Dialgebras and related triple systems

We consider some algebraical systems that lead to various nearly associative triple systems. We deal with a class of algebras which contains Leibniz–Poisson algebras, dialgebras, conformal algebras, and some triple systems. We describe all homogeneous structures of ternary Leibniz algebras on a dialgebra. For this purpose, in particular, we use the Leibniz–Poisson structure on a dialgebra. We then find a corollary describing the structure of a Lie triple system on an arbitrary dialgebra, a conformal associative algebra and a classical associative triple system. We also describe all homogeneous structures of an $(\varepsilon,\delta)$-Freudenthal–Kantor triple system on a dialgebra.