It is known that Lie superalgebra $\mathfrak{su}(2|2)$ (and only this one, not arbitrary $\mathfrak{su}(n|n)$) has the nontrivial central extension which forms an $\mathfrak{sl}_2$ triplet, let's call it $(\mathfrak{C},\mathfrak{P},\mathfrak{K})$. The easiest way to see this is to start from the superalgebra $\mathfrak{d}(2,1,\alpha)$ and take $\alpha=0$. An $\mathfrak{sl}_2$ automorphism allows one to rotate the vector $(\mathfrak{C},\mathfrak{P},\mathfrak{K})$ and transform it to e.g. $(\mathfrak{C}',0,0)$ for some $\mathfrak{C}'$. It turns out that representation theory of the algebra with the central extension $(\mathfrak{C}',0,0)$ (when two elements vanish) is much easier than the one with a full $(\mathfrak{C},\mathfrak{P},\mathfrak{K})$.

Now let us consider quantum deformation of of this algebra - $U_q(\mathfrak{su}(2|2))$. The question is how to generalize the outer automorphism to this case if one exists at all. I need this to build up a representation theory for the above superalgebra.

I understand that the question is technical, it's hard to realize its complexity without doing any explicit calculations, but, just in case, if anybody thought about something related, please let me know.


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