Download 3-Lie algebra Г27 over the prime field Z2

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].
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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
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[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 ,
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[X3 , X9 ] = X11 , [X4 , X6 ] = X10 , [X5 , X7 ] = X11 .
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(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
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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

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[X
,
X
1
2 ] = X2 , [X1 , X3 ] = X3 , [X1 , X4 ] = X4 , [X1 , X5 ] = X5 ,
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[X1 , X6 ] = X6 , [X1 , X7 ] = X7 , [X1 , X8 ] = X8 , [X1 , X9 ] = X9 ,
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[X2 , X8 ] = X10 , [X3 , X9 ] = X11 , [X4 , X6 ] = X10 , [X5 , X7 ] = X11 ,
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[X2 , X12 ] = X2 , [X3 , X12 ] = X3 , [X4 , X12 ] = X4 , [X5 , X12 ] = X5 ,
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 [X , X ] = X , [X , X ] = X , [X , X ] = X , [X , X ] = X ,
6
12
6
7
12
7
8
12
8
9
12
9
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[X
,
X
]
=
X
,
[X
,
X
]
=
X
,
[X
,
X
]
=
X
,
[X
,
X
]
=
X18 ,
2
13
2
6
13
6
10
13
10
18
13
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[X3 , X14 ] = X3 , [X7 , X14 ] = X7 , [X11 , X14 ] = X11 , [X16 , X14 ] = X16 ,
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[X4 , X15 ] = X4 , [X8 , X15 ] = X8 , [X10 , X15 ] = X10 , [X18 , X15 ] = X18 ,
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[X2 , X17 ] = X6 , [X3 , X17 ] = X7 , [X4 , X17 ] = X8 , [X5 , X17 ] = X9 ,
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[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