Cartan subalgebras of root-reductive Lie algebras

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dc.creatorDan-Cohen, Elizabeth
dc.creatorPenkov, Ivan
dc.creatorSnyder, Noah
dc.date2006-02-28
dc.date.accessioned2026-07-07T12:24:42Z
dc.date.available2026-07-07T12:24:42Z
dc.descriptionRoot-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.
dc.description28 pages, 1 figure
dc.identifierhttps://arxiv.org/abs/math/0603004
dc.identifierhttp://arxiv.org/abs/math/0603004
dc.identifierJ. Algebra 308 (2007) 583-611
dc.identifierdoi:10.1016/j.jalgebra.2006.05.012
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/214410
dc.subjectRepresentation Theory
dc.subject17B65; 17B20
dc.titleCartan subalgebras of root-reductive Lie algebras
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