The Hodge dual and the Moyal product related or just notation? The Hodge star operator $\star$ is a linear map between $\bigwedge ^pV$ and $\bigwedge ^{n-p}V$ for an inner product space $V$ of dimension $n$. 
So we can we write;
\begin{equation}
\lambda\in \bigwedge ^p V
\end{equation}
\begin{equation}
\star\lambda\in\bigwedge^{n-p}V
\end{equation}
I am wondering is this the same operation as used in the Moyal bracket for functions in phase space? Namely for two functions of the phase space $f$ and $g$, the Moyal bracket is given by;
\begin{equation}
\{f,g\}:=\frac{1}{i\hbar}(f\star g-g\star f)
\end{equation}
I think I'm wrong and that it is somehow a different operation with the same sign, but would really appreciate some help since I'm really not familiar with the Hodge operator other than what I have written above! 
Also if its not too much trouble, could anyone provide a bit of context to the Hodge star operation in physics? e.g. why should I really be interested in vectors in $\bigwedge ^{n-p}V$ space?  
 A: The Moyal bracket is the calculational manifestation of noncommutative geometry in rough terms. Essentially, to do math over a noncommutative geometry one simply replaces regular products with $\star$ products which encode the deformation considered. However, much of the calculation goes through more or less the same.
The Hodge $\star$ is completely different. For your interests, let's discuss spacetime. In that case $\star$ swaps a $p$-form for a $4-p$ form. For example, the $1$-form $j$ would have $\star j$ a $3$-form. On the other hand, $2$-forms Hodge dual to $4-2=2$-forms once again. Depending on some sign-conventions ( I could be off by a minus or two in what follows and I might be missing some other dimensional constants so consult your favorite physics text before doing anything with what I'm about to say). The Faraday tensor can be expressed as
$$ F = dt \wedge \omega_E+ \Phi_B $$
Where $\omega$ is the so-called work-form mapping with $\omega_E = E_xdx+E_ydy+E_zdz$ and $\Phi$ is the flux-form mapping given by $\Phi_B = B_xdy \wedge dz+B_y dz \wedge dx+B_zdx \wedge dy$. Basically, the Hodge dual takes $dt \wedge dx \wedge dy \wedge dz$ and whatever you give it gives you back whatever else is in the list. For example, $\star dx = \pm dt \wedge dy \wedge dz$ (again, the $\pm$ because I don't want to get into the signs here). Or $\star dt \wedge dx = dy \wedge dz$. It follows:
$$ \star F = dt \wedge \omega_B+ \Phi_E $$
The Hodge star swaps the $E$ and $B$ fields, well, not quite, there is a sign I am not making explicit here. But, perhaps this helps. I found the classic Gravitation by MTW to be helpful when first learning this material. It has quite a bit on the Hodge star.
