Lie algebras with di ff erential operators of any weights

: In this paper, we define a cohomology theory for di ff erential Lie algebras of any weight. As applications of the cohomology, we study abelian extensions and formal deformations of di ff erential Lie algebras of any weight. Finally, we consider homotopy di ff erential operators on L ∞ algebras and 2-di ff erential operators of any weight on Lie 2-algebras, and we prove that the category of 2-term L ∞ algebras with homotopy di ff erential operators of any weight is same as the category of Lie 2-algebras with 2-di ff erential operators of any weight.


Introduction
Derivations play a crucial role in studying deformation formulas [1], differential Galois theory [2] and homotopy algebras [3].They also are useful of control systems theory [4,5] and gauge algebras [6].The authors studied the operad of associative algebras with derivation in [7].Recently, the authors introduced Lie algebras with derivations, and studied their cohomology and deformations, extensions in [8].Later, Das [9] considered the similar results for Leibniz algebras with derivations.The authors studied cohomology of Leibniz triple systems with derivations in [10].
More and more scholars have begun to pay close attention to the structures of any weight thanks to the result of outstanding work [11,12], all kinds of Rota-Baxter algebras of any weight [13][14][15][16][17][18] appear successively.In order to study non-abelian extensions of Lie algebras.The notion of crossed homomorphisms of Lie algebras was introduced by Lue [19], which was applied to study the representations of Cartan Lie algebras [20].For λ ∈ k, the notion of a differential algebra of weight λ was first introduced by Guo and Keigher [21], which generalizes simultaneously the concept of the classical differential algebra and difference algebra [22].Applying the same method as for differential Lie algebras of weight λ.Later, the authors defined the cohomology of relative difference Lie algebras, and studied some properties in [23].Our aim in this paper is to consider Lie algebras with differential operators of weight λ (also known as differential Lie algebras).More precisely, we define a cohomology theory for differential Lie algebras and consider some properties.
The paper is organized as follows.In Section 2, we consider the representations of differential Lie algebras of any weight.In Section 3, we define a cohomology theory for differential Lie algebras of any weight.In Section 4, we study central extensions of differential Lie algebras of any weight.In Section 5, we study formal deformations of differential Lie algebras of any weight.In Section 6, we consider homotopy differential operators on L ∞ algebras and 2-differential operators of any weight on Lie 2-algebras.In Section 7, we prove that the category of 2-term L ∞ algebras with homotopy differential operator of any weight and the category of Lie 2-algebras with 2-differential operators of any weight are equivalent.
Throughout this paper, k denotes a field of characteristic zero.All the vector spaces, algebras, linear maps and tensor products are taken over k unless otherwise specified.

Representations of differential Lie algebras of any weight
For λ ∈ k.A differential operator of weight λ on a Lie algebra g is a linear operator We denote by Der λ (g) the set of differential operators of weight λ of the Lie algebra g.
We denote by LieD λ the category of differential Lie algebras and their morphisms.
To simply notations, for all the above notions, we will often suppress the mentioning of the weight λ unless it needs to be specified.Definition 2.3.(i) A representation over the differential Lie algebra (g, d g ) is a pair (V, d V ), where d V ∈ End k (V), and (V, ρ) is a representation over the Lie algebra g, such that ∀x ∈ g, v ∈ V, the following identity holds: One denotes by (g, d g )-Rep the category of representations over the differential Lie algebra (g, d g ).
Example 2.4.Any differential Lie algebra (g, d g ) is a representation over itself with It is called the adjoint representation over the differential Lie algebras (g, d g ).
Example 2.5.Let (V, ρ) be a representation of a Lie algebra g.Then the pair (V, Id V ) is a representation of the differential Lie algebra (g,Id g ) of weight −1.
Example 2.6.Let (g, d g ) be a differential Lie algebra of weight λ and (V, d V ) be a representation of it.Then for κ 0 ∈ k, the pair (V, κd V ) is a representation of the differential Lie algebra (g, κd g ) of any weight 1 κ λ.The following result is easily to check and we omit it.Proposition 2.7.Let (V, d V ) be a representation of the differential Lie algebra (g, d g ) of weight λ.Then

Cohomology of differential Lie algebras of any weight
Recall that the cochain complex of Lie algebra g with coefficients in representation V is the cochain complex , and the coboundary operator (−1) i+ j+n+1 f ([x i , x j ], x 1 , . . ., xi , . . ., xj , . . ., x n+1 ), for all f ∈ C n Lie (g, V), x 1 , . . ., x n+1 ∈ g.The corresponding cohomology is denoted by H * Lie (g, V).When V is the adjoint representation, we write H n Lie (g) = H n Lie (g, V), n ≥ 0. In the following, we will define the cohomology of the differential Lie algebra (g, d g ) of weight λ with coefficients in the representation (V, d V ).Define and define a linear map δ : for any f n ∈ C n LieD λ (g, V) and Given any f ∈ C n Lie (g, V), g ∈ C n−1 Lie (g, V) with n ≥ 1, we have Hence, the proof is finished.

Abelian extensions of differential Lie algebras of any weight
In this section, we show that abelian extensions of differential Lie algebras are classified by the second cohomology.
(ĝ, d ĝ) is called an abelian extension of (g, d g ) by (V, d V ).
Definition 4.2.Let (ĝ 1 , d ĝ1 ) and (ĝ 2 , d ĝ2 ) be two abelian extensions of (g, d g ) by (V, d V ).They are said to be isomorphic if there exists A section of an abelian extension (ĝ, d ĝ) of (g,  Proof.Firstly, we prove that ρ is a Lie algebra homomorphism, in fact, for any x, y ∈ g, v ∈ V, we have Moreover, we obtain Hence, (V, d V ) is a representation over (g, d g ).
We further consider linear maps ψ : g ⊗ g → V and χ : g → V by The differential Lie algebra structure on g⊕V with a multiplication [•, •] ψ and the differential operator d χ defined by Proof.For any x, y, z ∈ g, By (4.1), we have Since d χ satisfies Eq (2.1), we deduce that In the following, we will classify abelian extensions of differential Lie algebras.Theorem 4.5.Let V be a vector space and d V ∈ End k (V).Then abelian extensions of a differential Lie algebra (g, d g ) by (V, d V ) are classified by H 2 LieD λ (g, V) of (g, d g ).

Deformations of differential Lie algebras of any weight
In this section, we show that if H 2 LieD λ (g, g) = 0, then the differential Lie algebra (g, d g ) is rigid.Let (g, d g ) be a differential Lie algebra.Denote by µ g the multiplication of g.Consider the 1parameterized family Given any differential Lie algebra (g, d g ), interpret µ g and d g as the formal power series µ t and d t with µ i = δ i,0 µ g and d i = δ i,0 d g respectively for all i ≥ 0. Then (g[[t]], µ g , d g ) is a 1-parameter formal deformation of (g, d g ).
3) is equal to the Jabobi identity of µ g , and Eq (5.4) is equal to the fact that d g is a differential operator of weight λ.Proof.For n = 1, Eq (5. 3) is equal to ∂ Lie µ 1 = 0, and Eq (5.4) is equal to Thus for n = 1, Eqs (5.3) and (5.4) imply that (µ 1 , d 1 ) is a 2-cocycle.
If µ t = µ g in the above 1-parameter formal deformation of the differential Lie algebra (g, d g ), we obtain a 1-parameter formal deformation of the differential operator d g .Consequently, we have Corollary 5.4.Let d t be a 1-parameter formal deformation of the differential operator d g .Then d 1 is a 1-cocycle of the differential operator d g with coefficients in the adjoint representation (g, d g ).
Proof.In the special case when n = 1, Eq (5.4) is equal to ∂ Lie d 1 = 0, which implies that d 1 is a 1-cocycle of the differential operator d g with coefficients in the adjoint representation (g, d g ).
], µ t , d t ) be a formal isomorphism.For all x, y ∈ g, we have Furthermore, we obtain μ1 (x, y) = µ 1 (x, y) Thus, we have ( μ1 where ϕ i : g → g are linear maps with ϕ 0 = Id g , such that (5.8) Definition 5.9.A differential Lie algebra (g, d g ) is said to be rigid if every 1-parameter formal deformation is trivial.

Homotopy differential operators of any weight on 2-term L ∞ -algebras
In this section, we pay our attention to the homotopy differential operator of any weight on 2-term L ∞ -algebras introduced by [24].
for any a, b, c, w ∈ L 0 and u, v ∈ L 1 .
One denotes a 2-term L ∞ -algebra as above by (L 1 For any 2-term L ∞ -algebra L, the identity morphism Id L : L → L is given by the identity chain map L → L together with (Id L ) 2 = 0.
The collection of 2-term L ∞ -algebras and morphisms between them form a category.We denote this category by 2Lie ∞ .Definition 6.3.Let L = (L 1 d −→ L 0 , l 2 , l 3 ) be a 2-term L ∞ -algebra.A homotopy differential operator of weight λ on it consists of a chain map of the underlying chain complex (i.e., linear maps θ 0 : L 0 → L 0 and θ 1 : ) and a bilinear map θ 2 : L 0 ⊗ L 0 → L 1 such that for any a, b, c ∈ L 0 and u ∈ L 1 , the following identities are hold A 2-term L ∞ -algebra with a homotopy differential operator of weight λ as above denoted by the pair algebra with a homotopy differential operator of weight λ is said to be skeletal if the underlying 2-term L ∞ -algebra is skeletal, i.e., d = 0.
)) be two 2-term L ∞ -algebras with homotopy differential operators of weight λ.A morphism between them consists of a morphism ( f 0 , f 1 , f 2 ) between the underlying 2-term L ∞ -algebras and a linear map We denote the category of 2-term L ∞ -algebras with homotopy differential operators of weight λ and morphisms between them by 2LieD λ∞ .Theorem 6.5.There is a one-to-one correspondence between skeletal 2-term L ∞ -algebras with homotopy differential operators with weight λ and tuples ((g, d g ), (V, d V ), (θ, θ)), where (g, d g ) is a differential Lie algebra of weight λ , (V, d V ) is a representation and (θ, θ) is a 3-cocycle of the differential Lie algebra of weight λ with coefficients in the representation.
A 2-term L ∞ -algebra with a homotopy differential operator of weight λ is said to be strict if the underlying 2-term L ∞ -algebra is strict, i.e., θ 2 = 0. Next we introduce crossed modules of differential Lie algebras of weight λ and show that strict 2-term L ∞ -algebra with a homotopy differential operator of weight λ are in one-to-one correspondence with crossed module of differential Lie algebras of weight λ.Definition 6.6.A crossed module of differential Lie algebras of weight λ consist of ((g, d g ), (h, d h ), dt, Λ) where (g, d g ) and (h, d h ) are differential Lie algebras of weight λ, dt : g → h is a differential Lie algebra morphism and Theorem 6.7.There is a one-to-one correspondence between strict 2-term L ∞ -algebras with homotopy differential operators of weight λ and crossed module of differential Lie algebras of weight λ.

Categorification of differential Lie algebras of any weight
In this section, we study categorified differential operators of any weight (also called 2-differential operator) on Lie 2-algebras.
where Θ, R, P and Q are given by  We denote the category of Lie 2-algebras with 2-differential operators of weight λ and morphisms between them by LieD2 λ .
In the following, we will give our main result of this section.Given any 2-term L ∞ -algebra with a homotopy differential operator of weight λ morphism ( f 0 , f 1 , f 2 , Ψ) from L to L ′ , for any F 0 = f 0 , F 1 = f 1 and Moreover, S preserve the identity morphisms and composition of morphisms.Therefore, S is a functor from LieD2 λ to 2LieD λ∞ .
Finally, it is easy to prove that T • S 1 LieD2 λ , and the composite S • T 1 2LieD λ∞ and we omit them.

Definition 4 . 1 .
An abelian extension of differential Lie algebras is a short exact sequence of homomorphisms of differential Lie algebras 0

Proposition 4 . 3 .
With the above notations, (V, d V ) is a representation over the differential Lie algebra (g, d g ).

Proposition 5 . 3 .
Let (g[[t]], µ t , d t ) be a 1-parameter formal deformation of a differential Lie algebra (g, d g ).Then (µ 1 , d 1 ) is a 2-cocycle of the differential Lie algebra (g, d g ) with the coefficient in the adjoint representation (g, d g ).