Twisted Rota-Baxter families on Lie-Yamaguti algebras and NS-Lie-Yamaguti family algebras

In this paper, we first introduce twisted Rota-Baxter families on Lie-Yamaguti algebras indexed by a commutative semigroup $\mathrm{\Omega }$. Then, we study NS-Lie-Yamaguti family algebras as the underlying structures of twisted Rota-Baxter families. Finally, we investigate the cohomology of a twisted Rota-Baxter family. This cohomology can also be seen as the cohomology of a certain $\mathrm{\Omega }$-Lie-Yamaguti algebras with coefficients in an appropriate representation. As applications, we consider the deformations of twisted Rota-Baxter families from the cohomological points of view.