Hom-structures on simple graded Lie algebras of finite growth

A Hom-structure on a Lie algebra (g, [,]) is a linear map sigma : g -> g satisfying the Hom-Jacobi identity: [sigma(x), [y, z]] + [sigma(y), [z, x]] + [sigma(z), [x, y]] = 0 for all x, y, z is an element of g. A Hom-structure is referred to as multiplicative if it is also a Lie algebra homomorphism. In this paper, using a classification theorem due to Mathieu, we determine explicitly all the Hom-structures on the simple graded Lie algebras of finite growth. As a direct consequence, all the Hom-structures on any simple graded Lie algebras of finite growth constitute a Jordan algebra in the usual way.