Semisimple algebra (Killing form)

  • Thread starter droblly
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I have started learning Lie algebra and I can't understand one example given in the notes.
Given:
[tex] [h_{\alpha},e_{\alpha}] = 2 e_{\alpha} [/tex]
[tex] [h_{\alpha},f_{\alpha}] = -2 f_{\alpha} [/tex]
[tex] [e_{\alpha},f_{\alpha}] = h_{\alpha} [/tex]

and that
[tex]
[x,y] = K(x,y) t_{\alpha}
[/tex]
if [tex]\alpha[/tex] is a root and [tex] x \in L_{\alpha}, y \in L_{-\alpha} [/tex]
Now, the example is application of the theorem to [tex]A_2[/tex].
Generators are
[tex] h_{\alpha} = E_{11} -E_{22} [/tex]
[tex] h_{\beta} = E_{22} -E_{33} [/tex]
[tex] e_{\alpha} = E_{12} [/tex]
[tex] e_{\beta} = E_{23}[/tex]
[tex] e_{-\alpha} = E_{21} [/tex]
[tex]e_{-\beta} = E_{32} [/tex]
and Postive roots are {[tex] \alpha, \beta, \alpha+\beta [/tex]}.

I am meant to check that
1.[tex]\alpha(h_{\alpha}) = \beta(h_{\beta}) =2 [/tex]
2.[tex] \alpha(h_{\beta}) = \beta(h_{\alpha}) =-1 [/tex]

I can't do part (2). Part (1) seems simple:
[tex] \alpha(h_{\alpha}) = K(t_{\alpha},h_{\alpha} )= K(t_{\alpha},2\frac{t_{\alpha}}{K(t_{\alpha},t_{\alpha})}) = 2 [/tex]
My problem is with finding [tex]t_{\alpha}[/tex] and [tex] t_{\beta}[/tex] to calculate [tex]K(t_{\alpha},t_{\alpha})[/tex]. How would one go about doing it?
Because
[tex] \alpha(h_{\beta}) = K(t_{\alpha},h_{\beta} )= K(t_{\alpha},2\frac{t_{\beta}}{K(t_{\beta},t_{\beta})}) = \frac{2}{K(t_{\beta},t_{\beta})} K(t_{\alpha},t_{\beta}) [/tex]
Thanks.

EDIT: I hope I had posted in the right thread. Should I have posted this in HW help?
 
Last edited:
I have got it. It's just calculating the Cartan integers from the basis.
 

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