Cohomology of Leibniz Triple Systems and its applications

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 $(\mathfrak{L},B)$ 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.