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

Abstract

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 $S^1$.

63 pages, Latex; Introduction largely rewritten, a new section comparing ${\mathbf{U}}_{\upsilon }(\mathfrak{sl}(S_{\mathbb{Q}}^1))$ to other known infinite quantum groups is added, as well as an appendix by T. Kuwagaki giving a mirror dual construction of ${\mathbf{U}}_{\upsilon }(\mathfrak{sl}(S_{\mathbb{Q}}^1))$; v3: 64 pages, Final version published in Selecta Mathematica

References in corpus (4)

• Covariantly functorial wrapped Floer theory on Liouville sectors
• Moduli Spaces of Semistable Sheaves on Projective Deligne-Mumford Stacks
• Fock space representation of the circle quantum group
• Continuum Kac-Moody algebras

Cited by in corpus (1)

• Gluing semi-orthogonal decompositions