Abstract: Firstly,  by using the method of quasi-trace functions, the corresponding relationship between the cohomology group of the relative Rota-Baxter operator of a Lie representation pair and the cohomology group of the relative Rota-Baxter operator of the induced 3-Lie representation pair in the low-order cases was given. Secondly, by using the method of constructing chain maps, the homomorphism between the cohomology groups of the relative Rota-Baxter operator of a Lie representation pair and the induced 3-Lie representation pair with any order greater than or equal to 2 was obtained.

Key words:  , relative Rota-Baxter operator, quasi-trace function, cohomology, Lie algebra, 3-Lie algebra