The circle quantum group and the infinite root stack of a curve

In the present paper, we give a definition of the quantum group $mathbf{U}_\upsilon(mathfrak{sl}(S^1))$ of the circle $S^1: = mathbb{R}/mathbb{Z}$, and its fundamental representation. Such a definition is motivated by a realization of a quantum group $mathbf{U}_\upsilon(mathfrak{sl}(S_{mathbb{Q}}^1))$ associated to the rational circle $S_{mathbb{Q}}^1:=mathbb{Q}/mathbb{Z}$ as a direct limit of $mathbf{U}_\upsilon(widehat{mathfrak{sl}}(n))$’s, where the order is given by divisibility of positive integers. The quantum group $mathbf{U}_\upsilon(mathfrak{sl}(S_{mathbb{Q}}^1))$ arises as a subalgebra of the Hall algebra of coherent sheaves on the infinite root stack $X_infty$ over a fixed smooth projective curve $X$ defined over a finite field. Via this Hall algebra approach, we are able to realize geometrically the fundamental and the tensor representations, and a family of symmetric tensor representations, depending on the genus $g_X$, of $mathbf{U}_\upsilon(mathfrak{sl}(S_{mathbb{Q}}^1))$. Moreover, we show that $mathbf{U}_\upsilon(widehat{mathfrak{sl}}(+infty))$ and $mathbf{U}_\upsilon(widehat{mathfrak{sl}}(infty))$ are subalgebras of $mathbf{U}_\upsilon(mathfrak{sl}(S_{mathbb{Q}}^1))$. As proved by T. Kuwagaki in 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 of constructible sheaves on the circle $S^1$.