Mathematics – Representation Theory
Scientific paper
2009-04-02
Internat. Math. Res. Notices 2010, No. 6 (2010) 1062-1101
Mathematics
Representation Theory
36 pages
Scientific paper
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 $\gl_\infty$.
Dan-Cohen Elizabeth
Penkov Ivan
No associations
LandOfFree
Parabolic and Levi subalgebras of finitary Lie algebras does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Parabolic and Levi subalgebras of finitary Lie algebras, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Parabolic and Levi subalgebras of finitary Lie algebras will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-64015