Cohomology and Deformations of Compatible Lie Triple Systems

In this paper, we first introduce the notions of a compatible Lie triple system and its representation. We construct a bidifferential graded Lie algebra whose Maurer–Cartan elements are compatible Lie triple systems. We also obtain the bidifferential graded Lie algebra which controls deformations of a compatible Lie triple system. Then we investigate the cohomology theory of compatible Lie triple systems and consider the connection between the cohomology group of compatible Lie triple systems and the cohomology group of Lie triple systems. Furthermore, we develop the 1-parameter formal deformation theory of compatible Lie triple systems and prove that it is governed by the cohomology groups. At last, we study abelian extensions of compatible Lie triple systems and classify them by the second cohomology group.