International Journal of Algebra, Vol. 8, 2014, no. 15, 713 - 727 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2014.4881

Laplacian of Hom-Lie Quasi-bialgebras

Ibrahima BAKAYOKO

D´epartement de Math´ematiques UJNK/Centre Universitaire de N’Z´er´ekor´e, BP : 50, N’Z´er´ekor´e,Guinea

Copyright c 2014 Ibrahima BAKAYOKO. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribu- tion, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we introduce strictly exact Hom-Lie quasi-bialgebras and Hom-Lie quasi-bialgebras morphism. We show that, twisting strictly exact Lie quasi-bialgebras morphism one obtain strictly exact Hom-Lie quasi-bialgebras morphisms. Then we show that Hom-Lie algebras mor- phism extends to strictly exact Hom-Lie quasi-bialgebras morphism. We also establish the laplacian of Hom-Lie quasi-bialgebras and we study its properties. More precisely, we show that the laplacian of any Hom- Lie quasi-bialgebra is an α2-derivation. Moreover, we construct the laplacian of exact Hom-Lie quasi-bialgebras. Finally, we establish the relation between the laplacian of a Lie quasi-bialgebra and the Laplacian of the corresponding twisted Lie quasi-Lie bialgebra.

Mathematics Subject Classification: 17B62, 17A30

Keywords: Laplacian, α-derivation, Hom-Lie quasi-bialgebras, morphism, Hom-Schouten algebraic bracket, cohomology of Hom-Lie algebras

1 Introduction

Hom-Lie quasi-bialgebras are introduced in [8] as a twist generalization of Lie quasi-bialgebras [17], [18], [14]. More precisely, a Hom-Lie quasi-bialgebra is a Hom-Lie algebra [11], [2], [5] endowed with a Hom-Lie coalgebra structure [3], [8], both structures being connected by a compatibility condition. In fact 714 Ibrahima BAKAYOKO this condition means that the Hom-Lie coalgebra bracket is a 1-cocycle in the cohomology of Hom-Lie algebras [1], [8]. In other point of view, Hom-Lie quasi-bialgebras can be considered as a generalization of Hom-Lie bialgebras [6], [7]. Lie quasi-bialgebras [17] arise from the works of Drinfeld’s as classical limi- te of quasi-Hopf algebras. They have been investigated by many authors. In [18] the big bracket were introduced on the exterior algebra of the direct sum of a vector space and its dual to study all the various generalizations of Lie bialgebras. Geometrical objects, quasi-Poisson Lie groups, corresponding to Lie quasi-bialgebras are given as a generalization of the Poisson Lie groups. Moreover, the third Lie theorem for Lie quasi-bialgebras is also established i.e. the one-to-one correspondence between Lie quasi-bialgebras and simply connected quasi-Poisson Lie groups. A class of Lie bialgebras and Lie quasi- bialgebras related to a triangular decomposition of the underlying Lie algebras is discussed in [15]. While Lie algebras, Lie coalgebras, Lie bialgebras and Lie quasi-bialgebras are presented in [16] as solutions of Maurer-Cartan equa- tions on the corresponding governing differential graded Lie algebras using the big bracket construction of Kosmann-Schwarzbach [18]. These consid- erations allow the author to introduce L∞-(quasi)bialgebras (strongly homo- topy Lie (quasi)bialgebras) which are generalizations of L∞-algebras structure. An L∞-version of a Manin (quasi)triple and a correspondence theorem with L∞(quasi)-bialgebras are exposed. The representation of Lie quasi-bialgebras is studied in [9]. While the properties of the Laplacian of Lie quasi-bialgebras are studied in [13]. In the same paper, they establish the expression of the laplacian of Lie quasi-bialgebras by means of basis vectors and dual basis of the underlying vector space. In [8], it is shown that one can obtain a quasi- Hom-Lie bialgebra by twisting the Lie quasi-bialgebra structure. The purpose of this paper is to extend the notion of Lie quasi-bialgebras morphisms to that of Hom-Lie quasi-bialgebras morphisms, and to construct the laplacian of Hom-Lie quasi-bialgebras and to study some of its properties. This laplacian generalises that introduced by Kostant [4], J. H. Lu [10] and developped by Bangoura and Bakayoko [9]. In the second section, we introduce Hom-Lie quasi-bialgebras morphisms and prove that Lie quasi-bialgebras morphisms [14] become Hom-Lie quasi- bialgebras morphisms. Then, we show that Hom-Lie algebras morphisms ex- tend to strictly exact Hom-Lie quasi-bialgebras morphisms. In the third section, we associate to any Hom-Lie quasi-bialgebra some operators which are α-derivations. Then we give their expressions by means of the basis vectors and dual basis of the underlying vector space. The section four is devoted to definition of the laplacian of Hom-Lie quasi- bialgebras and its properties. In particular, we establish the Laplacian of axact Hom-Lie quasi-bialgebras. Then we put in relation the Laplacian of a Lie quasi- Laplacian of Hom-Lie quasi-bialgebras 715 bialgebra and the Laplacian of the corresponding Hom-Lie quasi-bialgebra. We end this section by fixing some conventions and notations. All the vector spaces are assumed to be finite dimensional. So, if (G, µ, α) is a Hom- Lie algebra and G∗ its dual vector space, the duality bracket between ΛG and ΛG∗ extending that between G and G∗ is defined by

n < ξ1 ∧ ξ2 ∧ ... ∧ ξm, x1 ∧ x2 ∧ ... ∧ xn >= δmdet(< ξi, xj >),

∗ ξi ∈ G , i = 1, ..., m, xj ∈ G, j = 1, ..., n.

In the sequel we use the following notations and abbreviations :

• α(x1 ∧ x2 ∧ ... ∧ xn−1 ∧ xn) = α(x1) ∧ α(x2) ∧ ... ∧ α(xn−1) ∧ α(xn) and

• α(x1 ∧ x2 ∧ ... ∧ xci ∧ ... ∧ xn−1 ∧ xn) = α(x1) ∧ α(x2) ∧ ... ∧ αd(xi) ∧ ... ∧ α(xn−1) ∧ α(xn), where xci means that we omit the i-th term of the list.

2 Morphism of Hom-lie quasi-bialgebras

In this section, we show that Lie quasi-bialgebras morphisms [14] turn to Hom- Lie quasi-bialgebras morphisms and Hom-Lie algebras morphisms extend to strictly exact Hom-Lie quasi-bialgebras morphisms. Let (G, µ, α) be a Hom-Lie algebra on a field K, supposed equal to R or C, where µ :Λ2G → G is the Hom-Lie bracket on G. Definition 2.1 The Hom-Schouten algebraic structure is a graded Hom-Lie µ,α L p+1 algebra bracket [·, ·] , on the exterior algebra ΛG = p≥−1 Λ G which : (i) vanish if one of the arguments is in K, (ii) extends the bracket µ, i.e. for any x, y ∈ G,

[x, y]µ,α = µ(x, y),

(iii) satisfies the following rule on the degree :

[X,Y ]µ,α ∈ Λp+q+1G, if X ∈ Λp+1G and Y ∈ Λq+1G, (iv) satisfies the graded skew-commutativity, i.e.

[X,Y ]µ,α = −(−1)pq[Y,X]µ,α, if X ∈ Λp+1G and Y ∈ Λq+1G, (v) satisfies the graded Hom-Leibniz rule, i.e.

[X,Y ∧ Z]µ,α = [X,Y ]µ,α ∧ α(Z) + (−1)p(q+1)α(Y ) ∧ [X,Z]µ,α, 716 Ibrahima BAKAYOKO if X ∈ Λp+1G, Y ∈ Λq+1G and Z ∈ ΛG, and (vi) satisfies the graded Hom-Jacobi identity, i.e.

(−1)pr[α(X), [Y,Z]µ,α]µ,α+(−1)pq[α(Y ), [Z,X]µ,α]µ,α+(−1)qr[α(Z), [X,Y ]µ,α]µ,α = 0, if X ∈ Λp+1G, Y ∈ Λq+1G and Z ∈ Λr+1G.

Definition 2.2 ([8]) A Hom-Lie quasi-bialgebra is a quintuple (G, µ, γ, φ, α) where (G, µ, α) is a Hom-Lie algebra, γ : G −→ Λ2G is a 1-cocycle and φ ∈ Λ3G such that :

µ,α Alt(γ ⊗ α)γ(x) = adx φ, (1) Alt(γ ⊗ α ⊗ α)φ = 0 (2) where Alt(x ⊗ y ⊗ z) = x ⊗ y ⊗ z + y ⊗ z ⊗ x + z ⊗ x ⊗ y and γ is a 1-cocycle means that γ(µ(x, y)) = x · γ(y) − y · γ(x) for any x, y ∈ G. If in addition α commutes with µ and γ, we say that the Hom-Lie quasi- bialgebra (G, µ, γ, φ, α) is multiplicative.

Remark. When α = IdG, we recover the definition of Lie quasi-bialgebras [17].

Example 2.3 ([8]) Let G be a diagonal matrix Lie group and let G be the Lie algebra of G. Let γ : G −→ Λ2G be a linear map and φ ∈ Λ3G such that (G, µ, γ, φ) be a Lie quasi-bialgebra. According to [5] Example 2.14, −1 the map Adx : G −→ G, g 7→ xgx , x ∈ G is a morphism of Lie al- gebra. Moreover, suppose that Adx commutes with γ. Then the quintuple

(G, µAdx , γAdx , φAdx , Adx) is a Hom-Lie quasi-bialgebra.

Now, we define morphisms for Hom-Lie quasi-bialgebras.

Definition 2.4 Let (G, µ, γ, φ, α) and (G0, µ0, γ0, φ0, α0) be two Hom-Lie quasi- bialgebras and f : G → G0 be a linear map. We say that f is a Hom-Lie quasi-bialgebras morphism if :

1. f :(G, µ, α) → (G0, µ0, α0) is a Hom-Lie algebras morphism,

2. f :(G, γ, α) → (G0, γ0, α0) is a Hom-Lie coalgebras morphism,

3. φ0 = f ⊗3(φ).

Remark. When α = IdG, we recover the definition of Lie quasi-bialgebras morphism [14]. Laplacian of Hom-Lie quasi-bialgebras 717

Lemma 2.5 ([8]) If (G, µ, γ, φ) is a Lie quasi-bialgebra and α is a mor- µ phism with respect to µ and γ, and which commutes with adx for any x ∈ G, then ⊗3 (G, µα = α ◦ µ, γα = γ ◦ α, φα = α φ) is a Hom-Lie quasi-bialgebra.

In the proposition below, we show that Lie quasi-bialgebras morphisms extend to Hom-Lie quasi-bialgebras morphisms.

Proposition 2.6 Let (G, µ, γ, φ) and (G0, µ0, γ0, φ0) be two Lie quasi-bialgebras 0 0 0 0 0 and Let (G, µα, γα, φα, α) and (G , µα, γα, φα, α ) be the associated Hom-Lie quasi-bialgebras as in Lemma 2.5. If f :(G, µ, γ, φ) → (G, µ0, γ0, φ0) is a Lie quasi-bialgebras morphism satisfying f ◦ α = α0 ◦ f, then

0 0 0 0 0 f :(G, µα, γα, φα, α) → (G , µα0 , γα0 , φα0 , α ) is a Hom-Lie quasi-bialgebras morphism.

Proof. We must prove conditions 1) − 3) in Definition 2.4. We have,

i) for any x, y ∈ G,

0 f(µα(x, y)) = f(µ(α(x), α(y))) = f(α(µ(x, y))) = α f(µ(x, y)) 0 0 0 = α (µ (f(x), f(y))) = µα0 (f(x), f(y)),

ii) for any x ∈ G,

0 0 0 0 γα0 (f(x)) = γ (α f(x)) = γ (f(α(x))) = (f ⊗ f)γ(α(x)) = (f ⊗ f)γα(x),

iii)

0 0⊗3 0 0⊗3 ⊗3 ⊗3 ⊗3 ⊗3 φα0 = α φ = α f φ = f α φ = f φα.

Therefore we conclude that f is a Hom-Lie quasi-bialgebras morphism.

Definition 2.7 A Hom-Lie quasi-bialgebra (G, µ, γ, φ, α) is said to be strictly exact if there exists a ∈ Λ2G such that : 1 γ(x) = adµ,αa, φ = − [a, a]µ,α, α⊗2(a) = a. x 2 Moreover, if α commutes with µ and γ, we say that the strictly exact Hom-Lie quasi-bialgebra (G, µ, γ, φ, α) is multiplicative. 718 Ibrahima BAKAYOKO

µ,α 1 µ,α 0 0 µ0,α0 0 1 0 0 µ0,α0 0 Theorem 2.8 Let (G, µ, ad a, − 2 [a, a] , α) and (G , µ , ad a , − 2 [a , a ] , α ) be two strictly exact Hom-Lie quasi-bialgebras, and f :(G, µ, α) → (G0, µ0, α0) be a Hom-Lie algebras morphism such that f ⊗2(a) = a0. Then f is a (strictly exact) Hom-Lie quasi-bialgebras morphism.

Proof. f being a Hom-Lie algebras morphism, it remains to prove conditions 2 and 3 in Definition 2.4.

i) For any x ∈ G,

µ,α (f ⊗ f)γ(x) = (f ⊗ f)(adx (a)) = (f ⊗ f)(µ(x, a1) ⊗ α(a2) + α(a1) ⊗ µ(x, a2)) 0 0 = µ (f(x), f(a1)) ⊗ f(α(a2)) + f(α(a1)) ⊗ µ (f(x), f(a2)) µ0 0 0 µ0 = adf(x)f(a1) ⊗ α f(a2) + α f(a1) ⊗ adf(x)f(a2) µ0,α0 = adf(x) (f(a1) ⊗ f(a2)) µ0,α0 0 0 = adf(x) (a1 ⊗ a2) = γ0(f(x)).

ii)

⊗3 ⊗3 µ,α ⊗3 µ,α −2f φ = f [a, a] = f ([a11 ⊗ a12, a21 ⊗ a22] ) 0 = f(α(a11)) ⊗ µ (f(a12), f(a21)) ⊗ f(α(a12)) 0 +f(α(a11)) ⊗ f(α(a21)) ⊗ µ (f(a12), f(a22)) 0 +µ (f(a11), f(a21)) ⊗ f(α(a22)) ⊗ f(α(a12)) 0 +f(α(a21)) ⊗ µ (f(a11), f(a22)) ⊗ f(α(a12)) 0 0 0 = α f(a11) ⊗ µ (f(a12), f(a21)) ⊗ α f(a12) 0 0 0 +α f(a11) ⊗ α f(a21) ⊗ µ (f(a12), f(a22)) 0 0 0 +µ (f(a11), f(a21)) ⊗ α f(a22) ⊗ α f(a12) 0 0 0 +α f(a21) ⊗ µ (f(a11), f(a22)) ⊗ α f(a12) 0 0 0 0 µ0,α0 = [a11 ⊗ a12, a21 ⊗ a22] = −2φ0.

Hence, f ⊗3φ = φ0. Thus the conclusion hold.

µ 1 µ Corollary 2.9 Let (G, µ, ad a, − 2 [a, a] ) be a strictly exact Lie quasi-bialgebra µ and α : G → G be a Lie algebras endomorphism which commutes with adx, for µα,α 1 µα,α any x ∈ G. Then (G, µα, ad a, − 2 [a, a] , α) is a strictly exact Hom-Lie quasi-bialgebra.

Proof. It follows from Lemma 2.5 and Theorem 2.8. Laplacian of Hom-Lie quasi-bialgebras 719

µ 1 µ 0 0 µ0 0 1 0 0 µ0 Corollary 2.10 Let (G, µ, ad a, − 2 [a, a] ) and (G , µ , ad a , − 2 [a , a ] ) be µα,α 1 µα,α two strictly exact Lie quasi-bialgebras, and (G, µα, ad a, − 2 [a, a] , α) and µ0 ,α0 1 µ0 ,α0 0 0 α0 0 0 0 α0 0 (G , µα0 , ad a , − 2 [a , a ] , α ) the associated Hom-Lie quasi-bialgebras as in Corollary 2.9 and f :(G, µ, α) → (G0, µ0, α0) be a Hom-Lie algebras morphism such that f ⊗2(a) = a0. Then

1 0 0 1 0 0 µα,α µα,α 0 0 µ 0 ,α 0 0 0 µ 0 ,α 0 f :(G, µ , ad a, − [a, a] , α) → (G , µ 0 , ad α a , − [a , a ] α , α ) α 2 α 2 is a Hom-Lie quasi-bialgebras morphism. Proof. It follows from Theorem 2.8 and Corollary 2.9. Proposition 2.11 Let (G, µ, γ, φ, α) be a Hom-Lie quasi-bialgebra. Then so are (G, kµ, kγ, kφ, α) and (G, kµop, kγ, kφop, α), where k ∈ R∗, µop = µ ◦ τ and φop = z ∧ y ∧ x when φ = x ∧ y ∧ z.

3 Derivations on Hom-Lie quasi-bialgebras

In this section, we associate to any Hom-Lie quasi-bialgebras some α-derivations. To simplify the typography, we denote indifferently α and its transpose.

The bracket µ of any Hom-Lie quasi-bialgebra (G, µ, γ, φ, α) allows to define Chevalley-Eilenberg operators [1], [5] (with trivial coefficients) :

k ∗ k+1 ∗ dµ,α :Λ G → Λ G ,

k k−1 ∂µ,α :Λ G → Λ G, where

X i+j (dµ,αξ)(x1∧...∧xk+1) = (−1) ξ(µ(xi, xj)∧α(x1)∧...∧αd(xi)∧...∧αd(xj)∧...∧α(xk+1)), i

X i+j ∂µ,α(x1∧...∧xk+1) = (−1) µ(xi, xj)∧α(x1)∧...∧αd(xi)∧...∧αd(xj)∧...∧α(xk+1), i

k k+1 dγ,α :Λ G → Λ G,

k ∗ k−1 ∗ ∂γ,α :Λ G → Λ G , where

X i+j (dγ,αX)(ξ1∧...∧ξk+1) = (−1) X(γ(ξi, ξj)∧α(ξ1)∧...∧αd(ξi)∧...∧αd(ξj)∧...∧α(ξk+1)), i

X i+j ∂γ,α(ξ1∧...∧ξk+1) = (−1) γ(ξi, ξj)∧α(ξ1)∧...∧αd(ξi)∧...∧αd(ξj)∧...∧α(ξk+1), i

Remark.

i) By using the definitions of operators ∂µ,α and dγ,α, one can show that they commute with α in the following sense

⊗k ⊗(k−1) (∂µ,α ◦ α )(X) = (α ◦ ∂µ,α)(X), (3) ⊗(k−1) ⊗k (dγ,α ◦ α )(X) = (α ◦ dγ,α)(X), (4) for any homogeneous element X ∈ ΛkG.

2 ii) ∂µ,α = 0 if and only if µ is a Hom-Lie algebra structure. Proposition 3.1 Let (G, µ, γ, φ, α) be a Hom-Lie quasi-bialgebras. Then

|A| dµ,α(A ∧ B) = (dµ,αA) ∧ α(B) + (−1) α(A) ∧ (dµ,αB),

|X| |X| µ,α ∂µ,α(X ∧ Y ) = (∂µ,αX) ∧ α(Y ) + (−1) α(X) ∧ (∂µ,αY ) + (−1) [X,Y ] , µ,α µ,α |X|−1 µ,α ∂µ,α[X,Y ] = [∂µ,αX, α(Y )] + (−1) [α(X), ∂µ,αY ] , ∗ for all X,Y ∈ ΛG et A, B ∈ ΛG . The operators dγ and ∂γ satisfy similar relations. Proof. They are proved by recurrence on the degrees of X, Y , A and B.

Proposition 3.2 Let (G, µ, γ, φ, α) be a Hom-Lie quasi-bialgebra. Then for any A, B ∈ ΛG∗ and X,Y ∈ ΛG, we have :

γ,α γ,α |A|−1 γ,α dµ,α([A, B] ) = [dµ,αA, α(B)] + (−1) [α(A), dµ,αB]

µ,α µ,α |X|−1 µ,α dγ,α([X,Y ] ) = [dγ,αX, α(Y )] + (−1) [α(X), dγ,αY ] Proof. If |X| = |Y | = 1 and |A| = |B| = 1, the two identities reduces to the 1-cocycle condition in the definition of Hom-Lie quasi-bialgebras. The general cases case are proved by recurrence on the degrees of X,Y,A and B.

We have the following proposition.

n Proposition 3.3 Let (G, µ, γ, φ, α) be a Hom-Lie quasi-bialgebra, {ei}i=1 i n a basis of G, and {ξ }i=1 its dual basis. The operators ∂µ,α and dγ,α can be written by means of the basis vectors as follows : n n 1 X µ,α 1 X γ∗,α ∂ = − ad ι i et d = ε ad . µ,α ei ξ γ,α ei ξi 2 i=1 2 i=1 Laplacian of Hom-Lie quasi-bialgebras 721

Proof. These relations are proved, analogouesly to that of Koszul in [12], Chapter II, by using the definitions of the operators ∂µ,α and dγ,α.

Proposition 3.4 Let (G, µ, γ, φ, α) be a multiplicative Hom-lie quasi-bialgebra. For any x ∈ G, ξ ∈ G∗, we have :

γ∗,α γ∗,α αadα(ξ) x = adξ α(x). (5)

Proof. Indeed, for any x ∈ G, ξ, η ∈ G∗,

γ∗,α γ∗,α γ,α hη, αadα(ξ) xi = hα(η), adα(ξ) xi = −hadα(ξ)α(η), xi

= −h[α(ξ), α(η)]G∗ , xi = −hα[ξ, η]G∗ , xi γ,α = −h[ξ, η]G∗ , α(x)i = −hadξ η, α(x)i γ∗,α = hη, adξ α(x)i.

4 Laplacian of Hom-Lie quasi-bialgebras

In this section we study properties of the laplacian of the Hom-Lie quasi- bialgebras.

Definition 4.1 Let (G, µ, γ, φ, α) be a Hom-Lie quasi-bialgebra. The operator k k L = ∂µ,αdγ,α + dγ,α∂µ,α :Λ G → Λ G is called the laplacian of the Hom-Lie quasi-bialgebra G.

We have the following result.

Theorem 4.2 Let (G, µ, γ, φ, α) be a Hom-Lie quasi-bialgebra with lapla- cian L.

1. The laplacian L is an α2-derivation of (ΛG, ∧), i.e.

L(X ∧ Y ) = L(X) ∧ α2(Y ) + α2(X) ∧ L(Y ), ∀X,Y ∈ ΛG. (6)

2. The laplacian L is an α2-derivation of (ΛG, [., .]µ,α), i.e.

L([X,Y ]µ,α) = [L(X), α2(Y )]µ,α + [α2(X),L(Y )]µ,α, ∀X,Y ∈ ΛG. (7)

Proof. The proofs follow from Definition 4.1 and Properties 3.1 and 3.2. 722 Ibrahima BAKAYOKO

1.

L(X ∧ Y ) = (∂µ,αdγ,α)(X ∧ Y ) + (dγ,α∂µ,α)(X ∧ Y ) =

|X| = ∂µ,α(dγ,αX ∧ α(Y ) + (−1) α(X) ∧ dγ,αY ) + dγ,α(∂µ,αX ∧ α(Y )

|X| |X| µ,α +(−1) α(X) ∧ ∂µ,αY + (−1) [X,Y ] )

2 |X|+1 = (∂µ,αdγ,α)(X) ∧ α (Y ) + (−1) α(dγ,αX) ∧ (∂µ,αα(Y ))

|X|+1 µ,α +(−1) [dγ,αX, α(Y )]

|X| |X| 2 +(−1) (∂µ,αα(X) ∧ α(dγ,αY ) + (−1) α (X) ∧ (∂µ,αdγ,α)(Y )

|X| µ,α 2 +(−1) [α(X), dγ,αY ] ) + (dγ,α∂µ,α)(X) ∧ α (Y )

|X|−1 |X| +(−1) α(∂µ,αX) ∧ (dγ,αY ) + (−1) (dγ,αα(X)) ∧ α(∂µ,αY )

|X| 2 |X| µ,α +(−1) α (X) ∧ (dγ,α∂µ,α)(Y )) + (−1) ([dγ,αX, α(Y )]

|X|−1 µ,α +(−1) [α(X), dγ,αY ] )

= L(X) ∧ α2(Y ) + α2(X) ∧ L(Y ).

2.

µ,α µ,α µ,α L([X,Y ] ) = (∂µ,αdγ,α)([X,Y ] ) + (dγ,α∂µ,α)([X,Y ] ) =

µ,α |X|−1 µ,α = ∂µ,α([dγ,αX, α(Y )] + (−1) [α(X), dγ,αY ] )

µ,α |X|−1 µ,α +dγ,α([∂µ,αX, α(Y )] + (−1) [α(X), ∂µ,αY ] )

2 µ,α |X| µ,α = [(∂µ,αdγ,α)(X), α (Y )] + (−1) [α(dγ,αX), ∂µ,αα(Y )]

|X|−1 µ,α +(−1) ([∂µ,αα(X), α(dγ,αY )]

|X|−1 2 µ,α +(−1) [α (X), (∂µ,αdγ,α)(Y )] ) Laplacian of Hom-Lie quasi-bialgebras 723

2 µ,α |X| µ,α +[(dγ,α∂µ,α)(X), α (Y )] + (−1) [α(∂µ,αX), dγ,α(α(Y ))]

|X|−1 µ,α +(−1) ([dγ,αα(X), α(∂µ,αY )]

|X|−1 2 µ,α +(−1) [α (X), (dγ,α∂µ,α)(Y )] )

= [L(X), α2(Y )]µ,α + [α2(X),L(Y )]µ,α.

∗ 2 ∗ Corollary 4.3 L = dµ,α∂γ,α + ∂γ,αdµ,α is an α -derivation of (ΛG , ∧) and an α2-derivation of (ΛG∗, [·, ·]γ,α).

In the following proposition, we establish a connection between the lapla- cian of the Lie quasi-bialgebra (G, µ, γ, φ) and the Laplacian of the Hom-Lie quasi-bialgebra (G, α ◦ µ, γ ◦ α, φα, α). It is traigthforward to show that the operators ∂µ, dγ on the Lie quasi-bialgebra and the operators ∂µ,α, dγ,α on the Hom-Lie quasi-bialgebra are linked by the relations : ∂µα,α = α ◦ ∂µ et dγα,α = α ◦ dγ. Moreover, the operators ∂µ and dγ commute with α.

Proposition 4.4 In the assumption of Lemma 2.5, suppose that L is the laplacian of the Lie quasi-bialgebra (G, µ, γ, φ) and Lα is the Laplacian of the Hom-Lie quasi-bialgebra (G, α ◦ µ, γ ◦ α, α). Then,

2 Lα = α ◦ L.

Proof. For any homogeneous element X ∈ ΛkG, we have :

Lα(X) = (∂µα,αdγα,α + dγα,α∂µα,α)(X) ⊗(k+1) ⊗(k−1) = (∂µα,α ◦ α ◦ dγ + dγα,α ◦ α ◦ ∂µ)(X) ⊗k ⊗k = (α ◦ ∂µα,α ◦ dγ + α ◦ dγα,α ◦ ∂µ)(X) ⊗k ⊗k = (α ◦ ∂µα,α)(dγ(X)) + (α ◦ dγα,α)(∂µ(X)) ⊗k ⊗k ⊗k ⊗k = (α ◦ α ◦ ∂µ)(dγ(X)) + (α ◦ α ◦ dγ)(∂µ(X)) 2 ⊗k = ((α ) ◦ (∂µ ◦ dγ + dγ ◦ ∂µ))(X) = ((α2)⊗k ◦ L)(X).

Proposition 4.5 Let (G, µ, γ, φ, α) be a Hom-Lie quasi-bialgebra.

i) The operators L and ∂µ,α commute, i.e.

[L, ∂µ,α] = 0. 724 Ibrahima BAKAYOKO

ii) KerL is stable under the operator ∂µ,α, i.e.

∀X ∈ KerL ⇒ ∂µ,α(X) ∈ KerL.

iii) ImL ⊆ Im∂µ,α + Imdγ,α.

iv) Ker∂µ,α ∩ Kerdγ,α ⊆ KerL.

Proof.

2 1. The proof follows from the Definition 4.1 and the fact that ∂µ,α = 0.

2. By (i), for any X ∈ KerL, we have

L∂µ,α(X) = ∂µ,αL(X) = 0, i.e. ∂µ,α(X) ∈ KerL.

3. According to the Definition 4.1, for any X ∈ ΛG, we have,

L(X) = (∂µ,αdγ,α+dγ,α∂µ,α)(X) = ∂µ,α(dγ,α(X))+dγ,α(∂µ,α(X)) ∈ Im∂µ,α+Imdγα.

Thus, ImL ⊆ Im∂µ,α + Imdγ,α.

4. For any X ∈ Ker∂µ,α ∩ Kerdγ,α one has ∂µ,α(X) = dγ,α(X) = 0 i.e. L(X) = 0 or X ∈ KerL.

In the rest of this section, we shall discuss of the laplacian of exact Hom-Lie quasi-bialgebras.

Definition 4.6 ([8]) A Hom-Lie quasi-bialgebra (G, µ, γ, φ, α) is said to be exact if there exists a ∈ Λ2G such that

α⊗2(a) = a and γ = adµ,α(a)

The following lemma allows to prove Theorem 4.8.

Lemma 4.7 If (G, µ, a, φ, α) is an exact Hom-Lie quasi-bialgebra. Then

dγ,α = εa ◦ ∂µ,α − ∂µ,α ◦ εa − εa0 ◦ α,

n X 0 where a0 = −∂µ,αa = µ(ai, ai). i=1 Laplacian of Hom-Lie quasi-bialgebras 725

Proof. By Proposition 3.1, µ,α ∂µ,α(a ∧ X) = (∂µ,αa) ∧ α(X) + α(a) ∧ (∂µ,αX) + [a, X] µ,α = (∂µ,αa) ∧ α(X) + a ∧ (∂µ,αX) + [a, X] . It follows that µ,α [a, X] = ∂µ,α(a ∧ X) − (∂µ,αa) ∧ α(X) − a ∧ (∂µ,αX)

= ∂µ,α(a ∧ X) + a0 ∧ α(X) − a ∧ (∂µ,αX)

= (∂µ,αεa)(X) + (εa0 ◦ α)(X) − (εa ◦ ∂µ,α)(X).

Since dγ,α is an α-derivation of ΛG, it suffises to find its expression on G. Then, µ,α µ,α µ,α dγ,α(x) = γ(x) = adx a = [x, a] = −[a, x] , which leads to

dγ,α(x) = −(∂µ,αεa)(x) − (εa0 ◦ α)(x) + (εa ◦ ∂µ,α)(x) i.e.

dγ,α = εa ◦ ∂µ,α − ∂µ,α ◦ εa − εa0 ◦ α. Theorem 4.8 Let (G, µ, a, φ, α) be an exact Hom-Lie quasi-bialgebra de- n X 0 2 fined by a = ai ∧ ai ∈ Λ G. Then, we have : i=1 µ,α L = ada0 ◦ α + ε(α(a0)−a0) ◦ ∂µ,α ◦ α n X 0 where a0 = −∂µa = µ(ai, ai). i=1 Proof. In fact, from the definition of L and using the Lemma 4.7, we have

L = ∂µ,αdγ,α + dγ,α∂µ,α

= ∂µ,α(εa ◦ ∂µ,α − ∂µ,α ◦ εa − εa0 ◦ α) + (εa ◦ ∂µ,α − ∂µ,α ◦ εa − εa0 ◦ α)∂µ,α

= −∂µ,α ◦ εa0 ◦ α − εa0 ◦ α ◦ ∂µ,α. Thus for any X in ΛG, we have :

L(X) = −∂µ,α(a0 ∧ α(X)) − a0 ∧ ∂µ,αα(X)

µ,α = (ada0 ◦ α)(X) + ((α(a0) − a0) ∧ ∂µ,α ◦ α)(X). Thus µ,α L = ada0 ◦ α + ε(α(a0)−a0) ◦ ∂µ,α ◦ α. By transposition, we get ∗ µ∗,α L = −α ◦ ada0 + α ◦ dµ,α ◦ ι(α(a0)−a0).

When α = IdG, we recover the laplacian of an exact Lie quasi-bialgebra [13]. 726 Ibrahima BAKAYOKO

References

[1] A. Makhlouf and S. Silvestrov, Notes on Formal Deformations of Hom- associative algebras and Hom-Lie algebras, Forum Math., 22 (2010), no. 4, 715-739.

[2] A. Makhlouf and S. Silvestrov, Hom-algebra structures, J. Gen. Lie Theory Appl. Vol.2 (2008), no. 2, 51-64.

[3] A. Makhlouf and S. Silvestrov, Hom-algebras and Hom-coalgebras, Journal of Algebra and Its Applications, Vol. 9 (4) (2010), 553-589.

[4] B. Kostant, Lie Algebra Cohomology and the Generalized Borel-Weil The- orem, Ann. of Mathematics, Second Series, Vol. 74, N 2 (Sep. 1961), pp 72-144.

[5] D. Yau, Hom-algebras and Homology, J. Lie Theory, 19 (2009), 409-421.

[6] D. Yau, The classical Hom-Yang-Baxter equation and Hom-Lie bialgebras, ArXiv:0905.1890v1, 12 May 2009.

[7] D. Yau, Infinitesimal Hom-bialgebras and Hom-Lie bialgebras, ArXiv:0911.5402v1 (2010).

[8] I. Bakayoko, L-modules, L-comodules and Hom-Lie quasi-bialgebras, African Diaspora Journal of Mathematics, Vol 17 (2014), 49-64.

[9] I. Bakayoko and M. Bangoura, Quasi-big`ebre de Lie et cohomologie des alg`ebres de Lie, Travaux Math´ematiquesde Luxembourg, Vol 21 (2012), 1-28.

[10] J. H. Lu, Lie bialgebras and Lie algebra cohomology, Preprint 1996, non publi´e.

[11] J. Hartwig, D. Larsson and S. Silvestrov, Deformations of Lie algebras using σ-derivations, J. Algebra 295 (2006), 314-361.

[12] J. L. Koszul, Homologie et cohomologie des alg`ebres de Lie, Bull. Soc. Math. France, 78 (1960) 65-127.

[13] M. Bangoura and I. Bakayoko, Laplacian d’une quasi-big`ebre de Lie, African Diaspora Journal of Mathematics, Vol 17 (2014), 10-31.

[14] M. Bangoura, Quasi-big`ebres jacobiennes et g´en´eralisations des groupes de Lie-Poisson, Th`ese,Universit´ede Lille 1 (France), 1995. Laplacian of Hom-Lie quasi-bialgebras 727

[15] N. Andruskiewitsch and A. Tiraboschi, Lie quasi-bialgebras with quasi- triangular decomposition, Journal of Lie theory, Vol10 (2000) 383-397.

[16] O. Kravchenko, Strongly Homotopy Lie Lie bialgebras and Lie quasi- bialgebras,

[17] V. G. Drinfeld, Quasi-Hopf algebras, Leningrad Math. Journal, 1 (6) (1990) 1419-1457.

[18] Y. Kosmann-Schwarzbach, Jacobian quasi-bialgebras and quasi-Poisson Lie groups, Contemporary Mathematics 132 (1992) 459-489.

Received: August 1, 2014