ON (n+ 2)-DIMENSIONAL n-LIE ALGEBRAS
DONALD W. BARNES
Abstract. I show that an (n + 2)-dimensional n-Lie algebra over an alge-
braically closed field must have a subalgebra of codimension 1.
R. Bai, X. Wang, H. An andW. Xiao [1] have been working on the classification of
the 5-dimensional 3-Lie algebras over an algebraically closed field of characteristic 2.
To complete their classification, they ask if such an algebra must have a subalgebra
of dimension 4. The following theorem answers that question.
Theorem 0.1. Let A be an (n+2)-dimensional n-Lie algebra over the algebraically
closed field F . Then A has a subalgebra of codimension 1.
Proof. We denote the derived algebra [A,A, . . . , A] of A by A(1). If A(1) < A,
then A has a subalgebra of codimension 1 since any subspace containing A(1) is a
subalgebra. Hence we may assume A(1) = A and so, that A is not nilpotent. Let H
be a minimal Engel subalgebra. Then n−1 ≤ dim(H) ≤ n+1. As F is infinite, H is
a Cartan subalgebra by Barnes [2, Theorem 4.3]. If dim(H) = n+1, the result holds,
so we may assume dim(H) ≤ n. This implies thatH is abelian and is represented on
A by commuting linear transformations. Since F is algebraically closed, they have
a common eigenvector u. Thus we have [h1, h2, . . . , hn−1, u] = α(h1, h2, . . . , hn−1)u
for all h1, h2, . . . , hn−1 ∈ H , where α is a linear map H
∧(n−1) → F . If dim(H) = n,
then 〈H,u〉 is an (n+ 1)-dimensional subalgebra.
Suppose dim(H) = n − 1, H = 〈a1, . . . , an−1〉. Let d be the inner derivation
d(a1, . . . , an−1) of A. For each eigenvalue λ of d, we have the λ-component Aλ =
{a ∈ A | (d− λI)n+2a = 0} of A, where I denotes the identity transformation. We
have A0 = H and A is the direct sum of the components for the eigenvalues of d.
Let λ1, . . . , λn be (not necessarily distinct) eigenvalues of d. Then [Aλ1 , . . . , Aλn ] ⊆
Aλ1+···+λn .
Since H ⊂ A(1), we either have two eigenvalues, say α, β with sum 0 or we
have α + β + γ = 0. Suppose first that α + β = 0. Suppose char(F ) 6= 2. We
have eigenvectors u, v for α, β. Then 〈H,u, v〉 is an (n+1)-dimensional subalgebra.
Suppose char(F ) = 2. Then we can choose u, v such that [a1, . . . , an−1, u] = αu
and [a1, . . . , an−1, v] = αv + θu for some θ. Again we have that 〈H,u, v〉 is an
(n+ 1)-dimensional subalgebra.
Now suppose α + β + γ = 0. Suppose char(F ) 6= 2. Then α + β is not an
eigenvalue of d, so [h1, . . . , hn−2, u, v] = 0 for all h1, . . . , hn−2 ∈ H . Thus 〈H,u, v〉
is an (n + 1)-dimensional subalgebra of A. Now suppose char(F ) = 2. We have
2000 Mathematics Subject Classification. Primary 17B05, 17B30.
Key words and phrases. n-Lie algebras, Engel subalgebras.
This work was done while the author was an Honorary Associate of the School of Mathematics
and Statistics, University of Sydney.
http://arxiv.org/abs/0704.1892v1
2 DONALD W. BARNES
the distinct non-zero eigenvalues α, β, γ = α + β and corresponding eigenvectors
u, v, w. If n = 2, 3, then H 6⊆ A(1), so we may suppose n ≥ 4.
For some re-ordering of the basis ofH , we have [a1, . . . , an−3, u, v, w] 6= 0. Denote
the string a1, . . . , an−3 by
a . We apply the Jacobi identity to the product P =
[−→a , an−2, u, [
a , an−1, v, w]]. Since [
a , an−1, v, w] ∈ 〈u〉, P = 0. But
P = [−→a , [−→a , an−2, u, an−1], v, w] + [
a , an−1, [
a , an−2, u, v], w]
+ [−→a , an−1, v, [
a , an−2, u, w]]
= [−→a , αu, v, w] + 0 + 0
since [−→a , an−2, u, v] ∈ 〈w〉 and [
a , an−2, u, w] ∈ 〈v〉. Therefore α = 0 contrary to
the definition of α. Thus this case cannot arise. �
References
1. R. Bai, X. Wang, H. An and W. Xiao, The classification of 5-dimensional 3-Lie algebras over
the field of characteristic 2, private communication to the author.
2. D. W. Barnes, Engel subalgebras of n-Lie algebra, Acta Math. Sinica, Series B, to appear.
arXiv:math.RA/0610347.
1 Little Wonga Rd, Cremorne NSW 2090 Australia
E-mail address: donwb@iprimus.com.au
http://arxiv.org/abs/math/0610347
	References

  I show that an (n+2)-dimensional n-Lie algebra over an algebraically closed
field must have a subalgeba of codimension 1.

ON (n+ 2)-DIMENSIONAL n-LIE ALGEBRAS
DONALD W. BARNES
Abstract. I show that an (n + 2)-dimensional n-Lie algebra over an alge-
braically closed field must have a subalgebra of codimension 1.
R. Bai, X. Wang, H. An andW. Xiao [1] have been working on the classification of
the 5-dimensional 3-Lie algebras over an algebraically closed field of characteristic 2.
To complete their classification, they ask if such an algebra must have a subalgebra
of dimension 4. The following theorem answers that question.
Theorem 0.1. Let A be an (n+2)-dimensional n-Lie algebra over the algebraically
closed field F . Then A has a subalgebra of codimension 1.
Proof. We denote the derived algebra [A,A, . . . , A] of A by A(1). If A(1) < A,
then A has a subalgebra of codimension 1 since any subspace containing A(1) is a
subalgebra. Hence we may assume A(1) = A and so, that A is not nilpotent. Let H
be a minimal Engel subalgebra. Then n−1 ≤ dim(H) ≤ n+1. As F is infinite, H is
a Cartan subalgebra by Barnes [2, Theorem 4.3]. If dim(H) = n+1, the result holds,
so we may assume dim(H) ≤ n. This implies thatH is abelian and is represented on
A by commuting linear transformations. Since F is algebraically closed, they have
a common eigenvector u. Thus we have [h1, h2, . . . , hn−1, u] = α(h1, h2, . . . , hn−1)u
for all h1, h2, . . . , hn−1 ∈ H , where α is a linear map H
∧(n−1) → F . If dim(H) = n,
then 〈H,u〉 is an (n+ 1)-dimensional subalgebra.
Suppose dim(H) = n − 1, H = 〈a1, . . . , an−1〉. Let d be the inner derivation
d(a1, . . . , an−1) of A. For each eigenvalue λ of d, we have the λ-component Aλ =
{a ∈ A | (d− λI)n+2a = 0} of A, where I denotes the identity transformation. We
have A0 = H and A is the direct sum of the components for the eigenvalues of d.
Let λ1, . . . , λn be (not necessarily distinct) eigenvalues of d. Then [Aλ1 , . . . , Aλn ] ⊆
Aλ1+···+λn .
Since H ⊂ A(1), we either have two eigenvalues, say α, β with sum 0 or we
have α + β + γ = 0. Suppose first that α + β = 0. Suppose char(F ) 6= 2. We
have eigenvectors u, v for α, β. Then 〈H,u, v〉 is an (n+1)-dimensional subalgebra.
Suppose char(F ) = 2. Then we can choose u, v such that [a1, . . . , an−1, u] = αu
and [a1, . . . , an−1, v] = αv + θu for some θ. Again we have that 〈H,u, v〉 is an
(n+ 1)-dimensional subalgebra.
Now suppose α + β + γ = 0. Suppose char(F ) 6= 2. Then α + β is not an
eigenvalue of d, so [h1, . . . , hn−2, u, v] = 0 for all h1, . . . , hn−2 ∈ H . Thus 〈H,u, v〉
is an (n + 1)-dimensional subalgebra of A. Now suppose char(F ) = 2. We have
2000 Mathematics Subject Classification. Primary 17B05, 17B30.
Key words and phrases. n-Lie algebras, Engel subalgebras.
This work was done while the author was an Honorary Associate of the School of Mathematics
and Statistics, University of Sydney.
http://arxiv.org/abs/0704.1892v1
2 DONALD W. BARNES
the distinct non-zero eigenvalues α, β, γ = α + β and corresponding eigenvectors
u, v, w. If n = 2, 3, then H 6⊆ A(1), so we may suppose n ≥ 4.
For some re-ordering of the basis ofH , we have [a1, . . . , an−3, u, v, w] 6= 0. Denote
the string a1, . . . , an−3 by
a . We apply the Jacobi identity to the product P =
[−→a , an−2, u, [
a , an−1, v, w]]. Since [
a , an−1, v, w] ∈ 〈u〉, P = 0. But
P = [−→a , [−→a , an−2, u, an−1], v, w] + [
a , an−1, [
a , an−2, u, v], w]
+ [−→a , an−1, v, [
a , an−2, u, w]]
= [−→a , αu, v, w] + 0 + 0
since [−→a , an−2, u, v] ∈ 〈w〉 and [
a , an−2, u, w] ∈ 〈v〉. Therefore α = 0 contrary to
the definition of α. Thus this case cannot arise. �
References
1. R. Bai, X. Wang, H. An and W. Xiao, The classification of 5-dimensional 3-Lie algebras over
the field of characteristic 2, private communication to the author.
2. D. W. Barnes, Engel subalgebras of n-Lie algebra, Acta Math. Sinica, Series B, to appear.
arXiv:math.RA/0610347.
1 Little Wonga Rd, Cremorne NSW 2090 Australia
E-mail address: donwb@iprimus.com.au
http://arxiv.org/abs/math/0610347
	References
