ISSN: 2041-6792

Clinical Investigation

Lie-triple-systems-Review

 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. This paper presents new results concerning the representation theory and homological algebra of Lie triple systems. We also include a development of these ideas for restricted Lie triple systems, introduced recently. If T is a Lie triple system of linear operators on the vector space V defined over a field k of positive characteristic p > 2, then T is restricted provided that xp ∈ T for all x ∈ T . Our original motivation came from the representation theory of algebraic groups, as we sketch below in connection with the results of Section 6. However, the results presented here have an independent interest, and they are largely of a foundational nature. Beyond the connections with algebraic groups, another possible application of these results is to the theory of quasigroups and loops.   

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