Lie n-algebras and cohomologies of relative Rota-Baxter operators on n-Lie algebras

TL;DR

This paper develops a cohomology theory for relative Rota-Baxter operators on n-Lie algebras, linking deformation theory with algebraic structures and extending to higher Lie algebra contexts.

Contribution

It introduces a Lie n-algebra framework for Rota-Baxter operators on n-Lie algebras and explores their cohomology and deformation properties.

Findings

Constructed a Lie n-algebra controlling deformations.

Defined cohomology for relative Rota-Baxter operators.

Established relations between cohomologies of different Lie algebra levels.

Abstract

Based on the differential graded Lie algebra controlling deformations of an $n$-Lie algebra with a representation (called an n-LieRep pair), we construct a Lie n-algebra, whose Maurer-Cartan elements characterize relative Rota-Baxter operators on n-LieRep pairs. The notion of an n-pre-Lie algebra is introduced, which is the underlying algebraic structure of the relative Rota-Baxter operator. We give the cohomology of relative Rota-Baxter operators and study infinitesimal deformations and extensions of order m deformations to order m+1 deformations of relative Rota-Baxter operators through the cohomology groups of relative Rota-Baxter operators. Moreover, we build the relation between the cohomology groups of relative Rota-Baxter operators on n-LieRep pairs and those on (n+1)-LieRep pairs by certain linear functions.