Cohomology and deformations of crossed homomorphisms between Lie-Yamaguti algebras

In this paper, we introduce the notion of crossed homomorphisms between Lie-Yamaguti algebras and establish the cohomology theory of crossed homomorphisms via the Yamaguti cohomology. Consequently, we use this cohomology to characterize linear deformations of crossed homomorphisms between Lie-Yamaguti algebras. We show that if two linear or formal deformations of a crossed homomorphism are equivalent, then their infinitesimals are in the same cohomology class in the first cohomology group. Moreover, we show that an order \(n\) deformation of a crossed homomorphism can be extended to an order \(n+1\) deformation if and only if the obstruction class in the second cohomology group is trivial.