A survey on deformations, cohomologies and homotopies of relative Rota-Baxter Lie algebras

TL;DR

This survey reviews the recent developments in deformation, cohomology, and homotopy theories of relative Rota-Baxter Lie algebras, highlighting the use of $L_$-algebras and higher derived brackets.

Contribution

It provides a comprehensive overview of the algebraic structures, deformation control methods, and homotopy theories for relative Rota-Baxter Lie algebras, including nonzero weight cases.

Findings

Construction of $L_$-algebras controlling deformations

Definition of cohomologies via twisted $L_$-algebras

Introduction of homotopy relative Rota-Baxter Lie algebras

Abstract

In this paper, we review deformation, cohomology and homotopy theories of relative Rota-Baxter Lie algebras, which have attracted quite much interest recently. Using Voronov's higher derived brackets, one can obtain an $L_\infty$-algebra whose Maurer-Cartan elements are relative Rota-Baxter Lie algebras. Then using the twisting method, one can obtain the $L_\infty$-algebra that controls deformations of a relative \RB Lie algebra. Meanwhile, the cohomologies of relative Rota-Baxter Lie algebras can also be defined with the help of the twisted $L_\infty$-algebra. Using the controlling algebra approach, one can also introduce the notion of homotopy relative Rota-Baxter Lie algebras with close connection to pre-Lie$_\infty$-algebras. Finally, we briefly review deformation, cohomology and homotopy theories of relative Rota-Baxter Lie algebras of nonzero weights.