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Mathematica Aeterna, Vol. 5, 2015, no. 5, 905 - 909 3-Lie algebra Γ27 over the prime field Z2 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 Abstract In this paper, the 8-dimensional 3-Lie algebra Γ27 over the prime field Z2 is constructed by 2-cubic matrix. It is proved that Γ27 is a solvable but non-nilpotent 3-Lie algebra. The inner derivation algebra ad(Γ27 ) is an 11-dimensional solvable Lie algebra, and the derivation algebra Der(Γ27 ) with dimension 18 is solvable but non-nilpotent. And the concrete expression of all derivations are given. 2010 Mathematics Subject Classification: 17B05 17D30 Keywords: N-cubic matrix, 3-Lie algebra, derivation, prime field. 1 Introduction The notion of n-Lie algebra (or Lie n-algebra, Filippov algebra, Nambu-Poisson algebra and so on) was introduced by Filippov in 1985 [1]. An n-Lie algebra A is a vector space A endowed with a n-ary skew-symmetric multiplication satisfying the n-Jacobi identity: [[x1 , · · · , xn ], y2 , · · · , yn ] = n X [x1 , · · · , [xi , y2, · · · , yn ], · · · , xn ]. i=1 The structure of n-Lie algebras is applied to the study of the supersymmetry and gauge symmetry transformations of the world-volume theory of multiple coincident M2- branes; the Bagger-Lambert theory has a novel local gauge symmetry which is based on a metric 3-Lie algebra[2]. 906 Bai Ruipu, Lin Lixin In papers [3-7], authors constructed 3-Lie algebras by well known algebras and N-cubic matrices over a field F with chF 6= 2, and the structure of 3-Lie algebras is studied. In this paper, we pay our main attention to 8-dimensional 3-Lie algebras which are constructed by 2-cubic matrix in the prime field Z2 = {0, 1}. In the following we suppose that Z2 = {0, 1} is the prime field with characteristic two, for a vector space V and a subset S, the subspace generated by S is denoted by (S). 2 Structure of 3-Lie algebras Γ27 An N-order cubic matrix A = (aijk ) over the field Z2 is an ordered object which the elements with 3 indices, and the element in the position (i, j, k) is (A)ijk = aijk , 1 ≤ i, j, k ≤ N and aijk = 0 or 1. Denote the set of all cubic matrix over Z2 by Ω2 . Then Ω2 is an N 3 -dimensional vector space with A + B = (aijk + bijk ) ∈ Ω2 , λA = (λaijk ) ∈ Ω2 , for ∀A = (aijk ), B = (bijk ) ∈ Ω2 , λ ∈ Z2 , that is, (A + B)ijk = aijk + bijk , (λA)ijk = λaijk . Denote Eijk = (eh1 h2 h3 ), where eh1 h2 h3 = δh1 i δh2 j δh3 k , that is when h1 = i, h2 = j, h3 = k, eh1 h2 h3 = 1, and elsewhere are zero. Then, {Eijk | 1 ≤ i, j, k ≤ n} is a basis of Ω2 . For all A = (aijk ), B = (bijk ) ∈ Ω2 , define the multiplication ∗27 in Ω2 by (A ∗27 B)ijk = n X aqjk bipk , 1 ≤ i, j, k ≤ n. (1) p,q=1 Denote hAi4 = N P (A)pqr = p,q,r=1 Ω2 to Z2 and satisfies N P p,q=1 apqr , Then h i4 is linear function from hA ∗27 Bi4 = hB ∗27 Ai4 . (2) Define the multiplication [, , ]27 : Ω2 ∧ Ω2 ∧ Ω2 → Ω2 as follows: [A, B, C]27 = hAi4 (B ∗27 C − C ∗27 B) +hBi4 (C ∗27 A − A ∗27 C) + hCi4 (A ∗27 B − B ∗27 A). (3) Theorem 2.1[4] The linear space Ω2 are 3-Lie algebra in the multiplication [, , ]27 , which is denoted by Γ27 , and the multiplication [, , ]27 simply denoted by [, , ]. From above discussion the dimension of Ω2 is eight and with a basis {E111 , E112 , E121 , E122 , E211 , E212 , E221 , E222 }. And for all A ∈ Ω2 , A = 2 P i,j,k=1 λijk Eijk , λijk = 1, 0 ∈ Z2 . 3-Lie algebra Γ27 over the prime field Z2 907 Theorem 2.2. The multiplication of the 3-Lie algebra Γ27 in the basis {E111 , E112 , E121 , E122 , E211 , E212 , E221 , E222 } is as follows [E111 , E112 , E121 ] = E121 + E111 , [E111 , E112 , E122 ] = E112 + E122 , [E111 , E121 , E122 ] = E122 + E112 , [E111 , E112 , E211 ] = E111 + E211 , [E112 , E121 , E211 ] = E221 + E111 , [E112 , E122 , E211 ] = E112 + E122 , [E121 , E122 , E211 ] = E111 + E221 , [E111 , E112 , E212 ] = E212 + E112 , [E111 , E121 , E212 ] = E111 + E121 , [E111 , E122 , E212 ] = E222 + E112 , [E121 , E122 , E212 ] = E222 + E112 , [E111 , E112 , E221 ] = E121 + E211 , [E112 , E121 , E221 ] = E221 + E121 , [E112 , E121 , E221 ] = E221 + E121 , [E112 , E122 , E221 ] = E112 + E122 , [E121 , E122 , E221 ] = E121 + E221 , [E111 , E112 , E222 ] = E212 + E122 , [E111 , E121 , E222 ] = E111 + E121 , [E111 , E122 , E222 ] = E222 + E122 , [E112 , E121 , E222 ] = E122 + E212 , [E211 , E222 , E111 ] = E111 + E211 , [E212 , E221 , E111 ] = E121 + E211 , [E212 , E222 , E111 ] = E212 + E222 , [E221 , E222 , E111 ] = E211 + E121 , [E211 , E212 , E112 ] = E112 + E212 , [E211 , E221 , E112 ] = E211 + E221 , [E211 , E222 , E112 ] = E122 + E212 , [E212 , E221 , E112 ] = E212 + E112 , [E211 , E222 , E121 ] = E221 + E111 , [E212 , E221 , E121 ] = E121 + E221 , [E212 , E222 , E121 ] = E212 + E222 , [E221 , E222 , E121 ] = E221 + E121 , [E211 , E212 , E122 ] = E112 + E222 , [E211 , E221 , E122 ] = E211 + E221 , [E211 , E222 , E122 ] = E122 + E222 , [E212 , E221 , E122 ] = E222 + E112 , [E221 , E222 , E112 ] = E122 + E212 , [E211 , E212 , E121 ] = E221 + E111 , [E121 , E122 , E222 ] = E222 + E122 , [E211 , E212 , E111 ] = E211 + E111 , [E111 , E122 , E221 ] = E121 + E211 , [E112 , E121 , E212 ] = E112 + E212 , [E111 , E122 , E211 ] = E211 + E111 , [E221 , E222 , E122 ] = E122 + E222 , [E212 , E222 , E112 ] = E112 + E122 + E222 + E212 , [E211 , E221 , E121 ] = E111 + E221 + E121 + E211 , [E112 , E122 , E222 ] = E222 + E212 + E112 + E122 , [E211 , E221 , E111 ] = E111 + E121 + E221 + E211 , [E111 , E121 , E221 ] = E221 + E211 + E111 + E121 , [E112 , E122 , E212 ] = E222 + E112 + E212 + E122 , [E111 , E121 , E211 ] = E221 + E211 + E111 + E121 , [E212 , E222 , E122 ] = E112 + E122 + E222 + E212 , (4) where the zero product of the basis vectors are omitted. Proof The result follows from the direct complication according to the definition of ∗27 and Eqs.(1), (2) and (3). Theorem 2.3 The 3-Lie algebra Γ27 is a non-nilpotent indecomposable 3-Lie algebra with a basis e1 = E111 , e2 = E112 + E111 , e3 = E111 + E121 , e4 = E112 + E122 , e5 = E211 + E111 ,e6 = E212 + E112 , e7 = E211 + E221 + E111 + E121 , e8 = E212 + E222 + E112 + E122 . And the multiplication in it is as follows: ( [e1 , e2 , e3 ] = e3 , [e1 , e2 , e4 ] = e4 , [e1 , e2 , e5 ] = e5 , [e1 , e3 , e5 ] = e7 , [e1 , e2 , e6 ] = e6 , [e1 , e4 , e6 ] = e8 . (5) 908 Bai Ruipu, Lin Lixin Proof It is clear that {e1 , · · · , e8 } is linearly independent, so it is a basis of Ω2 . By the definition of ∗27 , we obtain Eq.(5). Since Γ27 can not be written as the direct sum of two proper ideals, Γ27 is indecomposable. From Γ127 = [Γ27 , Γ27 , Γ27 ] = (e3 , e4 , e5 , e6 , e7 , e8 ), Γ227 = [Γ127 , Γ27 , Γ27 ] = (e3 , e4 , e5 , e6 , e7 , e8 ), then for all positive integer s > 1, we have Γs27 = Γ127 6= 0. Therefore, Γ27 is non-nilpotent. Theorem 2.4 The subalgebra H = (e1 , e2 , e7 , e8 ) is a Cartan subalgebra of the 3-Lie algebra Γ27 . And the decomposition of Γ27 associate to H is Γ27 = H +̇Γα +̇Γ−α , where α : (H ⊗ H) → Z2 , α(e1 , e2 ) = 1, and others are zero, Γα = (e3 , e6 ),Γ−α = (e4 , e5 ). Proof From Theorem 2.3, H = (e1 , e2 , e7 , e8 ) is a Cartan subalgebra of Γ27 . Denote α : H ⊗ H → Z2 , α(e1 , e2 ) = 1, α(e1 , e7 ) = α(e1 , e8 ) = α(e2 , e7 ) = α(e2 , e8 ) = α(e7 , e8 ) = 0, we have ad(e1 , e2 )(e3 ) = e3 , ad(e1 , e2 )(e4 ) = e4 , ad(e1 , e2 )(e5 ) = e5 , ad(e1 , e2 )(e6 ) = e6 , ad2 (e1 , e7 )ei = ad2 (e1 , e8 )(ei ) = ad2 (e2 , e7 )(ei ) = 0, ad2 (e2 , e8 )(ei ) = ad2 (e7 , e8 )(ei ) = 0, i = 3, 4, 5, 6. We obtain the result. Now we study the inner derivation algebra ad(Γ27 ). For ei , ej ∈ Ω2 , denote P ij ji ad(ei , ej )ek = 8l=1 aij 0 or 1 ∈ Z2 . Then the matrix kl el , where akl = −akl = P8 form of ad(ei , ej ) in the basis e1 , · · · , e8 is k,l=1 aij kl Ekl , where Ekl are the matrix units. Theorem 2.5 the inner derivation algebra ad(Γ27 ) is solvable but indecomposable Lie algebra with dimension 11, and X1 = E33 + E44 + E55 + E66 , X2 = E23 + E57 , X3 = E24 + E68 , X4 = E25 + E37 , X5 = E26 + E48 , X6 = E13 , X7 = E14 , X8 = E15 , X9 = E16 , X10 = E17 , X11 = E18 , is a basis. And the multiplication in it is [X1 , X2 ] = X2 , [X1 , X3 ] = X3 , [X1 , X4 ] = X4 , [X1 , X5 ] = X5 , [X1 , X6 ] = X6 , [X1 , X7 ] = X7 , [X1 , X8 ] = X8 , [X1 , X9 ] = X9 , [X2 , X8 ] = X10 , [X3 , X9 ] = X11 , [X4 , X6 ] = X10 , [X5 , X7 ] = X11 . (6) Proof By a direct computation according to Eq.(5) we have that ad(e1 , e2 ) = E33 + E44 + E55 + E66 , ad(e1 , e3 ) = E23 + E57 , ad(e1 , e4 ) = E24 + E68 , ad(e1 , e5 ) = E25 +E37 , ad(e1 , e6 ) = E26 + E48 , ad(e2 , e3 ) = E13 , ad(e2 , e4 ) = E14 , ad(e2 , e5 ) = E15 , ad(e2 , e6 ) = E16 , ad(e3 , e5 ) = E17 , ad(e4 , e6 ) = E18 . Then {X1 , · · · , X11 } is a basis of ad(Γ27 ). From [ad(ei , ej ), ad(ek , el )]= ad([ei , ej , ek ], el ) +ad(ek , [ei , ej el ]), we obtain Eq.(6). And ad1 (Γ27 ) = [ad(Γ27 ), ad(Γ27 )] = (X10 , X11 ), [X10 , X11 ] = 0 then ad(Γ27 ) is solvable. Since ad(X1 ) is nonnilpotent, and ad(Γ27 ) can not be written as the direct sum of two proper ideals, ad(Γ27 ) is indecomposable non-nilpotent. Now,we discuss the derivation algebra DerΓ27 . 3-Lie algebra Γ27 over the prime field Z2 909 Theorem 2.6 The derivation algebra Der(Γ27 ) with a basis {X1 , · · · , X18 }, where X12 = E11 + E22 + E77 + E88 , X13 = E33 + E77 , X14 = E44 + E88 ,X15 = E55 + E77 ,X16 = E28 , X17 = E12 , and X18 = E27 , Xi , 1 ≤ i ≤ 11 are in Theorem 2.5. The multiplication in it is [X , X 1 2 ] = X2 , [X1 , X3 ] = X3 , [X1 , X4 ] = X4 , [X1 , X5 ] = X5 , [X1 , X6 ] = X6 , [X1 , X7 ] = X7 , [X1 , X8 ] = X8 , [X1 , X9 ] = X9 , [X2 , X8 ] = X10 , [X3 , X9 ] = X11 , [X4 , X6 ] = X10 , [X5 , X7 ] = X11 , [X2 , X12 ] = X2 , [X3 , X12 ] = X3 , [X4 , X12 ] = X4 , [X5 , X12 ] = X5 , [X , X ] = X , [X , X ] = X , [X , X ] = X , [X , X ] = X , 6 12 6 7 12 7 8 12 8 9 12 9 [X , X ] = X , [X , X ] = X , [X , X ] = X , [X , X ] = X18 , 2 13 2 6 13 6 10 13 10 18 13 [X3 , X14 ] = X3 , [X7 , X14 ] = X7 , [X11 , X14 ] = X11 , [X16 , X14 ] = X16 , [X4 , X15 ] = X4 , [X8 , X15 ] = X8 , [X10 , X15 ] = X10 , [X18 , X15 ] = X18 , [X2 , X17 ] = X6 , [X3 , X17 ] = X7 , [X4 , X17 ] = X8 , [X5 , X17 ] = X9 , [X18 , X17 ] = X10 , [X16 , X17 ] = X11 . And Der(Γ27 ) = ad(Γ27 )+̇(X12 , · · · , X18 ) is solvable but non-nilpotent. Proof The result follows from a direct computation according to Theorem 2.4. 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). References [1] V.T. Filippov, n−Lie algebras, Sib. Mat. Zh., 26 (1985) 126-140 [2] J. Bagger, N. Lambert, Gauge symmetry and supersymmetry of multiple M2-branes, Phys. Rev. D 77 (2008) 065008. [3] A. Pozhidaev, Monomial n-Lie algebras, Algebra Log. 1998, 37(5):307-322. [4] R. Bai, H. Liu, M. Zhang, 3-Lie Algebras Realized by Cubic Matrices, Chin.Ann. Math., 2014, 35B(2): 261-270. [5] R. Bai, L. Lin, W. Guo, Structure of 8-dimensional 3-Lie algebra J21 , Mathematica Aeterna, 2015, 5(4): 599- 603. [6] R. Bai, W. Guo, L. Lin, Structure of the 3-Lie algebra J11 , Mathematica Aeterna, 2015, 5(4):593-597. Received: October, 2015