Allied Journal of Medical Research

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Allied Journal of Medical Research 44 7897 074717

Lie-triple-systems-Review-Articles

A Lie triple system consists of a space T of linear operators on a vector space V that is closed under the ternary product [x, y, z] = [[x, y], z], where [x, y] = xy −yx. Jacobson first introduced them in connection with problems from Jordan theory and quantum mechanics, viewing Lie triple systems as subspaces of Lie algebras g that are closed relative to the ternary product. (The two notions are equivalent.) For example, if θ is an involution (i.e., automorphism of order 2) of a Lie algebra g over a field of characteristic 6= 2, the corresponding −1-eigenspace T of θ is a Lie triple system in this sense. While the concept of a Lie triple system also has an abstract definition, all Lie triple systems have such realizations in terms of a Lie algebra and an involution. More recently, Lie triple systems have arisen in the study of symmetric spaces and have been connected with the study of the Yang-Baxter equations. Recently, Casas, Loday and Pirashvili have generalized the notion of a Leibniz algebra to n-ary Leibniz algebras; in the n = 3 case, Lie triple systems form a subclass of these algebras.

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