Modified Rota-Baxter operators of non-zero weight on $3$-Lie algebras

In this paper, we introduce the notion of modified Rota-Baxter operators of non-zero weight on $3$-Lie algebras and provide some examples. Next, we give various constructions of modified Rota-Baxter operators of non-zero weight according to constructions of $3$-Lie algebras. Furthermore, we define a cohomology of modified Rota-Baxter operators of non-zero weight on $3$-Lie algebras with coefficients in a suitable representation. As an application, we study formal deformations of modified Rota-Baxter operators of non-zero weight that are generated by the above-defined cohomology. In the final part of the paper, we construct two $L_\infty[1]$-algebra structures whose Maurer-Cartan elements correspond to relative and absolute modified Rota-Baxter $3$-Lie algebra structures of nonzero weight, respectively. Lastly, we compare our $L_\infty[1]$-algebraic approach with the deformation-controlling $L_\infty[1]$-algebra for relative Rota-Baxter $3$-Lie operators developed by Hou, Sheng, and Zhou.