locally reductive finitary Lie algebras and establishes the existence of Levi components in splittable subalgebras.

Findings

Description of parabolic subalgebras via joint stabilizers of taut couples

Introduction of trace conditions for characterizing parabolic subalgebras

Existence of Levi components in splittable subalgebras of _______________________________________

Abstract

Let $\g$ be a locally reductive complex Lie algebra which admits a faithful countable-dimensional finitary representation $V$. Such a Lie algebra is a split extension of an abelian Lie algebra by a direct sum of copies of $\sl_\infty$, $\so_\infty$, $\sp_\infty$, and finite-dimensional simple Lie algebras. A parabolic subalgebra of $\g$ is any subalgebra which contains a maximal locally solvable (that is, Borel) subalgebra. Building upon work by Dimitrov and the authors of the present paper, we give a general description of parabolic subalgebras of $\g$ in terms of joint stabilizers of taut couples of generalized flags. The main differences with the Borel subalgebra case are that the description of general parabolic subalgebras has to use both the natural and conatural modules, and that the parabolic subalgebras are singled out by further "trace conditions" in the suitable joint stabilizer. The technique of taut couples can also be used to prove the existence of a Levi component of an arbitrary subalgebra $\k$ of $\gl_\infty$. If $\k$ is splittable, we show that the linear nilradical admits a locally reductive complement in $\k$. We conclude the paper with descriptions of Cartan, Borel, and parabolic subalgebras of arbitrary splittable subalgebras of $\gl_\infty$.