Classification of metric Lie Algebras of dimension 7?
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I want to classify all metric two-step Lie algebras of dimension 7. Clearly, the dimension of the center of such Lie algebras is $leq 5$. So, I have to consider 5 cases separately, where the dimension of center is $1,2,cdots text{or}, 5$. Let $mathfrak{g}$ is a metric seven dimensional Lie algebra and $mathfrak{h}$ is its one dimensional center. We consider $mathfrak{a}_3$ is a $3-$dimensional vector subspace of $mathfrak{g}$ such that $[mathfrak{a}_3 ,mathfrak{a}_3]=mathfrak{h}$. There is a unique vector subspace $mathfrak{b}_3$ of $mathfrak{a}$ which is complementary subspace of $mathfrak{a}_3$ and commutes with $mathfrak{a}_3$. Then one can choose ${e_1,cdots ,e_7}$ an orthonormal basis of $mathfrak{g}$ such that $mathfrak{h}=text{span}langle e_7rangle$ and ${e_1,e_2,e_3}$ is an orthonormal basis of $mathfrak{a}_3$. We have
$$[e_i,e_j]=lambda_{ij} e_7,qquad i,jin {1,cdots,6}$$
where $0neq lambda_{ij}=-lambda_{ji}$. I want to find a new orthonormal basis to show that $mathfrak{g}$ is metric Heisenberg algebra.
Any suggestion is highly appreciated?
lie-groups lie-algebras
asked Jan 1 at 10:43
AlbertAlbert
344
$endgroup$
add a comment |
$begingroup$
I want to classify all metric two-step Lie algebras of dimension 7. Clearly, the dimension of the center of such Lie algebras is $leq 5$. So, I have to consider 5 cases separately, where the dimension of center is $1,2,cdots text{or}, 5$. Let $mathfrak{g}$ is a metric seven dimensional Lie algebra and $mathfrak{h}$ is its one dimensional center. We consider $mathfrak{a}_3$ is a $3-$dimensional vector subspace of $mathfrak{g}$ such that $[mathfrak{a}_3 ,mathfrak{a}_3]=mathfrak{h}$. There is a unique vector subspace $mathfrak{b}_3$ of $mathfrak{a}$ which is complementary subspace of $mathfrak{a}_3$ and commutes with $mathfrak{a}_3$. Then one can choose ${e_1,cdots ,e_7}$ an orthonormal basis of $mathfrak{g}$ such that $mathfrak{h}=text{span}langle e_7rangle$ and ${e_1,e_2,e_3}$ is an orthonormal basis of $mathfrak{a}_3$. We have
$$[e_i,e_j]=lambda_{ij} e_7,qquad i,jin {1,cdots,6}$$
where $0neq lambda_{ij}=-lambda_{ji}$. I want to find a new orthonormal basis to show that $mathfrak{g}$ is metric Heisenberg algebra.
Any suggestion is highly appreciated?
lie-groups lie-algebras
asked Jan 1 at 10:43
AlbertAlbert
344
$endgroup$
1
$begingroup$
A classification might be already available. In principle, one can use the work of Ines Kath. Here is a classification of metric nilpotent Lie algebras with commutator dimension $2$, up to dimension $10$. I will try to search more, if you want.
$endgroup$
– Dietrich Burde
Jan 1 at 13:17
add a comment |
$begingroup$
I want to classify all metric two-step Lie algebras of dimension 7. Clearly, the dimension of the center of such Lie algebras is $leq 5$. So, I have to consider 5 cases separately, where the dimension of center is $1,2,cdots text{or}, 5$. Let $mathfrak{g}$ is a metric seven dimensional Lie algebra and $mathfrak{h}$ is its one dimensional center. We consider $mathfrak{a}_3$ is a $3-$dimensional vector subspace of $mathfrak{g}$ such that $[mathfrak{a}_3 ,mathfrak{a}_3]=mathfrak{h}$. There is a unique vector subspace $mathfrak{b}_3$ of $mathfrak{a}$ which is complementary subspace of $mathfrak{a}_3$ and commutes with $mathfrak{a}_3$. Then one can choose ${e_1,cdots ,e_7}$ an orthonormal basis of $mathfrak{g}$ such that $mathfrak{h}=text{span}langle e_7rangle$ and ${e_1,e_2,e_3}$ is an orthonormal basis of $mathfrak{a}_3$. We have
$$[e_i,e_j]=lambda_{ij} e_7,qquad i,jin {1,cdots,6}$$
where $0neq lambda_{ij}=-lambda_{ji}$. I want to find a new orthonormal basis to show that $mathfrak{g}$ is metric Heisenberg algebra.
Any suggestion is highly appreciated?
lie-groups lie-algebras
asked Jan 1 at 10:43
AlbertAlbert
344
$endgroup$
I want to classify all metric two-step Lie algebras of dimension 7. Clearly, the dimension of the center of such Lie algebras is $leq 5$. So, I have to consider 5 cases separately, where the dimension of center is $1,2,cdots text{or}, 5$. Let $mathfrak{g}$ is a metric seven dimensional Lie algebra and $mathfrak{h}$ is its one dimensional center. We consider $mathfrak{a}_3$ is a $3-$dimensional vector subspace of $mathfrak{g}$ such that $[mathfrak{a}_3 ,mathfrak{a}_3]=mathfrak{h}$. There is a unique vector subspace $mathfrak{b}_3$ of $mathfrak{a}$ which is complementary subspace of $mathfrak{a}_3$ and commutes with $mathfrak{a}_3$. Then one can choose ${e_1,cdots ,e_7}$ an orthonormal basis of $mathfrak{g}$ such that $mathfrak{h}=text{span}langle e_7rangle$ and ${e_1,e_2,e_3}$ is an orthonormal basis of $mathfrak{a}_3$. We have
$$[e_i,e_j]=lambda_{ij} e_7,qquad i,jin {1,cdots,6}$$
where $0neq lambda_{ij}=-lambda_{ji}$. I want to find a new orthonormal basis to show that $mathfrak{g}$ is metric Heisenberg algebra.
Any suggestion is highly appreciated?
lie-groups lie-algebras
lie-groups lie-algebras
asked Jan 1 at 10:43
AlbertAlbert
344
asked Jan 1 at 10:43
AlbertAlbert
344
edited Jan 1 at 11:31
Albert
asked Jan 1 at 10:43
AlbertAlbert
344
asked Jan 1 at 10:43
AlbertAlbert
344
asked Jan 1 at 10:43
AlbertAlbert
344
344
1
$begingroup$
A classification might be already available. In principle, one can use the work of Ines Kath. Here is a classification of metric nilpotent Lie algebras with commutator dimension $2$, up to dimension $10$. I will try to search more, if you want.
$endgroup$
– Dietrich Burde
Jan 1 at 13:17
add a comment |
1
$begingroup$
A classification might be already available. In principle, one can use the work of Ines Kath. Here is a classification of metric nilpotent Lie algebras with commutator dimension $2$, up to dimension $10$. I will try to search more, if you want.
$endgroup$
– Dietrich Burde
Jan 1 at 13:17
1
1
$begingroup$
A classification might be already available. In principle, one can use the work of Ines Kath. Here is a classification of metric nilpotent Lie algebras with commutator dimension $2$, up to dimension $10$. I will try to search more, if you want.
$endgroup$
– Dietrich Burde
Jan 1 at 13:17
$begingroup$
A classification might be already available. In principle, one can use the work of Ines Kath. Here is a classification of metric nilpotent Lie algebras with commutator dimension $2$, up to dimension $10$. I will try to search more, if you want.
$endgroup$
– Dietrich Burde
Jan 1 at 13:17
add a comment |
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A classification might be already available. In principle, one can use the work of Ines Kath. Here is a classification of metric nilpotent Lie algebras with commutator dimension $2$, up to dimension $10$. I will try to search more, if you want.
$endgroup$
– Dietrich Burde
Jan 1 at 13:17