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The Cartan subalgebra of $\mathfrak{sl}_2(\mathbb{C})$ is $$\mathfrak{h} = \mathbb{C} \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$$ which is $1$-dimensional and hence can be identified with $\mathbb{C}$.

Now, given real numbers $\alpha,\beta \in \mathbb{R}$, how can one determine the value of $\langle \alpha,\beta \rangle$ where $\langle \cdot,\cdot \rangle$ is the Killing form?

I imagine the calculation would involve using the killing form to transfer $\mathfrak{h} \to \mathfrak{h}^*$. Is there a quick and easy way to do this?

nigel
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  • That is not really the Cartan subalgebra: it is only a Cartan subalgebra. If you are using that identification, then you should make it explicit in your question! (Or, even better not make it) – Mariano Suárez-Álvarez Sep 17 '15 at 03:08
  • I can't add a comment to the post so I'm 'commenting' here. Shouldn't $\alpha,\beta \in \mathfrak{sl}_2(\mathbb{C})$? The Killing form is a bilinear form on $\mathfrak{sl}_2(\mathbb{C})$ not on $\mathbb{R}$. Apologies if I am missing something here. – Sam Weatherhog Sep 17 '15 at 02:55
  • I am viewing $\alpha$ and $\beta$ as elements of $\mathfrak{h}$, using the identification $\alpha = \alpha \begin{pmatrix} 1 & 0 \ 0 & -1 \end{pmatrix}$ – nigel Sep 17 '15 at 03:06
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    What problem did you encounter using the definition of the Killing form in order to compute it? – Mariano Suárez-Álvarez Sep 17 '15 at 03:09

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With $\alpha$ and $\beta$ defined as $\alpha=\alpha \bigl( \begin{smallmatrix} 1&0\\0&-1\end{smallmatrix} \bigr)$, $\beta=\beta \bigl( \begin{smallmatrix} 1&0\\0&-1\end{smallmatrix} \bigr)$, and taking $x=\bigl(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\bigr)$, $h=\bigl(\begin{smallmatrix}1&0\\0&-1\end{smallmatrix}\bigr)$, $y=\bigl(\begin{smallmatrix}0&0\\1&0\end{smallmatrix}\bigr)$ as a basis for $\mathfrak{sl}_2(\mathbb{C})$, we have:

$$ ad_\alpha = \left(\begin{matrix} 2\alpha & 0 & 0\\ 0&0&0 \\ 0&0&-2\alpha \end{matrix} \right), ad_\beta=\left(\begin{matrix} 2\beta & 0 & 0\\ 0&0&0 \\ 0&0&-2\beta \end{matrix} \right) $$ The usual definition for the Killing form is $\kappa(x,y)=Tr(ad_x \cdot ad_y)$. Using this we get $\kappa(\alpha,\beta)=Tr(ad_\alpha \cdot ad_\beta)=8\alpha\beta$.

Sam Weatherhog
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