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Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
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Circle representations and regular representations
Write $\mathbb T$ for the circle group, $C_n$ for the cyclic group of order $n$, and fix embeddings $C_n \hookrightarrow \mathbb T$. Let $\lambda^i$ be the $\mathbb T$-representation with underlying v …