Cohomology, deformations and extensions of Rota-Baxter Leibniz algebras

Abstract

A Rota-Baxter Leibniz algebra is a Leibniz algebra $(\mathfrak{g},[~,~]_{\mathfrak{g}})$ equipped with a Rota-Baxter operator $T : \mathfrak{g} \rightarrow \mathfrak{g}$. We define representation and dual representation of Rota-Baxter Leibniz algebras. Next, we define a cohomology theory of Rota-Baxter Leibniz algebras. We also study the infinitesimal and formal deformation theory of Rota-Baxter Leibniz algebras and show that our cohomology is deformation cohomology. Moreover, We define an abelian extension of Rota-Baxter Leibniz algebras and show that equivalence classes of such extensions are related to the cohomology groups.

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].…”

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

“…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. Furthermore, Peng et al [44] introduced the concept of modified λ-differential Lie algebras.…”

The algebraic and geometric classification of nilpotent binary and mono Leibniz algebras

“…It appeared in some papers of Bloh [in the 1960 s] and Loday [in 1990 s]. Recently, they appeared in many geometric and physics applications (see, for example, [8,39,40,44,46] and references therein). A systematic study of algebraic properties of Leibniz algebras is started from the Loday paper.…”