<abstract><p>Let $ \mathfrak{L} $ and $ A $ be Lie triple systems, and let $ \theta_A $ be a representation of $ \mathfrak{L} $ on $ A. $ We first construct the third-order cohomology classes by derivations of $ A $ and $ \mathfrak{L}, $ then obtain a Lie algebra $ G_{\theta_A} $ with a representation $ \Phi $ on $ H^3(\mathfrak{L}, A), $ where $ \theta_A $ is given by an abelian extension</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ 0\longrightarrow A\longrightarrow {\tilde {\mathfrak{L}}} \xrightarrow{\pi} \mathfrak{L}\longrightarrow 0. $\end{document} </tex-math></disp-formula></p> <p>We study obstruction classes for extensibility of derivations of $ A $ and $ \mathfrak{L} $ to those of $ \tilde{\mathfrak{L}}. $ An application of $ \Phi $ is discussed.</p></abstract>

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In this paper, we introduce the first and third cohomology groups on Leibniz triple systems, which can be applied to extension theory and 1-parameter formal deformation theory. Specifically, we investigate the central extension theory for Leibniz triple systems and show that there is a one-to-one correspondence between equivalent classes of central extensions of Leibniz triple systems and the third cohomology group. We study the T * -extension of a Leibniz triple system and we determined that every even-dimensional quadratic Leibniz triple system (L, B) is isomorphic to a T * -extension of a Leibniz triple system under a suitable condition. We also give a necessary and sufficient condition for a quadratic Leibniz triple system to admit a symplectic form. At last, we develop the 1-parameter formal deformation theory of Leibniz triple systems and prove that it is governed by the cohomology groups.