Maurer-Cartan characterizations and cohomologies of crossed homomorphisms on Lie triple systems

In this paper, first we give the notion of a crossed homomorphism on a Lie triple system with respect to an action on another Lie triple system. We also construct an L-8 -algebra using the higher derived brackets, whose Maurer-Cartan elements are crossed homomorphisms on Lie triple systems. Consequently, we obtain the twisted L-8-algebra that controls deformations of a given crossed homomorphism on Lie triple systems. Next we construct a cohomology theory for a crossed homomorphism on Lie triple systems and classify infinitesimal deformations of crossed homomorphisms using the second cohomology group. Moreover we consider the relationship between cohomology groups of crossed homomorphisms on Lie algebras and associated Lie triple systems.