Actions of monoidal categories and representations of Cartan type Lie algebras

Using crossed homomorphisms, we show that the category of weak representations (resp. admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs) is a left module category over the monoidal category of representations of Lie algebras. In particular, the corresponding bifunctor of monoidal categories is established to give new weak representations (resp. admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs). This generalizes and unifies various existing constructions of representations of many Lie algebras by using this new bifunctor. We construct some crossed homomorphisms in different situations and use our actions of monoidal categories to recover some known constructions of representations of various Lie algebras, also to obtain new representations for generalized Witt algebras and their Lie subalgebras. The cohomology theory of crossed homomorphisms between Lie algebras is introduced and used to study linear deformations of crossed homomorphisms.

This paper aims to give a conceptual approach to unify various constructions of representations of certain Lie algebras and construct new representations of some Lie algebras using crossed homomorphisms, Lie-Rinehart algebras and Leibniz pairs. 1.1. Representations of Cartan type Lie algebras. The representation theory of Lie algebras is of great importance due to its own overall completeness, and applications in mathematics and mathematical physics. The Cartan type Lie algebras, originally introduced and studied by Cartan, consist of four classes of infinite-dimensional simple Lie algebras of vector fields with formal power series coefficients: the Witt algebras, the divergence-free algebras, the Hamiltonian algebras, and the contact algebras. The representation theory of Cartan type Lie algebras was first studied by Rudakov [40,41]. He showed that irreducible continuous representations can be described explicitly as induced representations or quotients of induced representations. Later Shen [42] studied graded modules of graded Lie algebras of Cartan type with polynomial coefficients of positive characteristic. Larsson constructed a class of representations for the Witt algebras with Laurent polynomial coefficients [24]. More precisely, Shen's modules, called mixed product, were constructed by certain monomorphism; while Larsson's modules, named conformal fields, came from physics background. Many other authors have contributed very much to the theory along these approaches for the last few decades. In particular, irreducible modules with finite-dimensional weight spaces over the Virasoro algebra (universal central extension of the Lie algebra W 1 of vector fields on a circle) was classified by Mathieu in [33], while Billig and Futorny recently gave the classification of irreducible modules over the Witt algebras W n (n ≥ 2) with finite-dimensional weight spaces [2]. Note that intrinsically there is a functor from the category of finite-dimensional irreducible representations of finite dimensional simple Lie algebras to the category of representations of Cartan type Lie algebras among these works. Actually there should be some essential part that applies to all those constructions (even more) of complicated modules over some classes of Lie algebras (not only Cartan type Lie algebras) as a whole regardless of any specific feature exhibited in each particular case. From this point of view it is of no surprise that earlier results in this direction due to many authors are fragments of the general theory. We find a unifying conceptual approach generalizing Shen's construction. This is one of the main purposes of the paper.

1.2.
Representations of Lie-Rinehart algebras and Leibniz pairs. Note that the above mentioned Cartan type Lie algebras are either Lie-Rinehart algebras, or Leibniz pairs. Lie-Rinehart algebras, which was originally studied by Rinehart in [39] in 1963, arose from a wide variety of constructions in differential geometry, and they have been introduced repeatedly into many areas under different terminologies, e.g. Lie pseudoalgebras. Lie-Rinehart algebras are the underlying structures of Lie algebroids. See [31] and references therein for more details. A Lie-Rinehart algebra is a quadruple (A, L, [·, ·] L , α), where A is a commutative associative algebra, L is an A-module, [·, ·] L is a Lie bracket on L and α : L → Der K (A) is an A-module homomorphism with some compatibility conditions involving the Lie brackets. Lie-Rinehart algebras have been further investigated in many aspects [6,19,20,21,32,34]. In particular, Rinehart constructed the universal enveloping algebra of a Lie-Rinehart algebra [39]. Huebschmann gave an alternative construction of the universal enveloping algebra U(A, L) of a Lie-Rinehart algebra (A, L, [·, ·] L , α) via the smash product, namely U(A, L) = (A#U(L))/J, where J is a certain two-sided ideal in A#U(L), and showed that there is a one-one correspondence between representations of a Lie-Rinehart algebra and representations of its universal enveloping algebra [19]. Representations of Lie-Rinehart algebras are deeply related to the theory of D-modules [38], which are modules over the algebra D of linear differential operators on a manifold. Since the algebra D is the universal enveloping algebra of the Lie-Rinehart algebra of vector fields, a D-module is the same as a module with a representation of the Lie-Rinehart algebra of vector fields. We introduce the notion of a weak representation of a Lie Rinehart algebra. The adjoint action is naturally a weak representation of a Lie-Rinehart algebra on itself. There is a one-to-one correspondence between weak representations of a Lie-Rinehart algebra and representations of the smash product A#U(L).
The notion of a Leibniz pair was originally introduced by Flato-Gerstenhaber-Voronov in [11], which consists of a K-Lie algebra (S, [·, ·] S ) and a K-Lie algebra homomorphism β : S → Der K (A). In this paper we only consider the case that A is a commutative associative algebra. A Leibniz pair was also studied by Winter [47], and called a Lie algop. Leibniz pairs were further studied in [18,23]. A Lie-Rinehart algebra (A, L, [·, ·] L , α) naturally gives rise to a Leibniz pair by forgetting the A-module structure on L. We introduce the notion of an admissible representation of a Leibniz pair. If WRep K (L) denotes the category of weak representations of a Lie-Rinehart algebra L, and ARep K (S) denotes the category of admissible representations of a Leibniz pair S, then we have the following category equivalence: where the right-hand side L is considered as the underlying Leibniz pair of a Lie-Rinehart algebra. On the other hand, a Leibniz pair also gives rise to a Lie-Rinehart algebra S ⊗ K A, known as the action Lie-Rinehart algebra. We show that an admissible representation of a Leibniz pair can be naturally extended to a representation of the corresponding action Lie-Rinehart algebra. Actually we have the following category equivalence: where Rep(S ⊗ K A) denotes the category of representations of the Lie-Rinehart algebra S ⊗ K A. See Remark 3.34 for more details about this equivalence.
1.3. Crossed homomorphisms. The concept of a crossed homomorphism of Lie algebras was introduced in [30] in the study of nonabelian extensions of Lie algebras in 1966. A special class of crossed homomorphisms are recently called a differential operator of weight 1 in [12,13]. A flat connection 1-form of a trivial principle bundle is naturally a crossed homomorphism. To the best of our knowledge this concept has not been investigated for many years. Now we have to use it in this paper. More precisely, by using crossed homomorphisms, we show that the category of weak representations (resp. admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs) is a left module category over the monoidal category of representations of Lie algebras. In particular, we obtain bifunctors among categories of certain representations: which we call the actions of monoidal categories, generalizing Shen-Larsson constructions of representations for Cartan type Lie algebras. Our construction sheds light on some difficult classification problems in representation theory of Lie algebras.
We have seen the importance of crossed homomorphisms in our above construction. To better understand crossed homomorphisms and our actions of monoidal categories, we also study deformations and cohomologies of crossed homomorphisms. The deformation of algebraic structures began with the seminal work of Gerstenhaber [16,17] for associative algebras and followed by its extension to Lie algebras by Nijenhuis and Richardson [36]. A suitable deformation theory of an algebraic structure needs to follow certain general principle: on one hand, for a given object with the algebraic structure, there should be a differential graded Lie algebra whose Maurer-Cartan elements characterize deformations of this object. On the other hand, there should be a suitable cohomology so that the infinitesimal of a formal deformation can be identified with a cohomology class. We successfully construct a differential graded Lie algebra such that crossed homomorphisms are characterized as Maurer-Cartan elements. The cohomology groups of crossed homomorphisms are also defined to control their linear deformations.
1.4. Outline of the paper. In Section 2, we recall the concept of crossed homomorphisms between Lie algebras and show that there is a one-to-one correspondence between crossed homomorphisms and certain Lie algebra homomorphisms (Theorem 2.7). This fact is the key ingredient in our later construction of the left module category.
In Section 3, we introduce the new concepts: weak representations (resp. admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs). Using crossed homomorphisms, we show that the category of weak representations (resp. admissible representations) of Lie-Rinehart algebras (resp. Leibniz pairs) is a left module category over the monoidal category of representations of Lie algebras. In particular, the corresponding bifunctor F H which we call the action of monoidal categories, is established to give new representations of Lie-Rinehart algebras (resp. Leibniz pairs). See Theorems 3.26 and 3.36. This generalizes and unifies various existing constructions of representations of many Lie algebras by using this new bifunctor.
In Section 4, to show the power of our action of monoidal categories F H established in Section 3, we construct some examples of crossed homomorphisms in different situations and using our action of monoidal categories to recover some known constructions representations of various Lie algebras (see Section 4.1-4.3), and to obtain new representations of generalized Witt algebras and their Lie subalgebras (See Corollaries 4.15,4.17,4.18). Certainly, our action of monoidal categories will be used to other situations to give new simple representations of suitable Lie algebras.
In Section 5, we characterize crossed homomorphisms as Maurer-Cartan elements in a suitable differential graded Lie algebra and introduce the cohomology theory of crossed homomorphisms. We use the cohomology theory of crossed homomorphisms that we established to study linear deformations of crossed homomorphisms, to prove that the linear deformation H t := H + td ρ H (−Hx) is trivial for any Nijenhuis element x (Theorem 5.14).
We conclude our paper in Section 6 by asking three questions. As usual, we denote by Z, Z + and C the sets of all integers, positive integers and complex numbers. All vector spaces are over an algebraically closed field K of characteristic 0.

Crossed homomorphisms between Lie algebras
Let (g, [·, ·] g ) and (h, [·, ·] h ) be Lie algebras. We will denote by Der(g) and Der(h) the Lie algebras of derivations on g and h respectively. A Lie algebra homomorphism ρ : g → Der(h) will be called an action of g on h in the sequel.
Remark 2.2. A crossed homomorphism from g to g with respect to the adjoint action is also called a differential operator of weight 1. See [12,13] for more details. Example 2.3. Let P be a trivial G-principle bundle over a differential manifold M, where G is a Lie group. Let ω ∈ Ω 1 (M, g) be a connection 1-form, where g is the Lie algebra of G. Then ω is flat if and only if dω Therefore, a flat connection 1-form, i.e. ω ∈ Ω 1 (M, g) = Hom(X(M), g ⊗ C ∞ (M)) satisfying the above equality, is a crossed homomorphism from the Lie algebra of vector fields X(M) to the Lie algebra g ⊗ C ∞ (M) with respect to the action ρ given by Example 2.4. If the action ρ of g on h is zero, then any crossed homomorphism from g to h is nothing but a Lie algebra homomorphism. If h is commutative, then any crossed homomorphism from g to h is simply a derivation from g to h with respect to the representation (h; ρ). Definition 2.5. Let H and H ′ be crossed homomorphisms from g to h with respect to the action ρ. A homomorphism from H ′ to H consists of two Lie algebra homomorphisms φ g : g −→ g and In particular, if φ g and φ h are invertible, then (φ g , φ h ) is called an isomorphism from H ′ to H.
The following result can be also found in [30]. Lemma 2.6. Let H be a crossed homomorphism from g to h with respect to the action ρ. Define ρ H : g −→ gl(h) by Then ρ H is also an action of g on h, i.e. ρ H : g → Der(h) is a Lie algebra homomorphism.
We use g ⋉ ρ H h and g ⋉ ρ h to denote the two semidirect products of g and h with respect to the actions ρ H and ρ respectively. More precisely, we have Theorem 2.7. Let H : g → h be a linear map and ρ : g → Der(h) an action of g on h.
(a) Suppose that ρ H given by (4) is an action of g on h. Then the linear mapĤ : if and only if (1) holds for H, which is equivalent to that H is a crossed homomorphism from g to h with respect to the action ρ.
(b) follows from the proof of (a) by taking u = v = 0.
Remark 2.8. In fact, crossed homomorphisms correspond to split nonabelian extensions of Lie algebras. More precisely, we consider the following nonabelian extension of Lie algebras: A section s : g → g ⊕ h must be of the form s(x) = (x, Hx), x ∈ g. Statement (b) says that s is a Lie algebra homomorphism if and only if H is a crossed homomorphism. Such an extension is called a split nonabelian extension. See [30] for more details.

Action of monoidal categories arising from Lie-Rinehart algebras and Leibniz pairs
In this section, first we introduce the notion of a weak representation of a Lie-Rinehart algebra, show that the category of weak representations of Lie-Rinehart algebras is a left module category over the monoidal category of representations of Lie algebras by using crossed homomorphisms. Then we introduce the notion of an admissible representation of a Leibniz pair and obtain similar results. In particular, the corresponding bifunctors are called the actions of monoidal categories for Lie-Rinehart algebras and Leibniz pairs.

Weak representations of Lie-Rinehart algebras.
Let A be a commutative associative algebra over K. We denote by Der K (A) the set of K-linear derivations of A, i.e.  We usually denote a Lie-Rinehart algebra over A by (A, L, [·, ·] L , α) or simply by L.    Remark 3.6. Let M be an A-module. It is straightforward to see that we have a semidirect product commutative associative algebra A ⋉ M, where the multiplication is given by Then (D, σ) is a first order differential operator on M if and only if (σ, D) is a derivation on the commutative associative algebra A ⋉ M. This result is the algebraic counterpart of the fact that a first order differential operator on a vector bundle E can be viewed as a linear vector field on the dual bundle E * . In fact, functions on the vector bundle E * → N are generated by C ∞ (N) and Γ(E), while the latter are fibre linear functions on E * . Since a first order differential operator maps a fibre linear function to a fibre linear function, so it is viewed as a linear vector field on E * .
, for all a ∈ A and x ∈ L.
Note that zero map from L to L ′ is not a Lie-Rinehart weak homomorphism if α 0.
Proof. This is easy to see.
We denote by WH(L, L ′ ) the set of weak homomorphisms from the Lie-Rinehart algebra That is, (D = ρ(x), σ = α(x)) is a first order differential operator on M. [35]) is a K-vector space A ⊗ U(L) with elements denoted by a#u , and with product defined for all a, b ∈ A and u, v ∈ U(L) by

Remark 3.11. In [19], Huebschmann showed that there is a one-one correspondence between representations of a Lie-Rinehart algebra and representations of its universal enveloping algebra U(A, L) := (A#U(L))/J, where J is a certain ideal of the smash product A#U(L). More explicitly, it is known that U(L) is a Hopf algebra and A is a U(L)-module algebra. Then the smash product A#U(L) (see
where we use the standard Sweedler notation ∆(u) = u (1) ⊗ u (2) for the coproduct ∆. The algebra A#U(L) is also called the Massey-Peterson algebra in [19]. It is not hard to see that there is a one-to-one correspondence between weak representations of a Lie-Rinehart algebra and representations of the smash product A#U(L).
Then ad is a weak representation of L on L. Note that ad is generally not a representation of L on itself.
Proof. This is easy to see.
We usually denote by M φ → M ′ a homomorphism between the weak representations (M; ρ) and (M ′ ; ρ ′ ), denote by WRep K (L) the category of weak representations of a Lie-Rinehart algebra (A, L, [·, ·] L , α) and Rep K (g) the category of representations of a K-Lie algebra (g, [·, ·] g ). It is obvious that the category of representations of a Lie-Rinehart algebra (A, L, [·, ·] L , α), denoted by Rep(L), is a full subcategory of the category WRep K (L). Please note the subtle difference of the two categories Rep K (L) and Rep(L).
Note that the Lie algebra L ⋉ ρ G acts on the Lie algebra (G, [·, ·] G ) bỹ Then using Theorem 2.7 (b) we can easily verify the following result.
Then ι H is a Lie-Rinehart injective weak homomorphism from L to L ⋉ ρ G.
Proof. By Propositions 3.8 and 3.17, we deduce that ρ Let (A, L, [·, ·] L , α) be a Lie-Rinehart algebra and (g, [·, ·] g ) a K-Lie algebra. Then G = g ⊗ K A is a Lie A-algebra, where the A-module structure and the Lie bracket [·, ·] G is given by Moreover, the Lie-Rinehart algebra (A, L, [·, ·] L , α) acts on the Lie A-algebra g ⊗ K A by α as follows: Consequently, we have the semidirect product Lie-Rinehart algebra (A, be a Lie-Rinehart algebra and (M; ρ) a Lie-Rinehart weak representation of (A, L, [·, ·] L , α). Let (g, [·, ·] g ) be a K-Lie algebra and (V; θ) a representation of g. Then V ⊗ K M has a natural A-module structure: Lemma 3.19. With the above notations, Proof. Since ρ is an A-module homomorphism, we have Thus, ρ ⊞ θ is also an A-module homomorphism.
Before we give the main result of the paper, we recall the notions of a monoidal category and a left module category over a monoidal category.

Definition 3.21. ([10])
A monoidal category is a 6-tuple (C, ⊗, a, 1, l, r) consists of the following data: • A category C; • A bifunctor ⊗ : C × C → C called the monoidal product; called the associativity isomorphism; • An object 1 ∈ Ob(C) called the unit object; • A natural isomorphism l : ⊗ • (1 × Id C ) → Id C called the left unit isomorphism and a natural isomorphism r : ⊗ • (Id C × 1) → Id C called the right unit isomorphism.
These data satisfy the following two axioms: (1) the pentagon axiom: the pentagon diagram commutes for the all W, X, Y, Z ∈ Ob(C).
(2) the triangle axiom: the triangle diagram The monoidal category C is strict if the associativity isomorphism, left unit isomorphism and right unit isomorphism a, l, r are all identities.
Example 3.22. Let C be a category and End(C) the category of endofunctors (the functors from C into itself). Then End(C) is a strict monoidal category with the composition of functors as the monoidal product and the identity functor as the unit object of this category.
Example 3.23. The category of representations Rep K (g) of a K-Lie algebra g is a monoidal category: the monoidal product of (V 1 ; θ 1 ) and (V 2 ; θ 2 ) is defined by and the unit object 1 is the 1-dimensional trivial representation (K; 0) of g. Moreover, the associativity isomorphism the left unit isomorphism l (V;θ) and the right unit isomorphism r (V;θ) are defined by Example 3.25. Any monoidal category (C, ⊗, a, 1, l, r) is a left module category over itself. More precisely, we set ⊗ C = ⊗, a C = a, l C = l. This left module category can be considered as a categorification of the regular representation of an associative algebra.
Let (A, L, [·, ·] L , α) be a Lie-Rinehart algebra and g a K-Lie algebra. Let H be a crossed homomorphism from the K-Lie algebra L to g ⊗ K A with respect to the action α given by (10). For all x ∈ L, we set • the natural isomorphism l (M;ρ) : F H (K; 0), (M; ρ)) → (M; ρ), which is defined by Proof. By Corollary 3.18 and Lemma 3.19, (V ⊗ K M; (ρ ⊞ θ) • ι H ) is a weak representation of L. Thus, F H is well-defined on the set of objects. To see that F H is also well-defined on the set of morphisms, we need to show that the linear map ψ ⊗ φ : For all x ∈ g, v ∈ V and m ∈ M, we have Thus, we obtain that F H (ψ, φ) = ψ⊗φ is a homomorphism of the weak representations. Moreover, by straightforward computations, we deduce that F H preserves identity morphisms and composite morphisms. Therefore, F H is a bifunctor.
Let (M; ρ) be a weak representation of the Lie-Rinehart algebra (A, L, [·, ·] L , α). We have For all x ∈ L, k ∈ K and m ∈ M, we have Thus, we deduce that l (M;ρ) is a homomorphism of weak representations. Moreover, by straightforward computations, we obtain that l (M;ρ) is a natural isomorphism and satisfies the triangle diagram in Definition 3.24. The proof is finished.
Since (A; α) is a representation of a Lie-Rinehart algebra (A, L, [·, ·] L , α), which is known as the natural representation, we obtain the following result.
We can also have a very useful functor on WRep K (L) as follows. A special but very interesting case of the above result is that (V; θ) = (g; ad). In the next section we will show that Corollary 3.28 is a very efficient way to construct interesting modules from easy modules.

3.3.
Admissible representations of Leibniz pairs. In this subsection, we introduce the notion of an admissible representation of a Leibniz pair. In the sequel, A is always a commutative associative algebra. The notion of a Leibniz pair was originally given in [11].   It is straightforward to obtain the following result. It is obvious that any Lie-Rinehart algebra is a Leibniz pair. A weak representation of a Lie-Rinehart algebra is naturally an admissible representation of the underlying Leibniz pair. Actually we have the following category equivalence: where the right-hand side L is considered as a Leibniz pair.
Conversely, given a Leibniz pair (A, S, [·, ·] S , β), we also have an action Lie-Rinehart algebra (A, S ⊗ K A, [·, ·], α), where the A-module structure and the K-Lie bracket [·, ·] are given by and an A-module homomorphism α : S ⊗ K A → Der K (A) is defined by α(x ⊗ a) := aβ(x) for all a, b ∈ A, x, y ∈ S. Furthermore, we obtain the following result. Proof. First it is obvious that ρ is an A-module homomorphism from S ⊗ K A to gl K (M). Then it is straightforward to deduce that ρ is a K-Lie algebra homomorphism. Finally, by (17), we deduce that Thus, (M; ρ) is a representation of the Lie-Rinehart algebra S ⊗ K A.

Denote this Leibniz pair by S ⋉ β (h ⊗ K A).
Let (M; ρ) be an admissible representation over S and (V; θ) a representation of a K-Lie algebra h. Then V ⊗ K M has a natural A-module structure: We define a K-linear map ρ ⊞ θ : Then it is straightforward to verify the following result. Let H be a crossed homomorphism from the K-Lie algebra S to h ⊗ K A. Then we have the Lie algebra homomorphism Similar to Theorem 3.26, we have the following result. Proof. We verify that the representation (V ⊗ K M; (ρ ⊞ θ) • ι H ) satisfies (17). For any The proof is similar to Theorem 3.26. So the details will be omitted.
Since (A; β) is an admissible representation of a Leibniz pair (A, S, [·, ·] S , β), we obtain the following result.
We can also have a very useful functor on WRep K (L) as follows.
A special but very interesting case of the above result is that (V; θ) = (h; ad). According to Corollaries 3.27 and 3.37, the bifunctors F H in Theorems 3.26 and F H given in Theorems 3.36 are the actions of monoidal categories.

Representations of Cartan type Lie algebras
From the definition of a crossed homomorphism we see that it is generally hard to find nontrivial crossed homomorphisms. Next we will show you some examples of crossed homomorphisms and their tremendous power in obtaining new irreducible modules via results in the previous section.

Shen-Larsson functors of Witt type.
For n ≥ 1, recall the Witt algebra W n = Der(A n ) over the Laurent polynomial algebra A n = C[x ±1 1 , · · · , x ±1 n ], which can be interpreted as the Lie algebra of (complex-valued) polynomial vector fields on an n-dimensional torus. Let ∂ i = ∂ ∂x i be the partial derivation with respect to the variable x i for i = 1, 2, . . . , n, denote d i = x i ∂ i , and x r = x r 1 1 x r 2 2 · · · x r n n for r = (r 1 , r 2 , · · · , r n ) T ∈ Z n . Then with the Lie bracket: Obviously, (A n , W n , [·, ·] W n , Id) is a Lie-Rinehart algebra. Certainly (A n ; Id) is the natural representation of the Lie-Rinehart algebra (A n , W n , [·, ·] W n , Id). Let g = gl n be the Lie algebra of all n × n complex matrices. Then G = gl n ⊗ A n is a Lie A n -algebra. For 1 ≤ i, j ≤ n, we use E i j to denote the n × n matrix with 1 at the (i, j) entry and zeros elsewhere.
By forgetting the A n -module structure, the corresponding functor F A n H is the wellknown Shen-Larsson functor of type (W n , gl n ), introduced by Shen [42] (over polynomial algebras), and Larsson [24] (over Laurent polynomial algebras), independently in different settings. For any simple gl n -module V the simplicity of the W n -module F A n H (V; θ) was determined in [9,15,26]. In particular, simple W n -modules of this class (with V to be simple finite-dimensional gl n -modules) are all the simple Harish-Chandra W n -modules [2].
1 , · · · , x ±1 n , ∂ 1 , · · · , ∂ n ] be the Weyl algebra, which is the universal enveloping algebra of the Lie-Rinehart algebra (A n , W n , [·, ·] W n , Id). Let (P; ρ) be a representation of A n . It is obvious that (P; ρ| W n ) is a W n -module. By Lemma 4.1 and Corollary 3.28, we obtain the following result. [25], is a generalization of the Shen-Larsson functor of type (W n , gl n ), which gives a class of new simple modules over W n . This class of simple W n -modules was used in the classification of simple W n -modules that are finitely generated as modules over its Cartan subalgebra (see [14]).

Remark 4.5. The functor F P H , introduced by Liu, Lu and Zhao in
Next we take g = C, the one-dimensional trivial Lie algebra. Let p = (p 1 , p 2 , · · · , p n ) ∈ , q ∈ C. Similar to the automorphism σ b in Section 2 of [45], we can easily see that the linear map W n → W n ⋉ Id A n , is a crossed homomorphism from W n to A n . In fact, H p,q ∈ Der C (W n , A n ). By Lemma 4.1 and Corollary 3.28, we obtain the following result. Remark 4.7. By forgetting the A n -module structure, the corresponding functor F p,q is just the twisting functor in the W n -module category introduced in [28,29,45], where a lot of new simple modules were obtained over the Virasoro algebra and W n .

Shen functors of divergence zero type.
In this section we assume that n ≥ 2. Let us recall the divergence map div : W n → A n with x r d i → r i x r , for all r ∈ Z n . It is well-known that is a Lie subalgebra of W n , called the Lie algebra of divergence zero vector fields on an ndimensional torus. Let d i j (r) = r j x r d i − r i x r d j . Then with the Lie bracket for r, s ∈ Z N , i, j, p, q = 1, · · · , n.
Note that S n is not a Lie-Rinehart subalgebra since S n is not an A n -module. It is straightforward to see that (A n , S n , [·, ·] W n , Id) is a Leibniz pair.
Recall that sl n is the Lie subalgebra of gl n consisting of all traceless complex matrices. The restriction H| S n of the crossed homomorphism H in Lemma 4.1 is a crossed homomorphism from S n to sl n ⊗ A n . By Corollary 3.37, we obtain the following result.
The functor F A n H is the well-known Shen-Larsson functor of type (S n , sl n ), introduced by Shen over polynomial algebras [42] and further studied in [5,44] over Laurent polynomial algebras.
Let (P; ρ) be a representation of A n . It follows that (P; ρ| S n ) is an admissible representation of S n since S n ⊂ A n . By Theorem 3.36, we obtain the following result.
It is well-known that H n = Span C {h(r) | r ∈ Z 2n } is a Lie subalgebra of W 2n , with This Lie algebra H n is called the Lie algebra of Hamiltonian vector fields on a 2n-dimensional torus. Note that H n is not a Lie-Rinehart algebra since H n is not an A 2n -module. It is straightforward to see that (A n , H n , [·, ·] W 2n , Id) is a Leibniz pair.
H is the well-known Shen-Larsson functor of type (H n , sp 2n ), introduced by Shen over polynomial algebras [42]. Note that there are no results for H n similar to those in [9,15,26].

4.4.
Actions of monoidal categories for generalized Cartan type. Let A be a commutative associative C-algebra, and let ∆ be a nonzero C-vector space of commuting C-derivations of A. Let us first recall the construction of the generalized Witt algebras from [37]. The tensor product Since A is commutative, this gives rise to a linear transformation α : A∆ → Der C (A). Define a bracket [·, ·] A∆ on A∆ by which gives a Lie algebra structure on A∆. Then α is clearly an action of A∆ on the commutative Lie algebra A. Assume that dim C ∆ < ∞. Then there are ∂ 1 , · · · , ∂ n ∈ ∆ such that A∆ is a free A-module with basis {∂ 1 , · · · , ∂ n } (see [49]). We denote this Lie algebra by W n (A, ∆). Note that (A, W n (A, ∆), [·, ·] A∆ , α) is a Lie-Rinehart algebra. Now we have a generalization of Lemma 4.1.
Lemma 4.14. The linear map H : Proof. It is straightforward but tedious to verify the above formula. We omit the details.
Similar to Corollary 4.2, by Lemma 4.14 and Theorem 3.26 we obtain the following result.  (1) If A = C[x ±1 1 , · · · , x ±1 n ] and ∆ = Span C {x 1 ∂ ∂x 1 , · · · , x n ∂ ∂x n }, W n (A, ∆) is the standard Witt algebra W n and the corresponding F A H is the Shen-Larsson functor of type (W n , gl n ).
1 , · · · , x ±1 n ] and∆ = Span C { ∂ ∂x 1 , · · · , ∂ ∂x n }, W n (A,∆) is also the standard Witt algebra W n . However, the corresponding Shen-Larsson functorF A H is different from the standard F A H except on the category of finite-dimensional gl n -modules. This was pointed out by Liu,Lu and Zhao in [25].
(3) If A is taken to be a polynomial algebra with finitely many variables x i together with some x −1 i , ∆ to be some mixed differential operators w.r.t x i , the Lie algebra W n (A, ∆) was introduced by Xu [48]. The corresponding Shen-Larsson functor F A H was introduced and studied by Zhao [50], generalizing Rao's results in [9].
is the Lie algebra S n of divergence zero vector fields on an n-dimensional torus.
Note that S n (A, ∆) is not a Lie-Rinehart subalgebra since S n (A, ∆) is not an A-module. It is straightforward to see that (A, S n (A, ∆), [·, ·] A∆ , Id) is a Leibniz pair.
It is clear that H| S n (A,∆) is a crossed homomorphism from S n (A, ∆) to sl n ⊗A. Similar to Corollary 4.6, by Lemma 4.14 and Corollary 3.37, we obtain the following result.
is the Lie algebra of Hamiltonian vector fields on a 2n-dimensional torus.
Note that H n (A, ∆) is not a Lie-Rinehart algebra since H n (A, ∆) is not an A-module. It is straightforward to see that (A, H n (A, ∆), [·, ·] A∆ , Id) is a Leibniz pair.
The restriction H| H n (A,∆) of the crossed homomorphism H in Lemma 4.14 is a crossed homomorphism from H n (A, ∆) to sp 2n ⊗ A. Similar to Corollary 4.12, by Lemma 4.14 and Corollary 3.37, we obtain the following result.

Deformation and cohomologies of crossed homomorphisms
In this section, first we give the Maurer-Cartan characterization of crossed homomorphisms of Lie algebras. In particular, we give the differential graded Lie algebra that control deformations of crossed homomorphisms. Then we define the cohomology groups of crossed homomorphisms, which can be applied to study linear deformations of crossed homomorphisms. 5.1. The differential graded Lie algebra controlling deformations.
Definition 5.1. A differential graded Lie algebra (g, [·, ·], d) is a Z-graded vector space g = ⊕ i∈Z g i together with a bilinear bracket [·, ·] : g ⊗ g → g and a linear map d : g → g satisfying the following conditions: for every a, b homogeneous.
• Every a, b, c homogeneous satisfy the Jacobi identity The map d is called the differential of g. We have used the notationā = i if a ∈ g i .
, d) be a differential graded Lie algebra. A degree 1 element θ ∈ g 1 is called a Maurer-Cartan element of g if it satisfies the following Maurer-Cartan equation: (18) dθ Proposition 5.3. ( [27]) Let (g = ⊕ k∈Z g k , [·, ·], d) be a differential graded Lie algebra and let µ ∈ g 1 be a Maurer-Cartan element. Then the map is a differential on the graded Lie algebra (g, [·, ·]). For any v ∈ g 1 , the sum µ + v is a Maurer-Cartan element of the differential graded Lie algebra (g, [·, ·], d) if and only if v is a Maurer-Cartan element of the differential graded Lie algebra (g, [·, ·], d µ ).
Note that for all Proposition 5.4. With the above notations, (C * (g, h), ·, · , d) is a differential graded Lie algebra. Its Maurer-Cartan elements are precisely crossed homomorphisms from g to h with respect to the action ρ.
Proof. In short, the graded Lie algebra (C * (g, h), ·, · ) is obtained via the derived bracket [22,46]. In fact, the Nijenhuis-Richardson bracket [·, ·] NR associated to the direct sum vector space g ⊕ V gives rise to a graded Lie algebra (⊕ k≥0 Hom( is an abelian subalgebra. We denote the Lie brackets [·, ·] g and [·, ·] h by µ g and µ h respectively. Since ρ is an action of the Lie algebra (g, [·, ·] g ). We deduce that µ g + ρ is a semidirect product Lie algebra structure on g ⊕ h. Thus µ g + ρ and µ h are Maurer-Cartan elements of the graded Lie Further, we define the derived bracket on the graded vector space ⊕ k≥0 Hom(∧ k g, h) by which is exactly the bracket given by (20). By [µ h , µ h ] NR = 0, we deduce that (C * (g, h), ·, · ) is a graded Lie algebra. Moreover, by Imρ ⊂ Der(h), we have [µ g +ρ, µ h ] NR = 0. We define a linear map d =: [µ g +ρ, ·] NR on the graded space C * (g ⊕ h, g ⊕ h). We obtain that d is closed on the subspace ⊕ k≥0 Hom(∧ k g, h), and is given by (19).
Finally, for a degree one element H ∈ Hom(g, h), we have Thus, Maurer-Cartan elements are precisely crossed homomorphisms from (g, [·, ·] g ) to (h, [·, ·] h ) with respect to the action ρ. The proof is finished.
Let H : g −→ h be a crossed homomorphism with respect to the action ρ. Since H is a Maurer-Cartan element of the differential graded Lie algebra (C * (g, h), ·, · , d) by Proposition 5.4, it follows from Proposition 5.3 that d H := d + H, · is a graded derivation on the graded Lie algebra (C * (g, h), ·, · ) satisfying d 2 H = 0. Therefore, (C * (g, h), ·, · , d H ) is a differential graded Lie algebra. This differential graded Lie algebra can control deformations of crossed homomorphisms. We have obtained the following result.
Theorem 5.5. Let H : g −→ h be a crossed homomorphism with respect to the action ρ. For a linear map H ′ : g −→ h, then H + H ′ is still a crossed homomorphism from g to h with respect to the action ρ if and only if H ′ is a Maurer-Cartan element of the differential graded Lie algebra (C * (g, h), ·, · , d H ).

5.2.
Cohomologies of crossed homomorphisms. In this subsection, we define cohomologies of a crossed homomorphism, which can be used to study linear deformations in Section 5.3.
Recall that ρ H defined by (4) is a representation of g on h. Let d ρ H : Hom(∧ k g, h) −→ Hom(∧ k+1 g, h) be the corresponding Chevalley-Eilenberg coboundary operator. More precisely, for all f ∈ Hom(∧ k g, h) and x 1 , · · · , x k+1 ∈ g, we have It is obvious that u ∈ h is closed if and only if ρ(x)u + [Hx, u] h = 0 for all x ∈ g, and f ∈ Hom(g, h) is closed if and only if Definition 5.6. Let H : g −→ h be a crossed homomorphism with respect to the action ρ. Denote by C k (g, h) = Hom(∧ k g, h) and (C * (g, h) = ⊕ k≥0 C k (g, h), d ρ H ) the above cochain complex. Denote the set of k-cocycles by Z k (g, h) and the set of k-coboundaries by B k (g, h). Denote by (22) H the k-th cohomology group which will be taken to be the k-th cohomology group for the crossed homomorphism H.
Proof. Indeed, for all x 1 , x 2 , · · · , x k+1 ∈ g and f ∈ Hom(∧ k g, h) , we have At the end of this section, we show that certain homomorphisms between crossed homomorphisms induce homomorphisms between the corresponding cohomology groups. Let H and H be two crossed homomorphisms from g to h with respect to the action ρ, and (φ g , φ h ) a homomorphism from H to H in which φ g is invertible. For all k ≥ 0, define Φ : Theorem 5.8. Let H and H be two crossed homomorphisms from g to h with respect to the action ρ of g on h, and (φ g , φ h ) a homomorphism from H to H in which φ g is invertible. Then the above Φ is a cochain map from the cochain complex (C * (g, h), between corresponding cohomology groups.
Proof. By the fact that (φ g , φ h ) is a homomorphism from H to H, we have which implies that Φ is a cochain map.  (29), it is obvious that a trivial linear deformation gives rise to a Nijenhuis element. The following result is in close analogue to the fact that the differential of a Nijenhuis operator on a Lie algebra generates a trivial linear deformation of the Lie algebra [7], justifying the notion of Nijenhuis elements.
Theorem 5.14. Let H be a crossed homomorphism from g to h with respect to the action ρ. Then for any x ∈ Nij(H), H t := H + tH with H := d ρ H (−Hx) is a linear deformation of the crossed homomorphism H. Moreover, this deformation is trivial.
For t sufficiently small, we see that Id g + tad x and Id h + tρ(x) are Lie algebra isomorphisms. Thus, we have By Lemma 5.15, we deduce that H t is a crossed homomorphism from g to h, for t sufficiently small. Thus, H given by Eq. (31) satisfies the conditions (23) and (24). Therefore, H t is a crossed homomorphism for all t, which means that H given by Eq. (31) generates a deformation. It is straightforward to see that this deformation is trivial.
It is generally not easy to find Nijenhuis elements associated to a crossed homomorphism H from a Lie algebra g to h. Next we give examples on some special Lie algebras where the Nijenhuis elements can be explicitly determined.
Example 5.16. Let g be a 2-step nilpotent Lie algebra, i.e., [g, [g, g]] = 0, and H : g → g a crossed homomorphism with respect to the adjoint action ad of g on g. It is easy to see that (25), (26), (29), (30) hold for any x ∈ g. Therefore Nij(H) = g for any crossed homomorphism H with respect to the adjoint action ad of g on g. For example we can take g to be any Heisenberg algebra. By a straightforward computation, we conclude that H is a crossed homomorphism if and only if a 21 = 0, (1 + a 11 )a 22 = 0. So we have the following two cases to consider.
(i) If a 22 = 0, then we deduce that any H = a 11 a 12 0 0 is a crossed homomorphism. In this case, x = t 1 e 1 + t 2 e 2 is a Nijenhuis element of H if and only if t 2 (t 1 a 11 + t 2 a 12 ) = 0. Then for any t 1 ∈ C, t 1 e 1 is a Nijenhuis element for the crossed homomorphism H = a 11 a 12 0 0 .
(ii) If 1 + a 11 = 0, then we deduce that any H = −1 a 12 0 a 22 is a crossed homomorphism. In this case, x = t 1 e 1 + t 2 e 2 is a Nijenhuis element of H if and only if t 2 (t 2 a 12 − t 1 a 22 − t 1 ) = 0. In particular, e 1 + e 2 is a Nijenhuis element for the crossed homomorphism H = −1 2 0 1 .
Example 5.18. For any crossed homomorphism H from a finite dimensional semisimple Lie algebra g over C to another Lie algebra h with respect to any action ρ, we claim that Nij(H) = 0. Let x ∈ g be a fixed nonzero vector and assume that g 0 = [x, g] is abelian, i.e. (25) holds. We will show that this is impossible.

Conclusion
We introduce the notions of weak representations of Lie-Rinehart algebras and admissible representations of Leibniz pairs. By using crossed homomorphisms between Lie algebras, we construct two actions of the monoidal category of representations of Lie algebras on the category of weak representations of Lie-Rinehart algebras and the category of admissible representations of Leibniz pairs respectively. In particular, the corresponding bifunctors, called the actions of monoidal categories, unify and generalize various constructions of modules over certain Cartan type Lie algebras. New representations of some Lie algebras are also constructed using the actions of monoidal categories. To better understand crossed homomorphisms and the actions of monoidal categories, we also give a systematic study of deformations and cohomologies of crossed homomorphisms.
There are some natural questions worthy to be considered in the future: