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I previously determined that the Lie algebra of $O(3,\mathbb C)$ is the set of skew symmetric matrices and that the Lie algebra of $SL_2(\mathbb C)$ is the set of traceless matrices. I am now trying to show that the two Lie algebras are isomorphic. But the calculations blew up and I'm now not so sure anymore if I'm on the right track.

Here is my idea:

Let $\mathfrak o$ denote the Lie algebra of $O(3,\mathbb C)$ and $\mathfrak g$ denote the Lie algebra of $SL_2$.

Define a map $\varphi : \mathfrak g \to \mathfrak o$ by defining $$ g = (p,q) \mapsto (P,Q,P\times Q) $$ where $p,q$ are the columns of $g$ and $P = (p,0), Q=(q,0)$. (that's padding with one $0$)

First, I am trying to show that it's in fact ahomomorphism:

$$ \begin{align} \varphi(p+p', q+q') &= (P+P', Q+Q', (P+P')\times (Q+Q')) \\ &= (P, Q+Q', (P+P')\times (Q+Q')) + (P', Q+Q', (P+P')\times (Q+Q')) \\ &= (P, Q, (P+P')\times (Q+Q')) + (P, Q', (P+P')\times (Q+Q')) + (P', Q, (P+P')\times (Q+Q')) + (P', Q', (P+P')\times (Q+Q')) \\ &= (P, Q, P\times (Q+Q')) +(P, Q, P' \times (Q+Q')) + (P, Q', P\times (Q+Q')) \\ &+ (P, Q', P'\times (Q+Q')) + (P', Q, P\times (Q+Q'))+(P', Q, P' \times (Q+Q')) \\ &+ (P', Q', P\times (Q+Q')) + (P', Q', P'\times (Q+Q')) \end{align}$$

But this is only getting bigger when in fact it should eventually become a sum of just 2 terms.

Does my idea work or is it wrong?

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  • $\begingroup$ The image of the map $\varphi$ doesn't appear to be contained in $\mathfrak{o}$, as the upper-left $2 \times 2$ minor of $\varphi(g)$ can be any tracefree matrix, but not all of these are skew-symmetric. $\endgroup$ – Travis Feb 1 '15 at 5:21
  • $\begingroup$ @Travis Good point, thank you for pointing it out. I will find a different candidate for a map. $\endgroup$ – learner Feb 1 '15 at 5:23
  • $\begingroup$ Without looking into the details, I'd say your idea is probably wrong. The point is you don't even start to give a motivation for doing things this way, so why should one expect it to be right? $\endgroup$ – Marc van Leeuwen Feb 1 '15 at 5:27
  • $\begingroup$ @learner: You're welcome. Since the dimension here is so small, it's a reasonable strategy just to write out a basis of, e.g., $\mathfrak{o}(3, \mathbb{C})$ and find linear combinations of these that obey the bracket relations of $\mathfrak{sl}(2, \mathbb{R})$, bearing in mind that the coefficients in these linear combinations need not be real. $\endgroup$ – Travis Feb 1 '15 at 5:28
  • $\begingroup$ @Travis That seems quite involved, a basis of $\mathfrak o$ contains 9 matrices, right? Could you please elaborate on what you mean by the bracket relations of $\mathfrak{sl}$? Do you mean that to find the map I have to think about how to make sure that $$ \varphi([X,Y])=[\varphi(X), \varphi(Y)]$$ for the Lie bracket? $\endgroup$ – learner Feb 1 '15 at 6:33

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