Hom-Lie algebras and Hom-Lie groups, integration and differentiation

In this work, we introduce the notion of a Hom-Lie group. In particular, we associate a Hom-Lie algebra to a Hom-Lie group and show that every regular Hom-Lie algebra is integrable. Then, we define a Hom-exponential ($\mathsf{Hexp}$) map from the Hom-Lie algebra of a Hom-Lie group to the Hom-Lie group and discuss the universality of this $\mathsf{Hexp}$ map. We also describe a Hom-Lie group action on a smooth manifold. Subsequently, we give the notion of an adjoint representation of a Hom-Lie group on its Hom-Lie algebra. At last, we integrate the Hom-Lie algebra $(\mathfrak{gl}(V),[\cdot,\cdot]_{\beta},\mathsf{Ad}_{\beta})$, and the derivation Hom-Lie algebra $\mathsf{Der}(\mathfrak{g})$ of a Hom-Lie algebra $(\mathfrak{g},[\cdot,\cdot]_{\mathfrak{g}},\phi_{\mathfrak{g}})$.


Introduction
The notion of a Hom-Lie algebra first appeared in the study of quantum deformations of Witt and Virasoro algebras in [5]. Hom-Lie algebras are generalizations of Lie algebras, where the Jacobi identity is twisted by a linear map, called the Hom-Jacobi identity. It is known that q-deformations of the Witt and the Virasoro algebras have the structure of a Hom-Lie algebra [5,8]. There is a growing interest in Hom-algebraic structures because of their close relationship with the discrete and deformed vector fields, and differential calculus [5,9,10]. In particular, representations and deformations of Hom-Lie algebras were studied in [1,13,15]; a categorical interpretation of Hom-Lie algebras was described in [2]; the categorification of Hom-Lie algebras was given in [16]; geometric and algebraic generalizations of Hom-Lie algebras were given in [3,12,14]; quantization of Hom-Lie algebras was studied in [19]; and the universal enveloping algebra of Hom-Lie algebras was studied in [11,18].
The notion of Hom-groups was introduced as a non-associative analogue of a group in [11]. The authors first gave a new construction of the universal enveloping algebra that is different from the one in [18]. This new construction leads to a Hom-Hopf algebra structure on the universal enveloping algebra of a Hom-Lie algebra. Moreover, one can associate a Hom-group to any Hom-Lie algebra by considering group-like elements in its universal enveloping algebra. Recently, M. Hassanzadeh developed representations and a (co)homology theory for Hom-groups in [6]. He also proved Lagrange's theorem for finite Hom-groups in [7]. The recent developments on Hom-groups (see [6,7,11]) make it natural to study Hom-Lie groups and to explore the relationship between Hom-Lie groups and Hom-Lie algebras.
In this paper, we introduce a (real) Hom-Lie group as a Hom-group (G, ⋄, e Φ , Φ), where the underlying set G is a (real) smooth manifold, the Hom-group operations (such as the product and the inverse) are smooth maps, and the underlying structure map Φ : G → G is a diffeomorphism. We associate a Hom-Lie algebra to a Hom-Lie group by considering the notion of left-invariant sections of the pullback bundle Φ ! T G. We define one-parameter Hom-Lie subgroups of a Hom-Lie group and discuss a Hom-analogue of the exponential map from the Hom-Lie algebra of a Hom-Lie group to the Hom-Lie group. Later on, we consider Hom-Lie group actions on a manifold M with respect to a map ι ∈ Diff(M ) and define an adjoint representation of a Hom-Lie group on its Hom-Lie algebra. Finally, we discuss the integration of the Hom-Lie algebra (gl(V ), [·, ·] β , Ad β ) and the derivation Hom-Lie algebra Der(g) of a Hom-Lie algebra (g, [·, ·] g , φ g ). The paper is organized as follows: In Section 2, we recall some basic definitions and results concerning Hom-Lie algebras and Hom-groups.
In Section 3, we define the notion of a Hom-Lie group with some useful examples. If (G, ⋄, e Φ , Φ) is a Hom-Lie group, then we show that the space of left-invariant sections of the pullback bundle Φ ! T G has a Hom-Lie algebra structure. Consequently, we deduce a Hom-Lie algebra structure on the fibre of pullback bundle Φ ! T G at e Φ . In this way, we associate a Hom-Lie algebra (g ! , [·, ·] g ! , φ g ! ) to the Hom-Lie group (G, ⋄, e Φ , Φ), where g ! := Φ ! T e Φ G. We also show that every regular Hom-Lie algebra is integrable. Next, we define one-parameter Hom-Lie subgroups of a Hom-Lie group (G, ⋄, e Φ , Φ) in terms of a weak homomorphism of Hom-Lie groups. We establish a one-to-one correspondence with g ! . In the end, we define Hom-exponential map Hexp : g ! → G and discuss its properties.
In Section 4, we study Hom-Lie group actions on a smooth manifold M with respect to a diffeomorphism ι ∈ Diff(M ). We define representations of a Hom-Lie group on a vector space V with respect to a map β ∈ GL(V ), which leads to the notion of an adjoint representation of a Hom-Lie group on the associated Hom-Lie algebra.
In the last section, we show that the derivation Hom-Lie algebra Der(g) is the Hom-Lie algebra of the Hom-Lie group of automorphisms of (g, [·, ·] g , φ g ).

Preliminaries
In this section, we first recall definitions of Hom-Lie algebras and Hom-groups.

Hom-Lie algebras
Definition 2.1. A (multiplicative) Hom-Lie algebra is a triple (g, [·, ·] g , φ g ) consisting of a vector space g, a skew-symmetric bilinear map (bracket) [·, ·] g : ∧ 2 g −→ g, and a linear map φ g : g → g preserving the bracket, such that the following Hom-Jacobi identity with respect to φ g is satisfied: A Hom-Lie algebra (g, [·, ·] g , φ g ) is called a regular Hom-Lie algebra if φ g is an invertible map. Lemma 2.2. Let (g, [·, ·] g , φ g ) be a regular Hom-Lie algebra. Then (g, [·, ·] Lie ) is a Lie algebra, where the Lie bracket [·, ·] Lie is given by [x, y] ] g for all x, y ∈ g. In the sequel, we always assume that φ g is an invertible map. That is, in this paper, all the Hom-Lie algebras are assumed to be regular Hom-Lie algebras.

Definition 2.4.
A representation of a Hom-Lie algebra (g, [·, ·] g , φ g ) on a vector space V with respect to β ∈ gl(V ) is a linear map ρ : g → gl(V ) such that for all x, y ∈ g, the following equations are satisfied: For all x ∈ g, let us define a map ad x : g → g by Then ad : g −→ gl(g) is a representation of the Hom-Lie algebra (g, [·, ·] g , φ g ) on g with respect to φ g , which is called the adjoint representation.
Let (ρ, V, β) be a representation of a Hom-Lie algebra (g, [·, ·] g , φ g ) on a vector space V with respect to the map β ∈ gl(V ). Then let us recall from [4] that the cohomology of the Hom-Lie algebra (g, [·, ·] g , φ g ) with coefficients in (ρ, V, β) is the cohomology of the cochain complex C k (g, V ) = Hom(∧ k g, V ) with the coboundary operator d : . The fact that d 2 = 0 is proved in [4]. Denote by Z k (g; ρ) and B k (g; ρ) the sets of k-cocycles and kcoboundaries respectively. We define the k-th cohomology group H k (g; ρ) to be Z k (g; ρ)/B k (g; ρ).

Definition 2.5.
A linear map D : g → g is called a derivation of a Hom-Lie algebra (g, [·, ·] g , φ g ) if the following identity holds: We denote the space of derivations of the Hom-Lie algebra (g, [·, ·] g , φ g ) by Der(g).
Let us denote the space of inner derivations by InnDer(g). It is immediate to see that Z 1 (g, ad) = Der(g); B 1 (g, ad) = InnDer(g).
Here, OutDer(g) denotes the space of outer derivations of the Hom-Lie algebra (g, [·, ·] g , φ g ). Let V be a vector space, and β ∈ GL(V ). Let us define a skew-symmetric bilinear bracket operation We also define a map Ad β : gl(V ) → gl(V ) by With the above notations, we have the following proposition. The Hom-Lie algebra (gl(V ), [·, ·] β , Ad β ) plays an important role in the representation theory of Hom-Lie algebras. See [17] for more details.

Hom-groups
Throughout this paper, we consider regular Hom-groups that is the case when the structure map is invertible. The axioms in the following definition of Hom-group is different from the one in [11,6,7]. However, we show that if the structure map is invertible, then some of the axioms in original definition are redundant and can be obtained from the Hom-associativity condition. Definition 2.8. A (regular) Hom-group is a set G equipped with a product ⋄ : G × G −→ G, a bijective map Φ : G −→ G such that the following axioms are satisfied: (ii) the product is Hom-associative, i.e., (iii) there exists a unique Hom-unit e Φ ∈ G such that (iv) for each x ∈ G, there exists an element x −1 ∈ G satisfying the following condition We denote a Hom-group by (G, ⋄, e Φ , Φ).

Remark 2.9.
Note that the definition of a Hom-group in [11,7] If Φ(e Φ ) = c, we get the following expression: So, c is a Hom-unit. Hence, by the uniqueness of the Hom-unit, The properties (ii) and (iii) easily follows from the Hom-associativity condition (for more details see [7]). We omit details.
Let us recall the Hom-invertibility condition in the definition of a Hom-group (G, Φ) in [11]: for each x ∈ G, there exists a positive integer k such that and the smallest such integer k is called the invertibility index of x ∈ G. In the regular case, it is immediate to see that the Hom-invertibility condition is equivalent to the condition (iv) in Definition 2.8.
Let us observe that for a homomorphism f : It follows by the definition of homomorphism and the identities: Φ(e Φ ) = e Φ , and Ψ(e Ψ ) = e Ψ . Furthermore, any homomorphism of Hom-groups is also a weak homomorphism, however, the converse may not be true.

Hom-Lie groups and Hom-Lie algebras
Let R be the field of real numbers. From here onwards, we consider all manifolds, vector spaces over the field R, and all the linear maps are considered to be R-linear unless otherwise stated.

Hom-Lie groups
Let V be a vector space and β ∈ GL(V ). Then let us define a product ⋄ : Proof. For all A, B ∈ GL(V ), we have Thus, Condition (i) in Definition 2.8 holds. For all A, B, C ∈ GL(V ), it easily follows that which implies that the product ⋄ is Hom-associative. Next, we have Similarly, Therefore, β is the Hom-unit. Finally, we have the following expression (i) the product ⋄ is given by the following equation: Let (G, ⋄, e Φ , Φ) be a Hom-Lie group and T G be the tangent bundle of the manifold G. Let us denote by Φ ! T G, the pullback bundle of the tangent bundle T G along the diffeomorphism Φ : G → G. Then we have the following one-to-one correspondence.

Lemma 3.5.
There is a one-to-one correspondence between the space of sections of the tangent bundle T G (i.e., Γ(T G)) and the space of sections of the pullback bundle Φ ! T G (i.e., Γ(Φ ! T G)).
there is a one-to-one correspondence between the sets Γ(T G) and Γ Φ ! (T G): For X ∈ Γ(T G), we denote the corresponding pullback section by X ! ∈ Γ(Φ ! T G). Through this paper, Thus, the space of sections Γ(Φ ! T G) can be identified with the space of (Φ * , Φ * )-derivations of on C ∞ (G), i.e., Der Φ * ,Φ * (C ∞ (G)). In the following theorem, we define a Hom-Lie algebra structure on the space of sections Γ(Φ ! T G).

The Hom-Lie algebra of a Hom-Lie group
Let (G, ⋄, e Φ , Φ) be a Hom-Lie group. For a ∈ G, let us define a smooth map Then the smooth map l a : G → G is a diffeomorphism (by Definition 3.1).
x satisfies the following equation: Let us denote by Γ L (Φ ! T G), the space of all left-invariant sections of the pullback bundle Φ ! T G. Next, we show that the space Γ L (Φ ! T G) carries a Hom-Lie algebra structure. In fact, we prove that ( is a Hom-Lie subalgebra of the Hom-Lie algebra (Γ(Φ ! T G), [·, ·] Φ , φ).
Proof. First, let us note that by using the Hom-associativity condition of the product ⋄, we get the following equation: By using the left-invariant property (8) of the section x, we have and Thus, by (10)-(12), we deduce that the desired identity (9) holds. Proof. First, let us prove that φ(x) ∈ Γ L (Φ ! T G) for any x ∈ Γ L (Φ ! T G). By (6) and (8), we have for all x, y ∈ Γ L (Φ ! T G), and a ∈ G. This, in turn, implies that φ(x) ∈ Γ L (Φ ! T G). Now we prove that [x, y] Φ ∈ Γ L (Φ ! T G). By (7) and (8), we have the following expressions: for all x, y ∈ Γ L (Φ ! T G) and a ∈ G. Thus, from Lemma 3.8, we have which implies that [x, y] Φ ∈ Γ L (Φ ! T G). The proof is finished.

Lemma 3.11. Let (G, ⋄, e Φ , Φ) be a Hom-Lie group. Let x be a section of Φ ! T G and X be the corresponding section of T G. Then x is left-invariant if and only if X is a left-invariant vector field of the associated
Lie group (G, ·, e Φ ) (by Remark 3.10).
Proof. If x ∈ Γ L (Φ ! T G), then by the definition of a left-invariant section, we get Let X be the corresponding section of T G, i.e., x = X • Φ. Then we obtain the following expression: Thus, X is a left invariant vector field of the Lie group (G, ·, e Φ ). Similarly, if X ∈ Γ(T G) is a left-invariant vector field of the Lie group (G, ·, e Φ ), then we can deduce that the corresponding section x ∈ Γ(Φ ! T G) is left-invariant. We omit the details.
Let (G, ⋄, e Φ , Φ) be a Hom-Lie group. Let us denote by g ! , the fibre of e Φ in the pullback bundle Φ ! T G.
Here, the map Φ * : T G → T G is the differential of the smooth map Φ : G → G.
At the end of this subsection, we show that every regular Hom-Lie algebra is integrable.
Proof. For the Lie algebra (g, [·, ·] Lie ) given in Lemma 2.2, it is easy to see that φ g is a Lie algebra isomorphism of (g, [·, ·] Lie ).
We have a unique simply connected Lie group (G, ·) such that (g, [·, ·] Lie ) is the Lie algebra of (G, ·). Since φ g is a Lie algebra isomorphism of (g, [·, ·] Lie ) and G is a simply connected Lie group, we have a unique isomorphism Φ of the Lie group (G, ·) such that Φ * e = φ g . By Example 3.2, the tuple (G, ⋄, e Φ , Φ) is a Hom-Lie group. Finally, by Lemma 3.12, it follows that
By Theorem 3.16, one-parameter Hom-Lie subgroup of Hom-Lie group (G, ⋄, e Φ , Φ) is in one-to-one correspondence with g ! . We denote by σ ! x (t) the one-parameter Hom-Lie subgroup of the Hom-Lie group (G, ⋄, e Φ , Φ), which corresponds with x.
Then, let us define a map Hexp : g ! → G by Theorem 3.17. Let (G, ⋄, e Φ , Φ) be a Hom-Lie group and (g ! , [·, ·] g ! , Φ g ! ) be the associated Hom-Lie algebra. Then the Hom-Lie bracket [·, ·] g ! can be expressed in terms of the Hexp : g ! → G map as follows: Proof. Let us denote for all x, y ∈ g ! . From Remark 3.10, the triple (G, ·, e Φ ) is a Lie group and g = g ! , where (g, [·, ·] g ) is the Lie algebra of the Lie group (G, ·, e Φ ). Next, we use (17), Theorem 3.16, and Lemma 3.12 to obtain the following expression: The proof is finished.  Then, we have which implies that f : (G, · G , e Φ ) → (H, · H , e Ψ ) is a Lie group homomorphism.
i.e., the following diagram commutes: Proof. By the definition of Hexp, it follows that and .

Actions of Hom-Lie groups and Hom-Lie algebras
Let (G, ⋄ G , e Φ , Φ) be a Hom-Lie group and M be a smooth manifold. Let θ : G × M → M be a smooth map that we denote by θ(a, x) = a ⊙ x, ∀ a ∈ G, x ∈ M.
Conversely, let us assume that L : (G, ⋄ G , e Φ , Φ) → (Diff(M ), ⋄, ι, Ad ι ) is a weak homomorphism of Hom-Lie groups. Then, it follows that and Therefore, we get the following identity: By (19) and (20), we deduce that θ : G × M → M is an action of the Hom-Lie group (G, ⋄ G , e Φ , Φ) on the smooth manifold M with respect to ι ∈ Diff(M ).
If (G, ⋄ G , e Φ , Φ) is a Hom-Lie group, then let us define a map Ad : Proof. For all x ∈ G, we have and Let us denote a ⊙ b := Ad(a, b), then we get the following expression: which implies that Thus, the map Ad : G × G → G gives an action of the Hom-Lie group (G, ⋄ G , e Φ , Φ) on the underlying manifold G with respect to the map Φ ∈ Diff(G) .
is called a representation of the Hom-Lie group (G, ⋄ G , e Φ , Φ) on the vector space V with respect to β ∈ GL(V ).
Let (g ! , [·, ·] ! , φ g ! ) be the Hom-Lie algebra of a Hom-Lie group (G, ⋄ G , e Φ , Φ). From Lemma 4.3, Ad gives an action of the Hom-Lie group (G, ⋄ G , e Φ , Φ) on G with respect to the map Φ. Now, let us denote Ad a := Ad(a, ·) for any a ∈ G. Then we observe that for all a ∈ G, the map Ad a : G → G is a weak isomorphism of Hom-Lie groups. Let us denote by ( Ad a ) ⊲ : g ! → g ! , the weak isomorphism of Hom-Lie algebra (g ! , [·, ·] ! , φ g ! ), obtained by Theorem 3.21. Subsequently, we have the following lemma. is a weak homomorphism from Hom-Lie group (G, ⋄ G , e Φ , Φ) to the Hom-Lie group (GL(g ! ), ⋄, φ g ! , Ad φ g ! ). Proof. For all a, b ∈ G and x ∈ g ! , it follows that , which implies that Ad is a Hom-Lie group (weak) homomorphism from G to GL(g ! ).
Let (G, θ, M, ι) be an action of the Hom-Lie group (G, ⋄ G , e Φ , Φ). By Theorem 3.16, for any x ∈ g ! , we have a unique map σ ! x : R → G. Now, we consider the curve γ : R → M given by Then we get a section of the pullback bundle ι * T M as follows: Thus, we have a map κ : g ! → Γ(ι * T M ) given by κ(x) = x for all x ∈ g ! . We denote κ(x)(m) simply by x • m for any x ∈ g ! , m ∈ M .
Proof. The proof of the proposition follows immediately from Theorem 3.22 and Lemma 4.5.

Integration of the Hom-Lie algebra (gl(V ), [·, ·] β , Ad β )
Let V be a vector space and β ∈ GL(V ). Let us define a map e β : gl(V ) → GL(V ) by Let us denote by e : gl(V ) → GL(V ), the usual exponential map. Then the map e β : gl(V ) → GL(V ) can be written as follows: The map e β : gl(V ) → GL(V ) gives rise to a one-parameter Hom-Lie subgroup of the Hom-Lie group (GL(V ), ⋄, Ad β ). More precisely, for any A ∈ gl(V ), let us define a map σ A : R → GL(V ) by σ A (t) = e tA β , ∀ t ∈ R.
Then we have the following lemma. which implies that (24) hold.

Integration of the derivation Hom-Lie algebra
Moreover,