Signatures of Roots and a New Characterization of Causal Symmetric Spaces
Identifieur interne : 001A49 ( Main/Exploration ); précédent : 001A48; suivant : 001A50Signatures of Roots and a New Characterization of Causal Symmetric Spaces
Auteurs : Soji Kaneyuki [Japon, États-Unis]Source :
- Progress in Nonlinear Differential Equations and Their Applications ; 1996.
Abstract
Abstract: Let g be a real semisimple Lie algebra, let τ be a Cartan involution of g. Choose a maximal abelian subspace a in the (−l)-eigenspace of τ, and let △ be the root system for (g, a). Oshima-Sekiguchi [11] introduced the notion of a signature function ∈ on △, which is a map of △∪(0) to {±1} satisfying a multiplicative property. Using ∈ and τ, they define a new involution △∈ of g by putting △∈(X) = ∈(a)τ(X), where X is a root vector for a root a ∈ △ ∪ (0). The resulting symmetric pair (a, τ∈) is said to be a symmetric pair of type K ∈. A (ℤ)-graded Lie algebra (or shortly GLA) $$g = \sum _{k = - v}^v{g_k}$$ is said to be of the v-th kind, if g −1 ≠ (0) and g ±v ≠ (0). In [7, 6], we have worked out the classification and construction of real semisimple GLA’s.
Url:
DOI: 10.1007/978-1-4612-2432-7_7
Affiliations:
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