Deformation maps in proto-twilled Leibniz algebras

This paper aims to find a unified approach to studying the cohomology theories of various operators on Leibniz algebras. We first introduce deformation maps in a proto-twilled Leibniz algebra to do this. Such maps generalize various well-known operators (such as homomorphisms, derivations, crossed homomorphisms, Rota-Baxter operators, modified Rota-Baxter operators, twisted Rota-Baxter operators, Reynolds operators etc) defined on Leibniz algebras and embedding tensors on Lie algebras. We define the cohomology of a deformation map unifying the existing cohomologies of some of the operators mentioned above. Then we construct a curved L ∞ -algebra whose Maurer-Cartan elements are precisely deformation maps in a given proto-twilled Leibniz algebra. In particular, we obtain the Maurer-Cartan characterizations of modified Rota-Baxter operators, twisted Rota-Baxter operators and Reynolds operators on a Leibniz algebra. Finally, given a proto-twilled Leibniz algebra and a deformation map r, we construct two governing L ∞ -algebras, the first one controls the deformations of the operator r while the second one controls the simultaneous deformations of both the proto-twilled Leibniz algebra and the operator r.