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Cartan subalgebras of root-reductive Lie algebras
- Source :
-
Journal of Algebra . Feb2007, Vol. 308 Issue 2, p583-611. 29p. - Publication Year :
- 2007
-
Abstract
- 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 , , and . As part of a structure theory program for root-reductive Lie algebras, Cartan subalgebras of the Lie algebra were introduced and studied in [K.-H. Neeb, I. Penkov, Cartan subalgebras of , 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 , Canad. Math. Bull. 46 (2003) 597–616] to the case of a general root-reductive Lie algebra . We prove that the Cartan subalgebras of 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 , , and . We conclude the paper with a characterization of the set of conjugacy classes of Cartan subalgebras of the Lie algebras , , , and with respect to the group of automorphisms of the natural representation which preserve the Lie algebra. [Copyright &y& Elsevier]
- Subjects :
- *MATHEMATICS
*LINEAR algebra
*MATHEMATICAL analysis
*INJECTIONS
Subjects
Details
- Language :
- English
- ISSN :
- 00218693
- Volume :
- 308
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- Journal of Algebra
- Publication Type :
- Academic Journal
- Accession number :
- 23277739
- Full Text :
- https://doi.org/10.1016/j.jalgebra.2006.05.012