Checking $sum_beta in Phi^+ langle beta, alpha^vee rangle = 2$ for a simple root $alpha$

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Let $mathfrak g$ be a semisimple complex Lie algebra and $Phi$ a set of simple roots. Let $alpha in Phi$, I want to know why $$ sum_beta in Phi^+ langle beta, alpha^vee rangle = 2$$



I checked it by hands for $mathfrak sl_n$ for $n=2,3,4$. I don't know how to generalize and this is stated without proof in several places for arbitrary semisimple Lie algebra so I'm sure I miss something. Any hints is appreciated.




lie-algebras root-systems




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    up vote
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    down vote

    favorite












    Let $mathfrak g$ be a semisimple complex Lie algebra and $Phi$ a set of simple roots. Let $alpha in Phi$, I want to know why $$ sum_beta in Phi^+ langle beta, alpha^vee rangle = 2$$



    I checked it by hands for $mathfrak sl_n$ for $n=2,3,4$. I don't know how to generalize and this is stated without proof in several places for arbitrary semisimple Lie algebra so I'm sure I miss something. Any hints is appreciated.




    lie-algebras root-systems




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Let $mathfrak g$ be a semisimple complex Lie algebra and $Phi$ a set of simple roots. Let $alpha in Phi$, I want to know why $$ sum_beta in Phi^+ langle beta, alpha^vee rangle = 2$$



      I checked it by hands for $mathfrak sl_n$ for $n=2,3,4$. I don't know how to generalize and this is stated without proof in several places for arbitrary semisimple Lie algebra so I'm sure I miss something. Any hints is appreciated.




      lie-algebras root-systems




      share|cite|improve this question











      Let $mathfrak g$ be a semisimple complex Lie algebra and $Phi$ a set of simple roots. Let $alpha in Phi$, I want to know why $$ sum_beta in Phi^+ langle beta, alpha^vee rangle = 2$$



      I checked it by hands for $mathfrak sl_n$ for $n=2,3,4$. I don't know how to generalize and this is stated without proof in several places for arbitrary semisimple Lie algebra so I'm sure I miss something. Any hints is appreciated.





      lie-algebras root-systems





      share|cite|improve this question










      share|cite|improve this question




      share|cite|improve this question









      asked Jul 26 at 11:29









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