I know that representations of a group $G$ are essentially $\mathbb{C}G$-modules. How is it that every $\mathbb{C}G-$module is also equivalent to a $\mathbb{C}^{G}-$comodule. I have not found this relationship in the literature, I would appreciate your help in pointing out how this construction is done.
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$\begingroup$ What is the definition of $\mathbf{C}^G$? $\endgroup$– StephenCommented May 27 at 14:44
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$\begingroup$ @Stephen functions from $G$ to $\mathbb{C}$. $\endgroup$– NoetherNerdCommented May 27 at 16:03
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1$\begingroup$ That's the usual definition of the group ring $\mathbf{C} G$ (with convolution as the multiplication). Perhaps you mean to take $\mathbf{C}^G$ to be the linear dual of the group ring, as a cogebra? In that case comodules for it are the same as modules for the group ring, essentially by definition. $\endgroup$– StephenCommented May 27 at 16:08
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