# The Killing forms of $\mathfrak{su}(n)$ and $\mathfrak{sl}(n,\Bbb R)$ are not isomorphic (as real Lie algebras)

Show that $\mathfrak{su}(n)$ and $\mathfrak{sl}(n,\Bbb R)$ are not isomorphic as real Lie algebras.

I calculated that both of them have the Killing form $$K(x,y)=2n \,\textrm{tr}(xy) .$$ I got the hint that I should consider the signature of both Killing forms. Can you tell me how I can use this hint?

• What's the question exactly? Have you computed the signatures of the Killing forms? – Travis May 30 '17 at 15:44
• I want to show that su(n) and sl(n,R) are not isomorphic. But I don't know how I can calculate the signature. Second I dont know why different signatures implie that these Lie algebras are not isomorphic. – user95 May 30 '17 at 15:52
• The Killing form is natural, so if you have a Lie algebra isomorphism $\Phi: \frak g \to \frak h$, then the Killing forms are related by $K_{\frak g} = \Phi^* K_{\frak_h}$. In particular, the signatures of $K_{\frak g}$ and $K_{\frak h}$ must coincide. – Travis May 30 '17 at 16:07

## 2 Answers

Hint For $X \in \mathfrak{su}(n)$ we have $X = -{\bar X}^{\top}$ and so $$K(X, X) = 2n\, \textrm{tr}(X^2) = -2n\, \textrm{tr}(\bar X^{\top} X) .$$ Can you show that this quantity is nonpositive?

• We have $tr(\bar{X}^TX)=\sum_{j=1}^n\sum_{i=1}^n \bar{x_{ij}}x_{ij}=\sum_{j=1}^n\sum_{i=1}^n |x_{ij}|^2\geq0$. Thus $K(X,X)\leq0$. But the Killingform of sl(2,R) is not negative definite and thus they are not isomorphic, right? – user95 May 30 '17 at 16:32
• Looks good to me. Note that to show that the Killing form of $\frak{sl}(2, \Bbb R)$ is not negative definite, it is enough find some $X \in \frak{sl}(2, \Bbb R)$ such that $K_{\frak{sl}(2, \Bbb R)}(X, X) > 0$. – Travis May 30 '17 at 17:51

One can also show that $\mathfrak{su}(n)$ and $\mathfrak{sl}(n,\Bbb R)$ are not isomorphic without using the Killing form. One algebra has subalgebras of a certain dimension, the other one does not - see the following question:

Furthermore, one algebra is compact, the other one not (here one could use the Killing form, which is negative definite for one algebra, but not for the other).