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Mathematica Aeterna, Vol. 6, 2016, no. 2, 159 - 163 3-Lie bialgebras (Lb, Cd) and (Lb, Ce) BAI Ruipu College of Mathematics and Information Science, Hebei University, Baoding, 071002, China email: [email protected] LIN Lixin College of Mathematics and Information Science, Hebei University, Baoding, 071002, China email: [email protected] Abstract In this paper, we continue to study the structure of four dimensional 3-Lie bialgebras. We discuss the existence of 3-Lie bialgebras of types (Lb , Cd ) and (Lb , Ce ). It is proved that there do not exist 3-Lie bialgebras of types (Lb1 , Ce ), (Lb2 , Ce ) and (Lb2 , Cd ). There exists only one class of 3-Lie bialgebras of type (Lb , Cd ), that is (Lb1 , Cd , ∆1 ) ( Theorem 3.4 ). 2010 Mathematics Subject Classification: 17B05 17D30 Keywords: 3-Lie algebra, 3-Lie coalgebra, 3-Lie bialgebra. 1 Introduction In 1985, Filippov provided n-Lie algebras [1]. Since then, the n-Lie algebra, especially, the 3-Lie algebra attracts more and more attention, and it is widely used in mathematics, mathematical physics and string theory. And some n-ary algebras such as n-Hom algebras, n-supper algebras, n-Rota-Baxter algebras, etc. are provided and studied (see [2, 3, 4]). Authors in paper [5] introduced 3-Lie coalgebras and 3-Lie bialgebras, and constructed some 4-dimensional 3Lie bialgebras. In paper [6], 4-dimensional 3-Lie bialgebras of type (Lb , Cb) are classified. It is proved that there exist seven classes of 4-dimensional 3-Lie bialgebras of type (Lb , Cb ). In this paper, we discuss 3-Lie bialgebras of types (Lb , Cd ) and (Lb , Ce ), that is, the 3-Lie algebras with one-dimensional derived algebra and 3-Lie coalgebras with three and four dimensional derived algebra, 160 Bai Ruipu, Lixin Lin which is denoted by (Lb , Cd ) and (Lb , Ce ), respectively. And we suppose that 3-Lie algebras and 3-Lie coalgebras are over a field F of characteristic zero, and omit the zero multiplication of basis vectors in 3-Lie algebras and 3-Lie coalgebras. 2 Preliminaries A 3-Lie algebra [1] is a vector space L endowed with a linear multiplication µ : L∧3 → L satisfying that, for all x, y, z, u, v ∈ L, µ(u, v, µ(x, y, z)) = µ(x, y, µ(u, v, z))+µ(y, z, µ(u, v, x))+µ(z, x, µ(u, v, y)). For defining 3-Lie coalgebras, we need to define following linear mapps ωi : L ⊗ L ⊗ L ⊗ L ⊗ L → L ⊗ L ⊗ L ⊗ L ⊗ L, 1 ≤ i ≤ 3, by ω1 (x1 ⊗ x2 ⊗ x3 ⊗ x4 ⊗ x5 ) = x3 ⊗ x4 ⊗ x1 ⊗ x2 ⊗ x5 , ω2 (x1 ⊗ x2 ⊗ x3 ⊗ x4 ⊗ x5 ) = x4 ⊗ x5 ⊗ x1 ⊗ x2 ⊗ x3 , ω3 (x1 ⊗ x2 ⊗ x3 ⊗ x4 ⊗ x5 ) = x5 ⊗ x3 ⊗ x1 ⊗ x2 ⊗ x4 . A 3-Lie coalgebra (L, ∆) [5] is a vector space L with a linear map ∆ : L → L ⊗ L ⊗ L satisfying Im(∆) ⊂ L ∧ L ∧ L, and (1 − ω1 − ω2 − ω3 )(1 ⊗ 1 ⊗ ∆)∆ = 0, where 1 : L⊗5 → L⊗5 is identity. Let (L1 , ∆1 ) and (L2 , ∆2 ) be 3-Lie coalgebras. If there is a linear isomorphism ϕ : L1 → L2 satisfying (ϕ ⊗ ϕ ⊗ ϕ)(∆1 (e)) = ∆2 (ϕ(e)), for all e ∈ L1 , then (L1 , ∆1 ) is isomorphic to (L2 , ∆2 ), and ϕ is called a 3-Lie coalgebra isoP P morphism, where (ϕ ⊗ ϕ ⊗ ϕ) (ai ⊗ bi ⊗ ci ) = ϕ(ai ) ⊗ ϕ(bi ) ⊗ ϕ(ci ). i i A 3-Lie bialgebra [5] is a triple (L, µ, ∆) such that (1) (L, µ) is a 3-Lie algebra with the multiplication µ : L ∧ L ∧ L → L, (2) (L, ∆) is a 3-Lie coalgebra with ∆ : L → L ∧ L ∧ L, (3) ∆ and µ satisfy the following identities, for x, y, u, v, w ∈ L, (3) (3) ∆µ(x, y, z) = ad(3) µ (x, y)∆(z) + adµ (y, z)∆(x) + adµ (z, x)∆(y), (3) (3) where ad(3) µ (x, y), adµ (z, x), adµ (y, z) : L ⊗ L ⊗ L → L ⊗ L ⊗ L are linear (3) maps defined by (similar for ad(3) µ (z, x) and adµ (y, z)) ad(3) µ (x, y)(u ⊗ v ⊗ w) = (adµ (x, y) ⊗ 1 ⊗ 1)(u ⊗ v ⊗ w) +(1 ⊗ adµ (x, y) ⊗ 1)(u ⊗ v ⊗ w) + (1 ⊗ 1 ⊗ adµ (x, y))(u ⊗ v ⊗ w) = µ(x, y, u) ⊗ v ⊗ w + u ⊗ µ(x, y, v) ⊗ w + u ⊗ v ⊗ µ(x, y, w). Two 3-Lie bialgebras (L1 , µ1 , ∆1 ) and (L2 , µ2 , ∆2 ) are called equivalent if there exists a linear isomorphism f : L1 → L2 such that (1) f : (L1 , µ1 ) → (L2 , µ2 ) is a 3-Lie algebra isomorphism, (2) f : (L1 , ∆1 ) → (L2 , ∆2 ) is a 3-Lie coalgebra isomorphism, that is, ∆2 (f (x)) = (f ⊗ f ⊗ f )∆1 (x) for all x ∈ L1 . Lemma 2.1[1] Let (L, µ) be a 4-dimensional 3-Lie algebra with dim L1 6= 0, 2, and e1 , e2 , e3 , e4 be a basis of L. Then L is isomorphic to one and only one of the following 3-Lie bialgebras (Lb , Cd ) and (Lb , Ce ) 161 Lb1 . µ(e2 , e3 , e4 ) = e1 . Lb2 . µ(e1 , e2 , e3 ) = e1 . Ld . µd (e2 , e3 , e4 ) = e1 , µd (e1 , e3 , e4 ) = e2 , µd (e1 , e2 , e4 ) = e3 . Le . µe (e2 , e3 , e4 ) = e1 , µe (e1 , e3 , e4 ) = e2 , µe (e1 , e2 , e4 ) = e3 , µe (e1 , e2 , e3 ) = e4 . Lemma 2.2 [5] Let L be a vector space over F , and µ : L ⊗ L ⊗ L → L be a 3-ary linear map. Then (L, µ) is a 3-Lie algebra if and only if (L∗ , µ∗ ) is a 3-Lie coalgebra with µ∗ : L∗ → L∗ ⊗ L∗ ⊗ L∗ , where µ∗ is the dual map of µ. 3 3-Lie bialgebras of types (Lb, Cd ) and (Lb, Ce) First We give the classification of 3-Lie coalgebras of the types (L, Cd ) and (L, Ce ). Lemma 3.1 Let (L, ∆) be a 4-dimensional 3-Lie coalgebra with m-dimensional derived algebra (3 ≤ m ≤ 4), and e1 , e2 , e3 , e4 be a basis of L. Then L isomorphic to one and only one of the following Cd . ∆d (e1 ) = e2 ∧ e3 ∧ e4 , ∆d (e2 ) = e1 ∧ e3 ∧ e4 , ∆d (e3 ) = e1 ∧ e2 ∧ e4 ; Ce . ∆e (e1 ) = e2 ∧ e3 ∧ e4 , ∆e (e2 ) = e1 ∧ e3 ∧ e4 , ∆e (e3 ) = e1 ∧ e2 ∧ e4 , ∆e (e4 ) = e1 ∧ e2 ∧ e3 . Proof The result follows from Lemma 2.1, Lemma 2.2 and a direct computation, we omit the computation process. For convenience, in the following, for a 3-Lie bialgebra (L, µ, ∆), if the 3-Lie algebra (L, µ) is the case (L, µbi ) in Lemma 2.1 and the 3-Lie coalgebra (L, ∆) is the case (L, ∆d ) and (L, ∆e ) in Lemma 3.1, then the 3-Lie bialgebra (L, µbi , ∆d ) and (L, µbi , ∆e ) are simply denoted by (Lbi , Cd ) and (Lbi , Ce ), which are called the 3-Lie bialgebras of type (Lb , Cd ), and (Lb , Ce ), respectively. For a given 3-Lie algebra L, in order to find all the 3-Lie bialgebra structures on L, we should find all the 3-Lie coalgebra structures on L which are compatible with the 3-Lie algebra L. Although a permutation of a basis of L gives isomorphic 3-Lie coalgebra, but it may lead to the non-equivalent 3-Lie bialgebra. Theorem 3.2 There do not exist 3-Lie bialgebras of the types (Lb1 , Ce ) and (Lb2 , Ce ). Proof By Lemma 2.1 and 2.2, we need to verify that 3-Lie algebras Lb1 and Lb2 are incompatible with the following six isomorphic 3-Lie coalgebras of the type Ce : (1).∆(e1 ) = e2 ∧e3 ∧e4 , ∆(e2 ) = e1 ∧e3 ∧e4 , ∆(e3 ) = e1 ∧e2 ∧e4 , ∆(e4 ) = e1 ∧e2 ∧e3 ; (2).∆(e1 ) = e2 ∧e3 ∧e4 , ∆(e2 ) = e1 ∧e3 ∧e4 , ∆(e3 ) = e2 ∧e1 ∧e4 , ∆(e4 ) = e2 ∧e1 ∧e3 ; (3).∆(e1 ) = e2 ∧e3 ∧e4 , ∆(e2 ) = e3 ∧e1 ∧e4 , ∆(e3 ) = e2 ∧e1 ∧e4 , ∆(e4 ) = e2 ∧e3 ∧e1 ; (4).∆(e1 ) = e2 ∧e4 ∧e3 , ∆(e2 ) = e1 ∧e4 ∧e3 , ∆(e3 ) = e2 ∧e1 ∧e4 , ∆(e4 ) = e2 ∧e1 ∧e3 ; (5).∆(e1 ) = e2 ∧e4 ∧e3 , ∆(e2 ) = e4 ∧e1 ∧e3 , ∆(e3 ) = e2 ∧e4 ∧e1 , ∆(e4 ) = e2 ∧e1 ∧e3 ; (6).∆(e1 ) = e2 ∧e4 ∧e3 , ∆(e2 ) = e4 ∧e3 ∧e1 , ∆(e3 ) = e2 ∧e4 ∧e1 , ∆(e4 ) = e2 ∧e3 ∧e1 . Here we only check the case (1) is incompatible with the 3-Lie algebra Lb1 . Since ∆µb1 (e2 , e3 , e4 ) = ∆(e1 ) = e2 ∧ e3 ∧ e4 , but (3) (3) (3) adµb1 (e2 , e3 )∆(e4 ) + adµb1 (e3 , e4 )∆(e2 ) + adµb1 (e4 , e2 )∆(e3 ) = 0, we obtain that (3) (3) (3) ∆µb1 e2 , e3 , e4 ) 6= adµb1 (e2 , e3 )∆(e4 ) + adµb1 (e3 , e4 )∆(e2 ) + adµb1 (e4 , e2 )∆(e3 ). Therefore, the case (1) of the type Ce is incompatible with the 3-Lie algebra Lb1 . Similar discussions for others cases, we get the result. 162 Bai Ruipu, Lixin Lin Theorem 3.3 There does not exist 3-Lie bialgebras of the type (Lb2 , Cd ). Proof By Lemma 2.1 and 2.2, we need to verify that 3-Lie algebra Lb2 is incompatible with following twenty-four isomorphic 3-Lie coalgebras of the type Cd : (1).∆(e1 ) = e2 ∧ e3 ∧ e4 , ∆(e2 ) = e1 ∧ e3 ∧ e4 , ∆(e3 ) = e1 ∧ e2 ∧ e4 ; (2).∆(e1 ) = e2 ∧ e3 ∧ e4 , ∆(e2 ) = e3 ∧ e1 ∧ e4 , ∆(e3 ) = e2 ∧ e1 ∧ e4 ; (3).∆(e1 ) = e2 ∧ e3 ∧ e4 , ∆(e2 ) = e1 ∧ e3 ∧ e4 , ∆(e3 ) = e2 ∧ e1 ∧ e4 ; (4).∆(e1 ) = e3 ∧ e2 ∧ e4 , ∆(e2 ) = e3 ∧ e1 ∧ e4 , ∆(e3 ) = e1 ∧ e2 ∧ e4 ; (5).∆(e1 ) = e3 ∧ e2 ∧ e4 , ∆(e2 ) = e3 ∧ e1 ∧ e4 , ∆(e3 ) = e2 ∧ e1 ∧ e4 ; (6).∆(e1 ) = e3 ∧ e2 ∧ e4 , ∆(e2 ) = e1 ∧ e3 ∧ e4 , ∆(e3 ) = e1 ∧ e2 ∧ e4 ; (7).∆(e1 ) = e2 ∧ e4 ∧ e3 , ∆(e2 ) = e1 ∧ e4 ∧ e3 , ∆(e4 ) = e2 ∧ e1 ∧ e3 ; (8).∆(e1 ) = e2 ∧ e4 ∧ e3 , ∆(e2 ) = e4 ∧ e1 ∧ e3 , ∆(e4 ) = e2 ∧ e1 ∧ e3 ; (9).∆(e1 ) = e2 ∧ e4 ∧ e3 , ∆(e2 ) = e1 ∧ e4 ∧ e3 , ∆(e4 ) = e1 ∧ e2 ∧ e3 ; (10).∆(e1 ) = e4 ∧ e2 ∧ e3 , ∆(e2 ) = e4 ∧ e1 ∧ e3 , ∆(e4 ) = e2 ∧ e1 ∧ e3 ; (11).∆(e1 ) = e4 ∧ e2 ∧ e3 , ∆(e2 ) = e1 ∧ e4 ∧ e3 , ∆(e4 ) = e1 ∧ e2 ∧ e3 ; (12).∆(e1 ) = e4 ∧ e2 ∧ e3 , ∆(e2 ) = e4 ∧ e1 ∧ e3 , ∆(e4 ) = e1 ∧ e2 ∧ e3 ; (13).∆(e1 ) = e3 ∧ e4 ∧ e2 , ∆(e3 ) = e4 ∧ e1 ∧ e2 , ∆(e4 ) = e3 ∧ e1 ∧ e2 ; (14).∆(e1 ) = e3 ∧ e4 ∧ e2 , ∆(e3 ) = e1 ∧ e4 ∧ e2 , ∆(e4 ) = e1 ∧ e3 ∧ e2 ; (15).∆(e1 ) = e3 ∧ e4 ∧ e2 , ∆(e3 ) = e1 ∧ e4 ∧ e2 , ∆(e4 ) = e3 ∧ e1 ∧ e2 ; (16).∆(e1 ) = e4 ∧ e3 ∧ e2 , ∆(e3 ) = e4 ∧ e1 ∧ e2 , ∆(e4 ) = e1 ∧ e3 ∧ e2 ; (17).∆(e1 ) = e4 ∧ e3 ∧ e2 , ∆(e3 ) = e4 ∧ e1 ∧ e2 , ∆(e4 ) = e3 ∧ e1 ∧ e2 ; (18).∆(e1 ) = e4 ∧ e3 ∧ e2 , ∆(e3 ) = e1 ∧ e4 ∧ e2 , ∆(e4 ) = e1 ∧ e3 ∧ e2 ; (19).∆(e2 ) = e4 ∧ e3 ∧ e1 , ∆(e3 ) = e4 ∧ e2 ∧ e1 , ∆(e4 ) = e3 ∧ e2 ∧ e1 ; (20).∆(e2 ) = e4 ∧ e3 ∧ e1 , ∆(e3 ) = e2 ∧ e4 ∧ e1 , ∆(e4 ) = e3 ∧ e2 ∧ e1 ; (21).∆(e2 ) = e4 ∧ e3 ∧ e1 , ∆(e3 ) = e4 ∧ e2 ∧ e1 , ∆(e4 ) = e2 ∧ e3 ∧ e1 ; (22).∆(e2 ) = e3 ∧ e4 ∧ e1 , ∆(e3 ) = e2 ∧ e4 ∧ e1 , ∆(e4 ) = e3 ∧ e2 ∧ e1 ; (23).∆(e2 ) = e3 ∧ e4 ∧ e1 , ∆(e3 ) = e2 ∧ e4 ∧ e1 , ∆(e4 ) = e2 ∧ e3 ∧ e1 ; (24).∆(e2 ) = e3 ∧ e4 ∧ e1 , ∆(e3 ) = e4 ∧ e2 ∧ e1 , ∆(e4 ) = e3 ∧ e2 ∧ e1 . The discussion is completely similar to Theorem 3.2. We omit the computing process. Theorem 3.4 The only non-equivalent 3-Lie bialgebras of the type (Lb1 , Cd ) is (Lb1 , Cd , ∆1 ) : ∆1 (e2 ) = e1 ∧ e4 ∧ e3 , ∆1 (e3 ) = e1 ∧ e4 ∧ e2 , ∆1 (e4 ) = e1 ∧ e3 ∧ e2 . Proof We need to discuss the compatibility of twenty-four isomorphic 3-Lie coalgebras of the type Ccd in Theorem 3.3 with the 3-Lie algebra Lb1 . By a direct computation, only cases (19), (20), (21), (22), (23) and (24) are compatible with the 3-Lie algebra Lb1 , respectively. Thanks to isomorphisms of 3-Lie bialgebras: (19) → (22) : f (e1 ) = −e1 , f (e2 ) = e3 , f (e3 ) = e2 , f (e4 ) = e4 ; (21) → (23), (23) → (24) : f (e1 ) = e1 , f (e2 ) = e√ 3 , f (e3 ) = e4 , f (e √4 ) = e2 ; −1e3 , f (e4 ) = −1e (19) → (20) : f (e1 ) = e1 , f (e2 ) = −e2 , f (e3 ) = √ √ 4; (19) → (21) : f (e1 ) = e1 , f (e2 ) = −e2 , f (e3 ) = − −1e3 , f (e4 ) = − −1e4 ; we get that the only non-equivalent 3-Lie bialgebras of the type (Lb1 , Cd ) is (Lb1 , Cd , ∆1 ). Acknowledgements The first author (R.-P. Bai) was supported in part by the Natural Science Foundation (11371245) and the Natural Science Foundation of Hebei Province (A2014201006). 3-Lie bialgebras (Lb , Cd ) and (Lb , Ce ) 163 References [1] V. Filippov, n-Lie algebras, Sib. Mat. Zh., 26 (1985) 126-140. [2] Y. Liu L. ChenY. Ma, Representations and module-extensions of 3-hom-Lie algebras. J. Geom. Phys., 98 (2015): 376-383. [3] B. SunL. Chen, RotaCBaxter multiplicative 3-ary Hom-Nambu Lie algebras. J. Geom. 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