Cohomology and deformations of crossed homomorphisms between Lie–Yamaguti superalgebras

In this study, we propose the idea of crossed homomorphisms between Lie–Yamaguti superalgebras and develop the Yamaguti cohomology theory of crossed homomorphisms. In light of this, we characterize linear deformations of crossing homomorphisms between Lie–Yamaguti superalgebras using this cohomology. We demonstrate that if two linear or formal deformations of a crossing homomorphism are similar, then their infinitesimals are in the same cohomology class in the first cohomology group. In addition, we show that an order n deformation of a crossing homomorphism can be extended to an order n+1 deformation if and only if the obstruction class in the second cohomology group is trivial.