Deformations and Homotopy Theory of Relative Rota-Baxter Lie Algebras

We determine the L-infinity-algebra that controls deformations of a relative RotaBaxter Lie algebra and show that it is an extension of the dg Lie algebra controlling deformations of the underlying LieRep pair by the dg Lie algebra controlling deformations of the relative Rota-Baxter operator. Consequently, we define the cohomology of relative Rota-Baxter Lie algebras and relate it to their infinitesimal deformations. Alarge class of relative Rota-Baxter Lie algebras is obtained from triangular Lie bialgebras and we construct amap between the corresponding deformation complexes. Next, the notion of a homotopy relative Rota-Baxter Lie algebra is introduced. We show that a class of homotopy relative Rota-Baxter Lie algebras is intimately related to pre-Lie(infinity)-algebras.