Post Lie-Yamaguti algebras, relative Rota-Baxter operators of nonzero weights, and their deformations

In this paper, we introduce the notions of relative Rota-Baxter operators of weight $1$ on Lie-Yamaguti algebras, and post-\LYA s, which is an underlying algebraic structure of relative Rota-Baxter operators of weight $1$. We give the relationship between these two algebraic structures. Besides, we establish the cohomology theory of relative Rota-Baxter operators of weight $1$ via the Yamaguti cohomology. Consequently, we use this cohomology to characterize linear deformations of relative Rota-Baxter operators of weight $1$ on Lie-Yamaguti algebras. We show that if two linear deformations of a relative Rota-Baxter operator of weight $1$ 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 relative Rota-Baxter operator of weight $1$ can be extended to an order $n+1$ deformation if and only if the obstruction class in the second cohomology group is trivial.