# Semisimple algebra (Killing form)

1. May 9, 2010

### droblly

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

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

I am meant to check that
1.$$\alpha(h_{\alpha}) = \beta(h_{\beta}) =2$$
2.$$\alpha(h_{\beta}) = \beta(h_{\alpha}) =-1$$

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

EDIT: I hope I had posted in the right thread. Should I have posted this in HW help?

Last edited: May 9, 2010
2. May 9, 2010

### droblly

I have got it. It's just calculating the Cartan integers from the basis.