  The representation and cohomology theory of Hom-Lie-Yamaguti algebras is
introduced. As an application, we study deformation and extension of
Hom-Lie-Yamaguti algebras. It proved that a 1-parameter infinitesimal
deformation of a Hom-Lie-Yamaguti algebra $T$ corresponds to a Hom-Lie-Yamaguti
algebra of deformation type and a (2,3)-cocycle of $T$ with coefficients in the
adjoint representation. We also prove that abelian extensions of
Hom-Lie-Yamaguti algebras are classified by the (2,3)-cohomology group.
