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 [Formula: see text]-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 [Formula: see text], we construct two governing [Formula: see text]-algebras, the first one controls the deformations of the operator [Formula: see text] while the second one controls the simultaneous deformations of both the proto-twilled Leibniz algebra and the operator [Formula: see text].