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

In the present paper, we give a definition of the quantum group ${\mathbf{U}}_{\upsilon }(\mathfrak{s}\mathfrak{l}(S^1))$ of the circle $S^1\colon =$\mathbb{R}$/$\mathbb{Z}$$, and its fundamental representation. Such a definition is motivated by a realization of a quantum group ${\mathbf{U}}_{\upsilon }(\mathfrak{s}\mathfrak{l}(S_{\mathbb{Q}}^1))$ associated to the rational circle $S_{\mathbb{Q}}^1\text{\hspace{0.17em}⁣}:=\mathbb{Q}\mathrm{/}\mathbb{Z}$ as a direct limit of ${\mathbf{U}}_{\upsilon }(\widehat{\mathfrak{s}\mathfrak{l}}(n))$'s, where the order is given by divisibility of positive integers. The quantum group ${\mathbf{U}}_{\upsilon }(\mathfrak{s}\mathfrak{l}(S_{\mathbb{Q}}^1))$ arises as a subalgebra of the Hall algebra of coherent sheaves on the infinite root stack $X_{\mathrm{\infty }}$ 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 ${\mathbf{U}}_{\upsilon }(\mathfrak{s}\mathfrak{l}(S_{\mathbb{Q}}^1))$. Moreover, we show that ${\mathbf{U}}_{\upsilon }(\widehat{\mathfrak{s}\mathfrak{l}}(+\mathrm{\infty }))$ and ${\mathbf{U}}_{\upsilon }(\widehat{\mathfrak{s}\mathfrak{l}}(\mathrm{\infty }))$ are subalgebras of ${\mathbf{U}}_{\upsilon }(\mathfrak{s}\mathfrak{l}(S_{\mathbb{Q}}^1))$. As proved by T. Kuwagaki in the appendix, the quantum group ${\mathbf{U}}_{\upsilon }(\mathfrak{s}\mathfrak{l}(S^1))$ naturally arises as well in the mirror dual picture, as a Hall algebra of constructible sheaves on the circle $S^1$.