Cartan subalgebras of root-reductive Lie algebras

Abstract

Root-reductive Lie algebras are direct limits of finite-dimensional reductive Lie algebras under injections which preserve the root spaces. It is known that a root-reductive Lie algebra is a split extension of an abelian Lie algebra by a direct sum of copies of finite-dimensional simple Lie algebras as well as copies of the three simple infinite-dimensional root-reductive Lie algebras sl_infty, so_infty, and sp_infty. As part of a structure theory program for root-reductive Lie algebras, Cartan subalgebras of the Lie algebra gl_infty were introduced and studied in a paper of Neeb and Penkov. In the present paper we refine and extend the results of [N-P] to the case of a general root-reductive Lie algebra g. We prove that the Cartan subalgebras of g are the centralizers of maximal toral subalgebras and that they are nilpotent and self-normalizing. We also give an explicit description of all Cartan subalgebras of the simple Lie algebras sl_infty, so_infty, and sp_infty. We conclude the paper with a characterization of the set of conjugacy classes of Cartan subalgebras of the Lie algebras gl_infty, sl_infty, so_infty, and sp_infty with respect to the group of automorphisms of the natural representation which preserve the Lie algebra.Comment: 28 pages, 1 figur

On an Infinite Limit of BGG Categories O

“…If g is root reducible, we are given an abelian subalgebra h = lim − → h n . Such subalgebras h ⊂ g are known as splitting maximal toral subalgebras of g. These subalgebras are also Cartan subalgebras of g, according to the definition and results in [DPS,Section 3]. We will simply use the term "Cartan subalgebra" when referring to splitting maximal toral subalgebras.…”

Parabolic subgroups of real direct limit Lie groups

“…In [4] the uniqueness issue was discussed for gl(∞, C), sl(∞, C), and sp(∞, C), but not for so(∞, C). In the orthogonal setting one can have three different self-taut generalized flags with the same stabilizer (see [3] and [7], where the non-uniqueness is discussed in special cases.) Theorem 2.8.…”

A Generalization of Lie's Theorem

“…In [3] it is shown that we cannot expect a 1-1 correspondence between maximal, locally solvable subalgebras and maximal generalized flags in this generality. Moreover, the work of Dan-Cohen in [1] shows that we cannot even expect a 1-1 correspondence between maximal, locally solvable subalgebras and maximal, closed generalized flags in an arbitrary locally finite Lie algebra.…”