Cohomologies of modified $\lambda$-differential Lie triple systems and applications

In this paper, we introduce the concept and representation of modified $\lambda$-differential Lie triple systems. Next, we define the cohomology of modified $\lambda$-differential Lie triple systems with coefficients in a suitable representation. As applications of the proposed cohomology theory, we study 1-parameter formal deformations and abelian extensions of modified $\lambda$-differential Lie triple systems.


Introduction
Jacobson [19,20] introduced the concept of Lie triple systems by quantum mechanics and Jordan theory.In fact, Lie triple systems originated from E.Cartan's research on symmetric spaces and totally geodesic submanifolds.Since then, the structure theory, representation theory, cohomology theory, deformation theory and extension theory of Lie triple systems were established in [16,18,21,24,25,37,39].
Derivations are useful for the study of algebraic structure.Derivations also play an important role in the study of homotopy algebras, deformation formulas, differential Galois theory, control theory and gauge theories of quantum field theory, see [1,2,4,5,27,33].Recently, associative algebras with derivations [26], Lie algebras with derivations [32], Leibniz algebras with derivations [7], Leibniz triple systems with derivations [35] and Lie triple systems with derivations [15,30,36] have been widely studied.All these results provide a good starting point for our further study.
In recent years, due to the outstanding work of [3,8,[11][12][13]34], more and more scholars begun to pay attention to the structure with arbitrary weights.Rota-Baxter Lie algebras of any weight [9], Rota-Baxter 3-Lie algebras of any weight [14,17] and Rota-Baxter Lie triple systems of any weight [6] appear successively.After that, for λ ∈ K, the cohomology, extension and deformation theory of Lie algebras with differential operators of weight λ are introduced by Li and Wang [22].In addition, the cohomology and deformation theory of modified Rota-Baxter associative algebras and modified Rota-Baxter Leibniz algebras of weight λ are given in [10,23,28].The concept of modified λ-differential Lie algebras are introduced in [29].The method of this paper is to follow the recent work [13,15,29,30,36].Our main objective is to consider modified λ-differential Lie triple systems.More precisely, we introduce the concept of a modified λ-differential Lie triple system, which includes a Lie triple system and a modified λ-differential operator.We define a cochain map Φ, and then give the cohomology of modified λ-differential Lie triple systems with coefficients in a representation by using a Yamaguti coboundary operator δ and a cochain map Φ.Next we study 1-parameter formal deformations of a modified λ-differential Lie triple system using the third cohomology group of modified λ-differential Lie triple system with the coefficient in the adjoint representation.Finally, we study abelian extensions of a modified λ-differential Lie triple system using the third cohomology group.All the results in this paper can be regarded as generalizations of Lie triple systems with derivations [15,30,36].
The paper is organized as follows.In Section 2, we introduce the concept of a modified λ-differential Lie triple system, and give its representation.In Section 3, we define a cohomology theory for modified λ-differential Lie triple systems with coefficients in a representation.In Section 4, we study 1-parameter formal deformations of a modified λ-differential Lie triple system.In Section 5, we study abelian extensions of a modified λ-differential Lie triple system.
Throughout this paper, K denotes a field of characteristic zero.All the algebras, vector spaces, algebras, linear maps and tensor products are taken over K.

Representations of modified λ-differential Lie triple systems
In this section, first, we recall some basic concepts of Lie triple systems from [19] and [37].Then, we introduce the concept of a modified λ-differential Lie triple system and its representation.
for all x, y, z, a, b ∈ L.
(ii) A homomorphism between two Lie triple systems Definition 2.2.
It is called the adjoint representation over the Lie triple system.
We denote the set of all derivations on L by Der(L).See [38] for various derivations of Lie triple systems.
Moreover, there is a close relationship between derivations and modified λ-differential operators.
Proposition 2.6.Let (L, [•, •, •]) be a Lie triple system.Then, a linear operator d : L → L is a modified λ-differential operator if and only if d + λ 2 id L is a derivation on L.
Proof.Eq. (2.6) is equivalent to The proposition follows.
From Eq. (2.7), we get One can refer to [15,30,36] for more information about Lie triple systems with derivations.
Moreover, the following result finds the relation between representations over modified λ-differential Lie triple systems and over Lie triple systems with derivations.Proposition 2.13.Let (V; θ) be a representation of the Lie triple system Proof.Eq. (2.7) is equivalent to The proposition follows.
Example 2.14.Let (V; θ) be a representation of the Lie triple system (L, [•, Next we construct the semidirect product in the context of modified λ-differential Lie triple systems.Proposition 2.16.Let (L, [•, •, •], d) be a modified λ-differential Lie triple system and (V; θ, d V ) be a representation of it.Then L ⊕ V is a modified λ-differential Lie triple system under the following maps: for all x, y, z ∈ L and u, v, w ∈ V.
Proof.First, as we all know, (L ⊕ V, [•, •, •] ⋉ ) is a Lie triple system.Next, for any x, y, z ∈ L, u, v, w ∈ V, by Eqs.(2.6), (2.7) and (2.8), we have and V * be a dual space of V. We define a bilinear map θ * : L × L → End(V * ) and a linear map Proposition 2.17.With the above notations, Proof.Following [31], we can easily get (V * ; −θ * τ ) is a representation of the Lie triple system (L, [•, •, •]).Moreover, for any a, b ∈ L, v ∈ V and u * ∈ V * , by Eqs.(2.7) and (2.9), we have Example 2.18.Let (L; R, d) be an adjoint representation of the modified λ-differential 3 Cohomology of modified λ-differential Lie triple systems In this section, we study the cohomology of a modified λ-differential Lie triple system with coefficients in its representation.
Next, we introduce a cohomology of a modified λ-differential Lie triple system with coefficients in a representation.
We first give the following lemma.
If ν t = ν in the above 1-parameter formal deformation of the modified λ-differential Lie triple system (L, ν, d), we obtain a 1-parameter formal deformation of the modified λ-differential operator d.So we have Proof.When n = 1, by ν 1 = 0 and Eq.(4.8), we have d 1 ∈ Der(L).That is, Eq. (4.8) is equivalent to δd 1 = 0, which implies that d 1 is a 1-cocycle of the modified λ-differential operator d with coefficients in the adjoint representation (L; R, d).
for any a, b ∈ L, v ∈ V and u * ∈ V * .Give the switching operator τ : L ⊗ L → L ⊗ L by τ (a ⊗ b) = τ (b ⊗ a), for any a, b ∈ L.

Corollary 4 . 3 .
Let d t be a 1-parameter formal deformation of the modified λ-differential operator d.Then d 1 is a 1-cocycle of the modified λ-differential operator d with coefficients in the adjoint representation (L; R, d).