A cohomological study of modified Rota-Baxter algebras

Abstract

A modified Rota-Baxter algebra is an algebra equipped with an operator that satisfies the modified Yang-Baxter equation. In this paper, we define the cohomology of a modified Rota-Baxter algebra with coefficients in a suitable bimodule. We relate our cohomology of a modified Rota-Baxter algebra with the known cohomology theory of a Rota-Baxter algebra. As applications of our cohomology, we study formal one-parameter deformations and abelian extensions of modified Rota-Baxter algebras. Contents 1. Introduction 1 2. Modified Rota-Baxter algebras 3 3. Cohomology of modified Rota-Baxter algebras 8 4. Deformations of modified Rota-Baxter algebras 11 5. Abelian extensions of modified Rota-Baxter algebras 13 References 15

Abelian Extensions of Modified λ-Differential Left-Symmetric Algebras and Crossed Modules

“…Inspired from [9,10], the notion of modified λ-differential Lie algebras was introduced in [11]. Subsequently, the algebraic structures with modified operators were widely studied in [12][13][14][15][16].…”

Abelian Extensions and Crossed Modules of Modified λ-Differential Left-Symmetric Algebras

“…Motivated by the modified r-matrices, in [11], Peng and his collaborators introduced the concept of modified λ-differential Lie algebras. Subsequently, the algebraic structures with modified operators have been widely studied in [12][13][14][15][16][17][18].…”

Deformations and Extensions of Modified λ-Differential 3-Lie Algebras

“…In recent years, scholars have increasingly focused on structures with arbitrary weights, thanks to the important work of [32][33][34][35][36][37]. The papers [38][39][40] established the cohomology, extensions and deformations of Rota-Baxter 3-Lie algebras with any weight λ, as well as the differential 3-Lie algebras with any weight λ. Additionally, the cohomology and deformation of modified Rota-Baxter algebras were studied by Das [41]. The works [42,43] provided insights into the cohomology and deformation of modified Rota-Baxter Leibniz algebras with weight λ.…”