Download 3-Lie algebra Г21 over the prime field Z2

Mathematica Aeterna, Vol. 5, 2015, no. 5, 739 - 743
3-Lie algebra Γ21 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
BAI Jin
College of Mathematics and Information Science,
Hebei University, Baoding, 071002, China
Abstract
The 8-dimensional 3-Lie algebra Γ21 over the prime field Z2 is constructed by 2-cubic matrix, and the structures of it is studied. It is
proved that Γ21 is a solvable but non-nilpotent 3-Lie algebra. The inner derivation algebra ad(Γ21 ) is a 12-dimensional solvable but nonnilpotent Lie algebra, and the derivation algebra Der(Γ21 ) with dimension 17 is unsolvable. 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
In papers [1, 2], authors constructed 3-Lie algebras by well known algebras
and linear functions, derivations and involutions. In paper [3], 3-Lie algebras
are constructed by N-cubic matrix over the field F with chF 6= 2. Bai, Guo,
and Lin in papers [4, 5] discussed 3-Lie algebras J11 and J21 which are realized
by 2-cubic matrix over a field F with characteristic chF 6= 2. 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
740
Bai Ruipu, Lin Lixin, Bai Jin
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 Γ21
First we introduce some notations. 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 Ω. Then Ω is an N 3 dimensional vector space with A + B = (aijk + bijk ) ∈ Ω, λA = (λaijk ) ∈ Ω,
for ∀A = (aijk ), B = (bijk ) ∈ Ω, 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 Ω.
For all A = (aijk ), B = (bijk ) ∈ Ω, define the multiplication ∗21 in Ω by
(A ∗21 B)ijk =
n
X
aqjp bipk , 1 ≤ i, j, k ≤ n.
(1)
p,q=1
Denote hAi1 =
N
P
(A)pqq =
p,q=1
N
P
p,q=1
apqq , Then h i1 is a linear function from
Ω to Z2 and satisfies
hA ∗21 Bi1 = hB ∗21 Ai1 .
(2)
Define the multiplication [, , ]21 : Ω ∧ Ω ∧ Ω → Ω as follows:
[A, B, C]21 = hAi1 (B ∗21 C − C ∗21 B)
+hBi1 (C ∗21 A − A ∗21 C) + hCi1 (A ∗21 B − B ∗21 A).
(3)
Theorem 2.1[3] The linear space Ω is a 3-Lie algebra in the multiplication
[, , ]21 , which is denoted by Γ21 , the multiplication [, , ]21 is simply denoted by
[, , ].
In the following we denote Ω2 the vector space over Z2 = {0, 1} which
consists of 2-cubic matrix. Then 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 .
Theorem 2.2.
The multiplication of the 3-Lie algebra Γ21 in the basis
741
3-Lie algebra Γ21 over the prime field Z2
{E111 , E112 , E121 , E122 , E211 , E212 , E221 , E222 } is as follows



































































[E111 , E112 , E121 ] = E111 + E122 ,
[E112 , E121 , E122 ] = E111 + E122 ,
[E212 , E221 , E222 ] = E211 + E222 ,
[E111 , E121 , E211 ] = E221 + E121 ,
[E111 , E121 , E212 ] = E222 + E111 ,
[E121 , E122 , E212 ] = E111 + E222 ,
[E111 , E122 , E221 ] = E221 + E121 ,
[E112 , E122 , E221 ] = E122 + E211 ,
[E111 , E122 , E222 ] = E222 − E122 ,
[E211 , E221 , E111 ] = E121 + E221 ,
[E221 , E222 , E111 ] = E121 + E221 ,
[E211 , E221 , E112 ] = E122 + E211 ,
[E211 , E212 , E121 ] = E111 + E222 ,
[E212 , E222 , E121 ] = E222 + E111 ,
[E211 , E212 , E122 ] = E112 + E212 ,
[E212 , E221 , E122 ] = E211 + E222 ,
[E111 , E122 , E211 ] = E111 + E211 ,
[E211 , E212 , E221 ] = E211 + E222 ,
[E211 , E222 , E111 ] = E211 + E111 ,
[E112 , E121 , E211 ] = E111 + E122 ,
[E111 , E122 , E212 ] = E112 + E212 ,
[E111 , E112 , E221 ] = E211 + E122 ,
[E121 , E122 , E211 ] = E121 + E221 ,
[E111 , E112 , E222 ] = E212 + E112 ,
[E112 , E121 , E222 ] = E111 + E122 ,
[E212 , E221 , E111 ] = E211 + E222 ,
[E112 , E122 , E222 ] = E212 + E112 ,
[E211 , E222 , E112 ] = E212 + E112 ,
[E211 , E222 , E121 ] = E121 + E221 ,
[E221 , E222 , E112 ] = E211 + E112 ,
[E211 , E222 , E122 ] = E122 + E222 ,
[E212 , E222 , E122 ] = E112 + E212 ,
(4)
where the zero product of the basis vectors are omitted.
Proof The result follows from the direct computation according to the
definition of ∗21 and Eqs.(1), (2) and (3).
For studying the structure of the 3-Lie algebra Γ21 , we need to find a special
basis to simplify its multiplication.
Theorem 2.3 The 3-Lie algebra Γ21 is a non-nilpotent indecomposable
3-Lie algebra with a basis e1 = E111 , e2 = E112 , e3 = E121 , e4 = E111 + E122 ,
e5 = E211 + E111 ,e6 = E212 + E112 , e7 = E221 − E121 , e8 = E122 + E222 . And the
multiplication in it is as follows:



[e1 , e2 , e3 ] = e4 , [e1 , e4 , e5 ] = e5 , [e1 , e3 , e6 ] = e8 ,
[e1 , e3 , e5 ] = e7 , [e1 , e4 , e6 ] = e6 , [e1 , e2 , e8 ] = e6 ,


[e1 , e2 , e7 ] = e5 , [e1 , e4 , e7 ] = e7 , [e1 , e4 , e8 ] = e8 .
(5)
Proof It is clear that {e1 , · · · , e8 } is a basis of Ω2 . By the definition of
∗21 , we obtain Eq.(5). Since Γ21 can not be written as the direct sum of two
proper ideals, Γ21 is indecomposable.
From Γ121 = [Γ21 , Γ21 , Γ21 ] = (e4 , e5 , e6 , e7 , e8 ), Γ221 = [Γ121 , Γ21 , Γ21 ] = (e5 , e6 ,
e7 , e8 ), and Γ321 = [Γ221 , Γ21 , Γ21 ] = (e5 , e6 , e7 , e8 ), then for all positive integer
s > 1, we have Γs21 = Γ221 6= 0. Therefore, Γ21 is non-nilpotent.
Theorem 2.4 The subalgebra H = (e1 , e2 , e3 , e4 ) is a Cartan subalgebra
of the 3-Lie algebra Γ21 . And the decomposition of Γ21 associate to H is
Γ21 = H +̇Γα , and Γα = (e5 , e6 , e7 , e8 ) = {e ∈ Γ21 | (ad(h1 , h2 ) +α(h1 , h2 ))2 (e) =
0, ∀h1 , h2 ∈ H}, where α ∈ (H ⊗ H)∗ , α(e1 , e4 ) = 1, and others are zero.
742
Bai Ruipu, Lin Lixin, Bai Jin
Proof From Theorem 2.3, H = (e1 , e2 , e3 , e4 ) is a Cartan subalgebra of
Γ21 . Denote α : H ⊗ H → Z2 , α(e1 , e4 ) = 1, α(e1 , e2 ) = α(e1 , e3 ) = α(e2 , e3 )
= α(e2 , e4 ) = α(e3 , e4 ) = 0, we have ad(e1 , e4 )(e5 ) = e5 , ad(e1 , e4 )(e6 ) =
e6 , ad(e1 , e4 )(e7 ) = e7 , ad(e1 , e4 )(e8 ) = e8 , ad2 (e1 , e2 )ei = ad2 (e1 , e3 )(ei ) =
ad2 (e2 , e3 )(ei ) = 0, ad2 (e2 , e4 )(ei ) = ad2 (e3 , e4 )(ei ) = 0, i = 5, 6, 7, 8. We
obtain the result.
Now we study the inner derivation algebra ad(Γ21 ). For ei , ej ∈ Ω2 , denote
P
ij
ji
ad(ei , ej )ek = 8l=1 aij
kl el , where akl = aP
kl = 0 or 1 ∈ Z2 . Then the matrix form
of ad(ei , ej ) in the basis e1 , · · · , e8 is 8k,l=1 aij
kl Ekl , where Ekl are the matrix
units.
Theorem 2.5 the inner derivation algebra ad(Γ21 ) is solvable but nonnilpotent Lie algebra with dimension 12, and X1 = E34 + E75 + E86 , X2 =
E24 + E57 + E68 , X3 = E55 + E66 + E77 + E88 , X4 = E37 + E45 , X5 = E38 +
E46 , X6 = E25 +E47 , X7 = E26 +E48 , X8 = E14 , X9 = E15 , X10 = E16 , X11 =
E17 , X12 = E18 is a basis. And the multiplication in it is

























[X2 , X1 ] = X3 ,
[X1 , X7 ] = X5 ,
[X2 , X4 ] = X6 ,
[X2 , X5 ] = X7 ,
[X3 , X4 ] = X4 ,
[X3 , X5 ] = X5 ,
[X4 , X8 ] = X9 ,
[X1 , X6 ] = X4 , [X7 , X8 ] = X12 ,
[X1 , X11 ] = X9 , [X3 , X12 ] = X12 ,
[X1 , X12 ] = X10 , [X3 , X10 ] = X10 ,
[X2 , X10 ] = X12 , [X3 , X11 ] = X11 ,
[X2 , X9 ] = X11 , [X3 , X6 ] = X6 ,
[X3 , X7 ] = X7 , [X3 , X9 ] = X9 ,
[X6 , X8 ] = X11 , [X5 , X8 ] = X10 .
(6)
Proof By a direct computation according to Eq.(5) we have ad(e1 , e2 ) =
E34 + E75 + E86 , ad(e1 , e3 ) = E24 + E57 + E68 , ad(e1 , e4 ) = E55 + E66 +
E77 + E88 , ad(e1 , e7 ) = E25 + E47 , ad(e1 , e6 ) = E38 + E46 , ad(e1 , e5 ) = E37 +
E45 , ad(e1 , e8 ) = E26 + E48 , ad(e2 , e3 ) = E14 , ad(e2 , e7 ) = E15 , ad(e2 , e8 ) =
E16 , ad(e3 , e5 ) = E17 , ad(e3 , e6 ) = E18 . Then {X1 , · · · , X12 } is a basis of
ad(Γ21 ). From [ad(ei , ej ), ad(ek , el )]= ad([ei , ej , ek ], el )+ad(ek , [ei , ej el ]), we obtain Eq.(6). And
ad1 (Γ21 ) = [ad(Γ21 ), ad(Γ21 )] = (X3 , X4 , X5 , X6 , X7 , X9 , X10 , X11 , X12 ),
ad2 (Γ21 ) = (X4 , X5 , X6 , X7 , X9 , X10 , X11 , X12 ), ads (Γ21 ) = ad2 (Γ21 ) 6= 0,
for s > 2, then ad(Γ21 ) is non-nilpotent.
By ad(1) (Γ21 ) = ad1 (Γ21 ), ad(2) (Γ21 ) = (X4 , X5 , X6 , X7 , X9 , X10 , X11 , X12 ),
ad(3) (Γ21 ) = (X9 , X10 , X11 , X12 ), ad(4) (Γ21 ) = 0, ad(Γ21 ) is solvable. The proof
is completed.
Now,we discuss the derivation algebra Der(Γ21 ).
Theorem 2.6 The derivation algebra Der(Γ21 ) is an unsolvable Lie algebra with dimension 17, and the multiplication in the basis {X1 , · · · , X17 },
where X1 = E34 + E75 + E86 , X2 = E24 + E57 + E68 , X3 = E55 + E66 + E77 + E88 ,
X4 = E37 + E45 , X5 = E38 + E46 , X6 = E25 + E47 , X7 = E26 + E48 , X8 = E14 ,
X9 = E15 , X10 = E16 , X11 = E17 , X12 = E18 , X13 = E11 + E44 + E77 + E88 ,
3-Lie algebra Γ21 over the prime field Z2
743
X14 = E22 +E33 +E77 +E88 , X15 = E55 +E77 , X16 = E56 +E78 , X17 = E65 +E87
is as follows











































[X2 , X1 ] = X3 , [X1 , X6 ] = X4 , [X1 , X13 ] = X1 , [X2 , X13 ] = X2 ,
[X1 , X7 ] = X5 , [X1 , X11 ] = X9 , [X4 , X13 ] = X4 , [X5 , X13 ] = X5 ,
[X2 , X4 ] = X6 , [X1 , X12 ] = X10 , [X9 , X13 ] = X9 , [X10 , X13 ] = X10 ,
[X2 , X5 ] = X7 , [X2 , X10 ] = X12 , [X3 , X4 ] = X4 , [X2 , X9 ] = X11 ,
[X3 , X5 ] = X5 , [X3 , X6 ] = X6 , [X4 , X15 ] = X4 , [X6 , X15 ] = X6 ,
[X3 , X7 ] = X7 , [X3 , X11 ] = X11 , [X9 , X15 ] = X9 , [X11 , X15 ] = X11 ,
[X3 , X9 ] = X9 , [X3 , X10 ] = X10 , [X4 , X16 ] = X5 , [X6 , X16 ] = X7 ,
[X4 , X8 ] = X9 , [X3 , X12 ] = X12 , [X9 , X16 ] = X10 , [X11 , X16 ] = X12 ,
[X6 , X8 ] = X11 , [X7 , X8 ] = X12 , [X5 , X17 ] = X4 , [X7 , X17 ] = X6 ,
[X5 , X8 ] = X10 , [X10 , X17 ] = X9 , [X12 , X17 ] = X11 , [X15 , X16 ] = X16 ,
[X15 , X17 ] = X17 , [X16 , X17 ] = X3 .
Proof By a direct computation according to the multiplication (5), X1 ,
· · · , X17 is a basis of Der(Γ21 ), and with the above multiplication. Since
Der (1) (Γ21 ) = (X1 , X2 , X3 , X4 , X5 , X6 , X7 , X9 , X10 , X11 , X12 , X16 , X17 ),
Der (2) (Γ21 ) = (X3 , X4 , X5 , X6 , X7 , X9 , X10 , X11 , X12 , X16 , X17 )
= Der (s) (Γ21 ) 6= 0 for all s > 2, we obtain the result.
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] R. Bai,Y. Wu, Constructions of 3-Lie algebras, Linear and Multilinear
Algebra, 2015, 63(11): 2171-2186.
[2] A. Pozhidaev, Monomial n-Lie algebras, Algebra Log. 1998, 37(5):307-322.
[3] R. Bai, H. Liu, M. Zhang, 3-Lie Algebras Realized by Cubic Matrices,
Chin.Ann. Math., 2014, 35B(2): 261-270.
[4] R. Bai, L. Lin, W. Guo, Structure of 8-dimensional 3-Lie algebra J21 ,
Mathematica Aeterna, 2015, 5(4): 599- 603.
[5] R. Bai, W. Guo, L. Lin, Structure of the 3-Lie algebra J11 , Mathematica
Aeterna, 2015, 5(4):593-597.
Received: October, 2015