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Laplacian of Hom-Lie Quasi-Bialgebras

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Laplacian of Hom-Lie Quasi-Bialgebras 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 L1-(quasi)bialgebras (strongly homo- topy Lie (quasi)bialgebras) which are generalizations of L1-algebras structure. An L1-version of a Manin (quasi)triple and a correspondence theorem with L1(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 2 G ; i = 1; :::; m; xj 2 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 2 G; [x; y]µ,α = µ(x; y); (iii) satisfies the following rule on the degree : [X; Y ]µ,α 2 Λp+q+1G; if X 2 Λp+1G and Y 2 Λq+1G, (iv) satisfies the graded skew-commutativity, i.e. [X; Y ]µ,α = −(−1)pq[Y; X]µ,α; if X 2 Λp+1G and Y 2 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 2 Λp+1G, Y 2 Λq+1G and Z 2 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 2 Λp+1G, Y 2 Λq+1G and Z 2 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 2 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 2 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 2 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 2 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 2 G, then ⊗3 (G; µα = α ◦ µ, γα = γ ◦ α; φα = α φ) is a Hom-Lie quasi-bialgebra.
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