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 $$\mathbf {U}_\upsilon (\mathfrak {sl}(S^1_{\mathbb {Q}}))$$ associated to the rational circle $$S^1_{\mathbb {Q}}:={\mathbb {Q}}/{\mathbb {Z}}$$ 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 $$\mathbf {U}_\upsilon (\mathfrak {sl}(S^1_{\mathbb {Q}}))$$ 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 $$\mathbf {U}_\upsilon (\mathfrak {sl}(S^1_{\mathbb {Q}}))$$. 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. 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 $$S^1$$.