Some Structure Theories of Leibniz Triple Systems

In this paper, we investigate the Leibniz triple system T and its universal Leibniz envelope U(T). The involutive automorphism of U(T) determining T is introduced, which gives a characterization of the (mathbb {Z}_{2})-grading of U(T). We show that the category of Leibniz triple systems is equivalent to a full subcategory of the category of (mathbb {Z}_{2})-graded Leibniz algebras. We give the relationship between the solvable radical R(T) of T and R a d(U(T)), the solvable radical of U(T). Further, Levi’s theorem for Leibniz triple systems is obtained. Moreover, the relationship between the nilpotent radical of T and that of U(T) is studied. Finally, we introduce the notion of representations of a Leibniz triple system, which can be described by using involutive representations of its universal Leibniz envelope.