Triality Groups Associated with Triple Systems and their Homotope Algebras

We introduce the notion of an (α, β, γ) triple system, which generalizes the familiar generalized Jordan triple system related to a construction of simple Lie algebras. We then discuss its realization by considering some bilinear algebras and vice versa. Next, as a new concept, we study triality relations (a triality group and a triality derivation) associated with these triple systems; the relations are a generalization of the automorphisms and derivations of the triple systems. Also, we provide examples of several involutive triple systems with a tripotent element.