Abstract. We find for each simple finitary Lie algebra g a category Tg of integrable modules in which the tensor product of copies of the natural and conatural modules are injective. The objects in Tg can be defined as the finite length absolute weight modules, where by absolute weight module we mean a module which is a weight module for every splitting Cartan subalgebra of g. The category Tg is Koszul in the sense that it is antiequivalent to the category of locally unitary finite-dimensional modules over a certain direct limit of finite-dimensional Koszul algebras. We describe these finite-dimensional algebras explicitly. We also prove an equivalence of the categories T o(∞) and T sp(∞) corresponding respectively to the orthogonal and symplectic finitary Lie algebras o(∞), sp(∞).

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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

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∞, so∞, sp ∞ , 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, [DP2], [D], 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 ∞ . 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 ∞ . MSC: 17B05 and 17B65

We characterize locally semisimple subalgebras l of sl∞, so∞, and sp ∞ which are Levi components of parabolic subalgebras. Given l, we describe the parabolic subalgebras p such that l is a Levi component of p. We also prove that not every maximal locally semisimple subalgebra of a finitary Lie algebra is a Levi component.When the set of self-normalizing parabolic subalgebras p with fixed Levi component l 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. Date