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How to write down a presentation of a Lie algebra if we know a set of generators in matrix form? For example, for $sl_2$, if we know $e=(0, 1; 0, 0)$, $f=(0, 0; 1, 0)$ , $h=(1, 0; 0, -1)$, how to write a presentation of $sl_2$ from these matrices?

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up vote 1 down vote accepted

What you need to compute is the Lie bracket of the generators, in this case this is just the commutator. Because of antisymmetry, you just have to compute one direction: \begin{align*} [e,f]&=ef-fe=\begin{pmatrix}0&1\\0&0\end{pmatrix}\begin{pmatrix}0&0\\1&0\end{pmatrix}-\begin{pmatrix}0&0\\1&0\end{pmatrix}\begin{pmatrix}0&1\\0&0\end{pmatrix}\\ &=\begin{pmatrix}1&0\\0&-1\end{pmatrix}=h\\ [e,h]&=eh-he=\begin{pmatrix}0&1\\0&0\end{pmatrix}\begin{pmatrix}1&0\\0&-1\end{pmatrix}-\begin{pmatrix}1&0\\0&-1\end{pmatrix}\begin{pmatrix}0&1\\0&0\end{pmatrix}\\ &=-2e \end{align*} I leave the computation for $[f,h]$ for you.

In googling this question I found a nice paper on the issue how to convert the different possibilities of presenting a Lie algebra into each other. It is Cohen, de Graaf, Ronyai: Computations in finite-dimensional Lie algebras, Discrete Mathematics and Theoretical Computer Science, 1997.

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