3-pre-Leibniz Algebras, Deformations and Cohomologies of Relative Rota-Baxter Operators on 3-Leibniz Algebras

In this paper, first we introduce the notions of 3-pre-Leibniz algebras and relative Rota-Baxter operators on 3-Leibniz algebras. We show that a 3-pre-Leibniz algebra gives rise to a 3-Leibniz algebra and a representation such that the identity map is a relative Rota-Baxter operator. Conversely, a relative Rota-Baxter operator naturally induces a 3-pre-Leibniz algebra. Then we construct a Lie 3-algebra, and characterize relative Rota-Baxter operators as its Maurer-Cartan elements. Consequently, we obtain the \(L_\infty\) $L_{\infty }$ -algebra that controls deformations of relative Rota-Baxter operators on 3-Leibniz algebras. Next we define the cohomology of relative Rota-Baxter operators on 3-Leibniz algebras and show that infinitesimal deformations of a relative Rota-Baxter operator are classified by the second cohomology group. Finally, we construct an \(L_\infty\) $L_{\infty }$ -algebra whose Maurer-Cartan elements are relative Rota-Baxter 3-Leibniz algebra structures, and define the cohomology of relative Rota-Baxter 3-Leibniz algebras.