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Is there any book or paper that discusses the differential operators corresponding to $SU(N)$ $(N=4,5,...)$ covariant derivatives in details? At least for the case of $N=4,$ the explicit form is of high importance in my work. Recall that these covarinat derivatives basically correspond to generators of the $SU(N)$ right (left) rotations.

Thanks in advance

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    $\begingroup$ I have no idea what you are asking. Do you want the expressions for the left (resp. right) invariant vector fields on the $SU(N)$ manifold? $\endgroup$
    – José Figueroa-O'Farrill
    Commented Jun 28, 2013 at 18:50
  • $\begingroup$ As you are a physicist, I give you the following example. Suppose you want to define topological charge operator on $CP^1$ manifold. To do so, one identifies the $R_{\pm}:=R_1\pm iR_2$ of the $SU(2)$ right rotations as some well-known operators in terms of derivatives wrt coordinates on any given patch of $CP^1$. Then $[R_{+},R_{-}]\Psi=n\Psi$ where $\Psi$ is a line section on the patch. (Take QHE example, and $n$ becomes the monopole charge and etc.). I'm looking for these generators on the $SU(N)$ manifold specially the case $N=4.$ $\endgroup$
    – Alireza Behtash
    Commented Jun 28, 2013 at 23:04
  • $\begingroup$ But in your example what you have are differential operators on $\mathbb{C}P^1$ not on the the 3-sphere. So do you want differential operators on $SU(N)$ or on some other manifold on which $SU(N)$ acts transitively, e.g., $\mathbb{C}P^{N-1}$? $\endgroup$
    – José Figueroa-O'Farrill
    Commented Jun 29, 2013 at 10:42
  • $\begingroup$ As I said, I want differential operators on $SU(N)$. I already have the differential operators on the coset space, which is a Grassmanian (thus Kahlarian)! $\endgroup$
    – Alireza Behtash
    Commented Jun 29, 2013 at 13:54

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