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 $T :\mathfrak{g} \rightarrow \mathfrak{g}$. We define representation and dualrepresentation of Rota-Baxter Leibniz algebras. Next, we define a cohomologytheory of Rota-Baxter Leibniz algebras. We also study the infinitesimal andformal deformation theory of Rota-Baxter Leibniz algebras and show that ourcohomology is deformation cohomology. Moreover, We define an abelian extensionof Rota-Baxter Leibniz algebras and show that equivalence classes of suchextensions are related to the cohomology groups.Comment: 25 Page