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Transactions of the American Mathematical Society

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Cartan subspaces of symmetric Lie algebras

Authors: J. Lepowsky and G. W. McCollum
Journal: Trans. Amer. Math. Soc. 216 (1976), 217-228
MSC: Primary 17B05
MathSciNet review: 0404361
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Abstract: A symmetric Lie algebra is defined, following J. Dixmier, to be a Lie algebra $ \mathfrak{g}$ with a decomposition $ \mathfrak{g} = \mathfrak{k} \oplus \mathfrak{p}$ such that $ \mathfrak{k}$ is a subalgebra of $ \mathfrak{g},[\mathfrak{k},\mathfrak{p}] \subset \mathfrak{p}$ and $ [\mathfrak{p},\mathfrak{p}] \subset \mathfrak{k}$. A definition of Cartan subspace of a symmetric Lie algebra is given, and a theory is presented which parallels the standard theory of Cartan subalgebras of Lie algebras, and which generalizes the classical results for real and complex semisimple symmetric Lie algebras.

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Keywords: Cartan subspaces, symmetric Lie algebras, weight theory for nil sets, nil subspaces, natural $ \mathfrak{p}$-subalgebras, splitting Cartan subspaces, reductive symmetric Lie algebras
Article copyright: © Copyright 1976 American Mathematical Society