Modified Rota-Baxter operators of non-zero weight on $3$-Lie algebras

TL;DR

This paper introduces modified Rota-Baxter operators of non-zero weight on 3-Lie algebras, explores their cohomology, deformations, and related algebraic structures, advancing the theoretical understanding of these operators.

Contribution

It defines and constructs modified Rota-Baxter operators on 3-Lie algebras, develops their cohomology and deformation theory, and connects these to L-infinity algebra structures.

Findings

Defined cohomology for modified Rota-Baxter operators on 3-Lie algebras

Studied formal deformations via cohomology

Constructed L-infinity algebra structures related to these operators

Abstract

In this paper, we introduce the notion of modified Rota-Baxter operators of non-zero weight on $3$-Lie algebras and provide some examples. Next, we give various constructions of modified Rota-Baxter operators of non-zero weight according to constructions of $3$-Lie algebras. Furthermore, we define a cohomology of modified Rota-Baxter operators of non-zero weight on $3$-Lie algebras with coefficients in a suitable representation. As an application, we study formal deformations of modified Rota-Baxter operators of non-zero weight that are generated by the above-defined cohomology. In the final part of the paper, we construct two \(L_\infty[1]\)-algebra structures whose Maurer-Cartan elements correspond to relative and absolute modified Rota-Baxter \(3\)-Lie algebra structures of nonzero weight, respectively. Lastly, we compare our \(L_\infty[1]\)-algebraic approach with the deformation-controlling \(L_\infty[1]\)-algebra for relative Rota-Baxter \(3\)-Lie operators developed by Hou, Sheng, and Zhou.