Deformation maps in proto-twilled Leibniz algebras

TL;DR

This paper introduces a unified cohomology framework for various operators on Leibniz algebras using deformation maps and $L_$-algebras, enabling a comprehensive study of their deformations and interrelations.

Contribution

It develops a unified cohomology theory for multiple operators on Leibniz algebras via deformation maps and constructs $L_$-algebras to characterize their deformations and Maurer-Cartan elements.

Findings

Unified cohomology theory for operators on Leibniz algebras.

Maurer-Cartan characterizations of Rota-Baxter and Reynolds operators.

Construction of governing $L_$-algebras for deformations.

Abstract

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 all the operators mentioned above. Then we construct a curved $L_\infty$-algebra whose Maurer-Cartan elements are precisely deformation maps in a given proto-twilled Leibniz algebra. In particular, we get 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 $L_\infty$-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 $r$.