Deformations and homotopy theory of relative Rota-Baxter Lie algebras

TL;DR

This paper develops a homotopy theory for relative Rota-Baxter Lie algebras, establishing their deformation theory via $L_$-algebras, and connects them to pre-Lie$_$-algebras, enriching the algebraic structure landscape.

Contribution

It introduces the $L_$-algebra controlling deformations, defines cohomology for these algebras, and relates homotopy relative Rota-Baxter Lie algebras to pre-Lie$_$-algebras, expanding their theoretical framework.

Findings

Deformation controlled by an $L_$-algebra extension.

Cohomology relates to infinitesimal deformations.

Homotopy relative Rota-Baxter Lie algebras connect to pre-Lie$_$-algebras.

Abstract

We determine the \emph{$L_\infty$-algebra} that controls deformations of a relative Rota-Baxter 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 {\em cohomology} of relative Rota-Baxter Lie algebras and relate it to their infinitesimal deformations. A large class of relative Rota-Baxter Lie algebras is obtained from triangular Lie bialgebras and we construct a map between the corresponding deformation complexes. Next, the notion of a \emph{homotopy} relative Rota-Baxter Lie algebra is introduced. We show that a class of homotopy relative Rota-Baxter Lie algebras is intimately related to \emph{pre-Lie$_\infty$-algebras}.