ISSN: 2041-6792

Clinical Investigation

Lie-triple-systems

 Throughout this paper, Lie triple systems T are considered of arbitrary dimension and over an arbitrary field K. It is worth to mention that, unless otherwise stated, there is not any restriction on dim Tα or {k ∈ K: kα ∈ 1 for a fixed α ∈ 1}, where Tα denotes the root space associated to the root α, and 1 the set of nonzero roots of T . Following the ideas of Lister and Faulkner to study finite dimensional Lie triple systems, and motivated by the successful development over the recent years of a theory of split Lie algebras , the author and Forero introduced in the concept of split Lie triple system of arbitrary dimension and studied the locally finite ones. We also introduced in techniques of connection of roots in the framework of split Lie algebras. In the present paper we extend these techniques to the framework of split Lie triple systems so as to obtain a generalization of the results in [1]. We consider the wide class of split Lie triple systems (which contains the class of split Lie algebras), having a symmetric root system, and begin the study of this class of triple systems by considering those with a coherent 0-root space. In §2, we establish the preliminaries on split Lie triple systems. In §3, we introduce the notion of connection of roots in the framework of split Lie triple systems and study its properties.   

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