Subgraphs of Dynkin Diagrams
Am I right in thinking that if we have two semisimple Lie Algebras $mathfrak{g} $ and $mathfrak{h}$ with respective Dynkin Diagrams $A$ and $B$, we may find an injective homomorphism of Lie Algebras $phi: mathfrak{g} rightarrow mathfrak{h}$ iff $A$ is a subgraph of $B$?
lie-algebras root-systems dynkin-diagrams
asked yesterday
user366818
877410
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Am I right in thinking that if we have two semisimple Lie Algebras $mathfrak{g} $ and $mathfrak{h}$ with respective Dynkin Diagrams $A$ and $B$, we may find an injective homomorphism of Lie Algebras $phi: mathfrak{g} rightarrow mathfrak{h}$ iff $A$ is a subgraph of $B$?
lie-algebras root-systems dynkin-diagrams
asked yesterday
user366818
877410
add a comment |
Am I right in thinking that if we have two semisimple Lie Algebras $mathfrak{g} $ and $mathfrak{h}$ with respective Dynkin Diagrams $A$ and $B$, we may find an injective homomorphism of Lie Algebras $phi: mathfrak{g} rightarrow mathfrak{h}$ iff $A$ is a subgraph of $B$?
lie-algebras root-systems dynkin-diagrams
asked yesterday
user366818
877410
Am I right in thinking that if we have two semisimple Lie Algebras $mathfrak{g} $ and $mathfrak{h}$ with respective Dynkin Diagrams $A$ and $B$, we may find an injective homomorphism of Lie Algebras $phi: mathfrak{g} rightarrow mathfrak{h}$ iff $A$ is a subgraph of $B$?
lie-algebras root-systems dynkin-diagrams
lie-algebras root-systems dynkin-diagrams
asked yesterday
user366818
877410
asked yesterday
user366818
877410
asked yesterday
user366818
877410
asked yesterday
user366818
877410
asked yesterday
user366818
877410
877410
add a comment |
add a comment |
1 Answer
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No. For example, the Lie algebra of type $D_4$ is the algebra $mathfrak{so}_8$ of $8times 8$ skew-symmetric matrices. It is contained in the Lie algebra of type $A_7$, that is, the algebra $mathfrak{sl}_8$ of $8times 8$ trace-zero matrices, even though $D_4$ is not a subgraph of $A_7$.
answered yesterday
Pierre-Guy Plamondon
8,75011639
Thank you for this response. In general, is there any information that you can gain from comparing Dynkin Diagrams as to whether or not one may be injected into the other?
– user366818
22 hours ago
@user366818 Apart from what you say in your post, I don't see any general rule that can be applied.
– Pierre-Guy Plamondon
15 hours ago
add a comment |
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1 Answer
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1 Answer
1
active
oldest
votes
active
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active
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No. For example, the Lie algebra of type $D_4$ is the algebra $mathfrak{so}_8$ of $8times 8$ skew-symmetric matrices. It is contained in the Lie algebra of type $A_7$, that is, the algebra $mathfrak{sl}_8$ of $8times 8$ trace-zero matrices, even though $D_4$ is not a subgraph of $A_7$.
answered yesterday
Pierre-Guy Plamondon
8,75011639
Thank you for this response. In general, is there any information that you can gain from comparing Dynkin Diagrams as to whether or not one may be injected into the other?
– user366818
22 hours ago
@user366818 Apart from what you say in your post, I don't see any general rule that can be applied.
– Pierre-Guy Plamondon
15 hours ago
add a comment |
No. For example, the Lie algebra of type $D_4$ is the algebra $mathfrak{so}_8$ of $8times 8$ skew-symmetric matrices. It is contained in the Lie algebra of type $A_7$, that is, the algebra $mathfrak{sl}_8$ of $8times 8$ trace-zero matrices, even though $D_4$ is not a subgraph of $A_7$.
answered yesterday
Pierre-Guy Plamondon
8,75011639
Thank you for this response. In general, is there any information that you can gain from comparing Dynkin Diagrams as to whether or not one may be injected into the other?
– user366818
22 hours ago
@user366818 Apart from what you say in your post, I don't see any general rule that can be applied.
– Pierre-Guy Plamondon
15 hours ago
add a comment |
No. For example, the Lie algebra of type $D_4$ is the algebra $mathfrak{so}_8$ of $8times 8$ skew-symmetric matrices. It is contained in the Lie algebra of type $A_7$, that is, the algebra $mathfrak{sl}_8$ of $8times 8$ trace-zero matrices, even though $D_4$ is not a subgraph of $A_7$.
answered yesterday
Pierre-Guy Plamondon
8,75011639
No. For example, the Lie algebra of type $D_4$ is the algebra $mathfrak{so}_8$ of $8times 8$ skew-symmetric matrices. It is contained in the Lie algebra of type $A_7$, that is, the algebra $mathfrak{sl}_8$ of $8times 8$ trace-zero matrices, even though $D_4$ is not a subgraph of $A_7$.
answered yesterday
Pierre-Guy Plamondon
8,75011639
answered yesterday
Pierre-Guy Plamondon
8,75011639
answered yesterday
Pierre-Guy Plamondon
8,75011639
answered yesterday
Pierre-Guy Plamondon
8,75011639
8,75011639
Thank you for this response. In general, is there any information that you can gain from comparing Dynkin Diagrams as to whether or not one may be injected into the other?
– user366818
22 hours ago
@user366818 Apart from what you say in your post, I don't see any general rule that can be applied.
– Pierre-Guy Plamondon
15 hours ago
add a comment |
Thank you for this response. In general, is there any information that you can gain from comparing Dynkin Diagrams as to whether or not one may be injected into the other?
– user366818
22 hours ago
@user366818 Apart from what you say in your post, I don't see any general rule that can be applied.
– Pierre-Guy Plamondon
15 hours ago
Thank you for this response. In general, is there any information that you can gain from comparing Dynkin Diagrams as to whether or not one may be injected into the other?
– user366818
22 hours ago
Thank you for this response. In general, is there any information that you can gain from comparing Dynkin Diagrams as to whether or not one may be injected into the other?
– user366818
22 hours ago
@user366818 Apart from what you say in your post, I don't see any general rule that can be applied.
– Pierre-Guy Plamondon
15 hours ago
@user366818 Apart from what you say in your post, I don't see any general rule that can be applied.
– Pierre-Guy Plamondon
15 hours ago
add a comment |
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