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Note: For experts, to save your time, you can just read the highlighted texts and Eqs directly. The ordinary group 2-cocycle condition: Let us remind the usual so-called homogeneous group 2-cocycle $...

Let $R_G$ be the representation ring of $G$ a connected, simply connected Lie group, $I_G$ the augmentation ideal and $\mathbb{Z}=R_G/I_G$. $R_G$ acts on $\mathbb{Z}$ via $V \cdot n = (\dim V ) n$. I ...