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The following is related to the article "Review of AdS/CFT Integrability, Chapter VI.1: Superconformal Symmetry" here

The superconformal group is ${\rm PSU}(2,2|4)$ with corresponding Lie algebra $\mathfrak{psu}(2,2|4)$

In physical models, multiplets of states transform under unitary representations of the symmetry algebra.

Question: Perhaps this is just confusion arising from the physicist convention of using groups and algebras interchangeably, but shouldn't we be looking for unitary representations of the group, or rather its universal cover, instead of the algebra?

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What is meant is that the algebra is the set of transformations, called derivations sometimes, which define a unitary group. The elementary case is where Hermitean operators generate unitary groups. This is seen with $U~=~e^{-iHt/\hbar}$ so that $U^\dagger U~=~1$ for the parameter or time very small we have $U~\simeq~1~-~iHt/\hbar$ then $$ U^\dagger U~=~1~+~i(H^\dagger~-~H)t/\hbar~+~O(t^2). $$ The $O(t)$ term must be zero which means $H^\dagger~=~H$.

I am not sure what level of answer is needed here. The projective group $PSU(2,2|4)$ is graded and has signature $+,+,-,-$ on its rank. The algebra then involve Bogoliubov or related transformations that in a sense make the group unitary.

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