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 Jan 4 at 22:21
user366818user366818
933410
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 Jan 4 at 22:21
user366818user366818
933410
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 Jan 4 at 22:21
user366818user366818
933410
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 Jan 4 at 22:21
user366818user366818
933410
asked Jan 4 at 22:21
user366818user366818
933410
asked Jan 4 at 22:21
user366818user366818
933410
asked Jan 4 at 22:21
user366818user366818
933410
asked Jan 4 at 22:21
user366818user366818
933410
933410
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 Jan 4 at 22:29
Pierre-Guy PlamondonPierre-Guy Plamondon
8,76011639
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
Jan 5 at 14:27
@user366818 Apart from what you say in your post, I don't see any general rule that can be applied.
– Pierre-Guy Plamondon
Jan 5 at 21:14
add a comment |
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1 Answer
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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 Jan 4 at 22:29
Pierre-Guy PlamondonPierre-Guy Plamondon
8,76011639
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
Jan 5 at 14:27
@user366818 Apart from what you say in your post, I don't see any general rule that can be applied.
– Pierre-Guy Plamondon
Jan 5 at 21:14
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 Jan 4 at 22:29
Pierre-Guy PlamondonPierre-Guy Plamondon
8,76011639
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
Jan 5 at 14:27
@user366818 Apart from what you say in your post, I don't see any general rule that can be applied.
– Pierre-Guy Plamondon
Jan 5 at 21:14
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 Jan 4 at 22:29
Pierre-Guy PlamondonPierre-Guy Plamondon
8,76011639
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 Jan 4 at 22:29
Pierre-Guy PlamondonPierre-Guy Plamondon
8,76011639
answered Jan 4 at 22:29
Pierre-Guy PlamondonPierre-Guy Plamondon
8,76011639
answered Jan 4 at 22:29
Pierre-Guy PlamondonPierre-Guy Plamondon
8,76011639
answered Jan 4 at 22:29
Pierre-Guy PlamondonPierre-Guy Plamondon
8,76011639
8,76011639
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
Jan 5 at 14:27
@user366818 Apart from what you say in your post, I don't see any general rule that can be applied.
– Pierre-Guy Plamondon
Jan 5 at 21:14
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
Jan 5 at 14:27
@user366818 Apart from what you say in your post, I don't see any general rule that can be applied.
– Pierre-Guy Plamondon
Jan 5 at 21:14
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
Jan 5 at 14:27
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
Jan 5 at 14:27
@user366818 Apart from what you say in your post, I don't see any general rule that can be applied.
– Pierre-Guy Plamondon
Jan 5 at 21:14
@user366818 Apart from what you say in your post, I don't see any general rule that can be applied.
– Pierre-Guy Plamondon
Jan 5 at 21:14
add a comment |
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