Let $G$ be an algebraic group over a field $k$. Then we can define the loop group $LG$ to be the sheaf which takes a $k$-algebra $R$ and spits out $G(R((t)))$. My question is, why is this called the loop group? If one takes $k = \mathbb{C}$, then is there a relation between this group and the "topological" loop group $\mathrm{Hom}(S^1, G(\mathbb{C}))$?
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1I think what you are asking about is a result of Pressley--Segal. For instance, see Theorem 1.6.1 of Xinwen Zhu's notes: https://arxiv.org/pdf/1603.05593.pdf – Alex Youcis Dec 11 '23 at 01:44
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I see thank you. – Calculus101 Dec 11 '23 at 04:17