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I am asked to prove that the Lie algebras $\mathfrak{sl}_2(\mathbb C), \; \mathfrak{so}_3(\mathbb C)$ are Lie isomorphic to each other.

We have the standard basis for $\mathfrak{sl}_2(\mathbb C)$ to be $\{e = \left(\begin{matrix}0 & 1 \\ 0 & 0\end{matrix}\right), \; f = \left(\begin{matrix}0 & 0 \\ 1 & 0\end{matrix}\right), \; h = \left(\begin{matrix}1 & 0 \\ 0 & -1\end{matrix}\right)\}$

which satisfy the bracket relations $[e,f] = h, \; [h,e] = 2e,\; [h,f] = - 2f$

After a little work, I found the basis of $\mathfrak{so}_3(\mathbb C)$ to be: $\{a = \left(\begin{matrix}0&1&0 \\ 0&0&0 \\ -1&0&0 \end{matrix}\right), \; b = \left(\begin{matrix}0&0&1 \\ -1&0&0 \\ 0&0&0 \end{matrix}\right), \; c = \left(\begin{matrix}0&0&0 \\ 0&1&0 \\ 0&0&-1 \end{matrix}\right) \}$

which I found to have the bracket relations: $[a, b] = c , \; [c, a] = -a, \; [c , b] = b$

I'm now stuck, because I don't know to map the basis $\{e, f, h\}$ to a basis expressed in terms of $\{a, b, c\}$ such that the bracket relations are maintained.

I would appreciate it if someone could highlight how I should augment the basis of $\mathfrak{so}_3(\mathbb C)$ so that I can construct the Lie isomorphism, as well as explain what the general method for finding it might be for similar situations.

Thank you in advance.

user366818
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  • Can you adjust/redefine a,b,c to obey the same Lie algebra as e,f,h? – Cosmas Zachos Oct 12 '18 at 16:42
  • @CosmasZachos I want to try to do that but I don't know how I would go about doing it – user366818 Oct 12 '18 at 17:46
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    Why not try looking for (nonzero) numbers $\lambda, \mu, \nu$ such that $\lambda a, \mu b, \nu c$ obey the same relations as $e,f,h$? Then the map $e\mapsto \lambda a, f \mapsto \mu b, h \mapsto \nu c$ would be a Lie isomorphism. The motivation here is that the relations for a,b,c look like the relations for e,f,h, except for some different scalars, so a map like that is worth a try. – Matthew Towers Oct 12 '18 at 18:02

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