I posted the following question on MSE but I was advised to ask the question here.


Let $\mathfrak{g}$ be a lie algebra. Consider the anti-automomorphism 


$$\phi: \mathfrak{g} \rightarrow \mathfrak{g} $$ such that $$\phi(X) = X^T, \quad \forall \quad X \in \mathfrak{g}$$ where $X^T$ is the usual transpose of $X$.


Observe that $$\phi([X,Y]) = \phi(XY-YX) = (XY-YX)^T = (XY)^T-(YX)^T.$$ Thus, we have that
$$\phi([X,Y)] = Y^TX^T- X^TY^T = [\phi(Y), \phi(X)].$$


Note that we can extend this anti-automorphism to the universal enveloping algebra $\mathfrak{Ug}$ such that $\phi : \mathfrak{Ug} \rightarrow \mathfrak{Ug}$ still denoted by $\phi.$


**My question:**
 Show that $$\phi(a)= a \quad$$ for all $a \in Z\mathcal{g} $ where $Z\mathcal{g}$ denotes the centre of $\mathfrak{Ug}.$


I thought about using the Harish-Chandra isomorphim but I dont seem to see how this helps.



On the centre of the universal enveloping algebra
$\newcommand{\g}{\mathfrak g}$
François is right that it doesn't quite make sense to talk about transpose for a general Lie algebra. Since you mention the Harish-Chandra isomorphism I assume you want $\g$ to be a simple Lie algebra. In that case there is an anti-involution $\omega$ given by $e\_i\mapsto f\_i$, $f\_i \mapsto e\_i$, $h\_i \mapsto h\_i$ for a choice of Chevalley generators $e\_i,f\_i,h\_i$. If $\g$ is $\mathfrak{gl}\_n$ this recovers the transpose, so I'll assume your question is about that.


I feel there should be a much simpler argument but here's an idea involving the Harish-Chandra map as you suggest. Let $\g=\mathfrak{n}^+ \oplus \mathfrak{h}\oplus \mathfrak{n}^-$ be the decomposition induced by your choice of generators. By PBW this induces a decomposition
$$U(\g)= U(\mathfrak{h})\oplus (\mathfrak{n}^-U(\g)+ U(\g)\mathfrak{n}^+).$$
Let $p$ be the projection on $U(\mathfrak{h})$ relative to this decomposition. 


I claim that $p\circ \omega=p$. Now, the restriction of $p$ to the center of $U(\g)$ is actually injective: this is essentially a reformulation of Harish-Chandra. The result you want follows.


