Restricted Lie Triple Systems

Lie triple systems arose initially in studies of Riemannian geometry;the totally geodesic manifolds;Jordan algebras and Lie algebras.We mainly apply the knowledge of Lie triple systems and restricted Lie algebras to study restricted Lie triple systems (the Lie triple systems of character p > 2.) By now, the theory of restricted Lie triple systems are just in threshold. As a new academic field, restricted Lie triple systems attract more and more attentions. In the paper, some properties of restricted Lie triple systems are firstly obtained .And then, restrictable Lie triple systems are defined, the relation between the two is elaborated. Finally , the unique of decomposition of restricted Lie triple systems is given.The main results in this paper are the follows:Theorem 1 Let T be a restricted Lie triple system over k. Let if be a subsystem of T and let H_p be the intersection of all p-subsystem of T containing H. Suppose that (ej)j∈J a basis of H. Then the following statements hold:(1)(2)(3) If H is an ideal of T, then (H_p)ⁿ = Hⁿ H_pⁿ = Hⁿ (n ≥ 1)(4) If H is an ideal of T, H_p is solvable (nilpotent) if and only if H is solvable (nilpotent).(5) If I is an ideal of T, then I_p is a p-ideal of T.(6) If I is an ideal of H, then I is also an -ideal of H_p.(7) The nilradical (maximal nilpotent ideal) and the solvableradical (maximal solvable ideal) of T are p-ideal of T.Theorem 2 Let T be a subsystem of a restricted Lie triple system (G, [p])and Let [p]₁ : T → T be a mapping. Then the following statements are equivalent:(1) [p]₁ is a p-mapping on T.(2) There exists a p-semilinear mapping f : T → Z_G(T)suchthat[p]₁ = [p] + f.Theorem 3 T is restrictable if and only if there is a mapping [p] : T —* T which makes T into a restricted Lie triple system.Theorem4 Let / : 7\ —> Ti be a surjective homomorphism of Lie triple systems. If T\ is restrictable , so is T2-Theorem5 Let T be a restricted lie triple system. If Z(T) — 0, T has a decomposition of p-ideals as follows:T = mi ?? ? ? 9T = m ? ni ? ? ? ? ? mwhere mi, m2, ? ? ?,rns and n\,ri2, â–  â–  ? ,nt are indecomposable, then s = t, and m* 1,2, ? ? â– , s after changing the orders.