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

In the present paper, we give a definition of the quantum group 𝐔_υ (𝔰𝔩(S^1)) of the circle S^1:=ℝ/ℤ , and its fundamental representation. Such a definition is motivated by a realization of a quantum group 𝐔_υ (𝔰𝔩(S^1_ℚ)) associated to the rational circle S^1_ℚ:=ℚ/ℤ as a direct limit of 𝐔_υ (𝔰𝔩(n)) ’s, where the order is given by divisibility of positive integers. The quantum group 𝐔_υ (𝔰𝔩(S^1_ℚ)) arises as a subalgebra of the Hall algebra of coherent sheaves on the infinite root stack X_∞ 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 𝐔_υ (𝔰𝔩(S^1_ℚ)) . Moreover, we show that 𝐔_υ (𝔰𝔩(+∞ )) and 𝐔_υ (𝔰𝔩(∞ )) are subalgebras of 𝐔_υ (𝔰𝔩(S^1_ℚ)) . As proved by T. Kuwagaki in the appendix, the quantum group 𝐔_υ (𝔰𝔩(S^1)) naturally arises as well in the mirror dual picture, as a Hall algebra of constructible sheaves on the circle S^1 .