Abstract

We determine the 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 mathsf {Lie}mathsf {Rep} 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. 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 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_infty -algebras.

Highlights

  • In this paper we initiate the study of deformations and cohomology of relative Rota– Baxter Lie algebras and their homotopy versions.1.1

  • We show that infinitesimal deformations of Rota–Baxter Lie algebras and triangular Lie bialgebras are classified by the corresponding second cohomology groups

  • We show that strict homotopy relative Rota–Baxter operators give rise to pre-Lie∞-algebras, and the identity map is a strict homotopy relative Rota–Baxter operator on the subadjacent L∞-algebra of a pre-Lie∞-algebra

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Summary

Introduction

In this paper we initiate the study of deformations and cohomology of relative Rota– Baxter Lie algebras and their homotopy versions.1.1. Let T : V −→ g be a relative Rota–Baxter operator on a Lie algebra (g, μ) with respect to a representation (V ; ρ).

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