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In the present paper, we give a definition of the quantum group$\mathbf{U}_\upsilon(\mathfrak{sl}(S^1))$ of the circle $S^1\colon=\mathbb{R}/\mathbb{Z}$, and its fundamental representation. Such a definitionis motivated by a realization of a quantum group$\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$ associated to the rationalcircle $S^1_\mathbb{Q}\colon= \mathbb{Q}/\mathbb{Z}$ as a direct limit of$\mathbf{U}_\upsilon(\widehat{\mathfrak{sl}}(n))$'s, where the order is givenby divisibility of positive integers. The quantum group$\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$ arises as a subalgebra ofthe Hall algebra of coherent sheaves on the infinite root stack $X_\infty$ overa fixed smooth projective curve $X$ defined over a finite field. Via this Hallalgebra approach, we are able to realize geometrically the fundamental and thetensor representations, and a family of symmetric tensor representations,depending on the genus $g_X$, of$\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$. Moreover, we show that$\mathbf{U}_\upsilon(\widehat{\mathfrak{sl}}(+\infty))$ and$\mathbf{U}_\upsilon(\widehat{\mathfrak{sl}}(\infty))$ are subalgebras of$\mathbf{U}_\upsilon(\mathfrak{sl}(S^1_\mathbb{Q}))$. As proved by T. Kuwagakiin the appendix, the quantum group $\mathbf{U}_\upsilon(\mathfrak{sl}(S^1))$naturally arises as well in the mirror dual picture, as a Hall algebra ofconstructible sheaves on the circle $S^1$.

The circle quantum group and the infinite root stack of a curve (with an appendix by Tatsuki Kuwagaki)