Nijenhuis Operators, Product Structures and Complex Structures on Leibniz Triple Systems

In this paper, we first introduce the notion of a Nijenhuis operator on Leibniz triple systems, which can generate a trivial deformation. Then we use Nijenhuis operators to define product structures on a Leibniz triple system. There exists a product structure on a Leibniz triple system if and only if the Leibniz triple system is the direct sum of two subalgebras. There are some special product structures, each of which corresponds to a special decomposition of a Leibniz triple system. Parallelly, we study a complex structure on a Leibniz triple system. Finally, we add a compatibility condition between a product structure and a complex structure to introduce the notion of a complex product structure on a Leibniz triple system.