Parabolic and Levi Subalgebras of Finitary Lie Algebras

Let $\mathfrak g$ be a locally reductive complex Lie algebra that 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 $\mathfrak {sl}_{\infty}, \mathfrak {so}_{\infty}, \mathfrak {sp}_{\infty}$, and finite-dimensional simple Lie algebras. A parabolic subalgebra of $\mathfrak g$ is any subalgebra that contains a maximal locally solvable (that is, Borel) subalgebra. Building upon the work by Dimitrov and the authors of the present article [4, 8], we give a general description of parabolic subalgebras of $\mathfrak g$ 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 $\mathfrak k$ of $\mathfrak {gl}_{\infty}$. If $\mathfrak k$ is splittable, we show that the linear nilradical admits a locally reductive complement in $\mathfrak k$. We conclude the article with descriptions of Cartan, Borel, and parabolic subalgebras of arbitrary splittable subalgebras of $\mathfrak {gl}_{\infty}$.