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 ∞ , so ∞ , and sp ∞ . As part of a structure theory program for root-reductive Lie algebras, Cartan subalgebras of the Lie algebra gl ∞ were introduced and studied in [K.-H. Neeb, I. Penkov, Cartan subalgebras of gl ∞ , Canad. Math. Bull. 46 (2003) 597–616]. In the present paper we refine and extend the results of [K.-H. Neeb, I. Penkov, Cartan subalgebras of gl ∞ , Canad. Math. Bull. 46 (2003) 597–616] 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 ∞ , so ∞ , and sp ∞ . We conclude the paper with a characterization of the set of conjugacy classes of Cartan subalgebras of the Lie algebras gl ∞ , sl ∞ , so ∞ , and sp ∞ with respect to the group of automorphisms of the natural representation which preserve the Lie algebra.