Deformation Cohomology of Lie-Yamaguti algebras and Free Lie-Yamaguti algebras

Infinitesimal deformation theory of Lie-Yamaguti algebras was introduced by Tao Zhang and Juan Li . We extend their theory to develop formal one-parameter deformation theory of Lie-Yamaguti algebras. It turns out that the right deformation cohomology which controls deformations in this context, is a variant of Lie-Yamaguti algebra cohomology which we introduce in this paper. At the end, we introduce the free object in the category of Lie-Yamaguti algebras and prove that it is rigid.