Deformations and Homotopy Theory of Relative Rota\u2013Baxter Lie Algebras

Andrey Lazarev,Yunhe Sheng,Rong Tang

doi:10.1007/s00220-020-03881-3

Andrey Lazarev, Yunhe Sheng + Show 1 more

Open Access

https://doi.org/10.1007/s00220-020-03881-3

Copy DOI

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

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).

Results

Conclusion