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

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