Subgraphs of Dynkin Diagrams












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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 $B$?






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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







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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







      share|cite|improve this question













      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






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      asked Jan 4 at 22:21









      user366818user366818

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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 $A_7$.






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          • 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











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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 $A_7$.






          share|cite|improve this answer





















          • 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
















          2














          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 $A_7$.






          share|cite|improve this answer





















          • 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














          2












          2








          2






          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 $A_7$.






          share|cite|improve this answer












          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 $A_7$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          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


















          • 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


















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