Cohomology, deformations and extensions of Rota-Baxter Leibniz algebras

A Rota-Baxter Leibniz algebra is a Leibniz algebra $(\mathfrak{g},[,{]}_{\mathfrak{g}})$ equipped with a Rota-Baxter operator $T:\mathfrak{g}\to \mathfrak{g}$. We define representation and dual representation of Rota-Baxter Leibniz algebras. Next, we define a cohomology theory of Rota-Baxter Leibniz algebras. We also study the infinitesimal and formal deformation theory of Rota-Baxter Leibniz algebras and show that our cohomology is deformation cohomology. Moreover, We define an abelian extension of Rota-Baxter Leibniz algebras and show that equivalence classes of such extensions are related to the cohomology groups.