Representations and cohomologies of relative Rota-Baxter Lie algebras and applications

Abstract

In this paper, first we give the notion of a representation of a relative Rota-Baxter Lie algebra and introduce the cohomologies of a relative Rota-Baxter Lie algebra with coefficients in a representation. Then we classify abelian extensions of relative Rota-Baxter Lie algebras using the second cohomology group, and classify skeletal relative Rota-Baxter Lie 2-algebras using the third cohomology group as applications. At last, using the established general framework of representations and cohomologies of relative Rota-Baxter Lie algebras, we give the notion of representations of Rota-Baxter Lie algebras, which is consistent with representations of Rota-Baxter associative algebras in the literature, and introduce the cohomologies of Rota-Baxter Lie algebras with coefficients in a representation. Applications are also given to classify abelian extensions of Rota-Baxter Lie algebras and skeletal Rota-Baxter Lie 2-algebras. Contents 1. Introduction 1 2. Representations of relative Rota-Baxter Lie algebras 3 3. Cohomology of relative Rota-Baxter Lie algebras 7 4. Classification of abelian extensions of relative Rota-Baxter Lie algebras 13 5. Classification of skeletal relative Rota-Baxter Lie 2-algebras 18 6. Representations and cohomologies of Rota-Baxter Lie algebras 21 References 25and cohomologies of Rota-Baxter Lie algebras using the above general framework of relative Rota-Baxter Lie algebras and give various applications.Acknowledgements. This research is supported by NSFC (11922110). Representations of relative Rota-Baxter Lie algebrasIn this section, we introduce the notion of a representation of a relative Rota-Baxter Lie algebra. We give the semidirect products characterization of representations of relative Rota-Baxter Lie algebras. The dual representation is given. A representation of a relative Rota-Baxter Lie algebra also induces a representation of the underlying pre-Lie algebra.Definition 2.1.Lie algebra, ρ : g → gl(V) is a representation of g on a vector space V and T : V → g is a relative Rota-Baxter operator, i.e.Note that a Rota-Baxter operator on a Lie algebra is a relative Rota-Baxter operator with respect to the adjoint representation. Definition 2.2. Let ((g, [•, •] g ), (V, ρ), T ) and ((g ′ , [•, •] g ′ ), (V ′ , ρ ′ ), T ′ ) be two relative Rota-Baxter Lie algebras. A homomorphism from ((g, [•, •] g ), (V, ρ), T ) to ((g ′ , [•, •] g ′ ), (V ′ , ρ ′ ), T ′ ) consists of a Lie algebra homomorphism φ : g → g ′ and a linear map ϕ : V → V ′ such that

Factorizable Lie bialgebras, quadratic Rota-Baxter Lie algebras and Rota-Baxter Lie bialgebras

“…Then (g, B) and (g * , B * ) are Rota-Baxter Lie algebras of weight λ. By Proposition 4.6, (g ⊲⊳ g * , B) is a Rota-Baxter Lie algebra of weight λ, where B is defined by (21). Moreover, it is straightforward to deduce that (g ⊲⊳ g * , B, S ) is a quadratic Rota-Baxter Lie algebra of weight λ, where S is given by ( 22).…”

“…For the purpose of defining matched pairs of Rota-Baxter Lie algebras, we introduce the notion of representations of Rota-Baxter Lie algebras of weight λ. Note that the concept of representations of Rota-Baxter Lie algebras of weight 0 was already given in [21] in the study of cohomologies of Rota-Baxter Lie algebras, and the notion of representations of Rota-Baxter associative algebras was introduced in [23], and further studied in [31]. Definition 3.6.…”

Rota-Baxter Lie $2$-algebras

“…Definition 2.5. ( [11] ) A representation of a Rota-Baxter Lie algebra (g, [•, •] g , R) on a vector space V with respect to a linear map R ∈ gl(V) is a representation ρ of the Lie algebra g on V, satisfying…”

“…A Lie algebra equipped with a Rota-Baxter operator is called a Rota-Baxter Lie algebra. Recently, cohomologies, deformations and extensions of Rota-Baxter Lie algebras are studied in [11,14,18]. See [10] fore details on Rota-Baxter operators.…”

Deformations and cohomology theory of weighted Rota-Baxter $3$-Lie algebras

“…which is easily seen to be isomorphic to the original one (16). Now we investigate the influence of different choices of sections.…”