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, 25:77 | Cite as

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

  • Francesco SalaEmail author
  • Olivier Schiffmann
Article
  • 17 Downloads

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 \(\mathbf {U}_\upsilon (\mathfrak {sl}(S^1_{\mathbb {Q}}))\) 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 \(S^1\).

Keywords

Hall algebras Quantum groups Shuffle algebras Root stacks 

Mathematics Subject Classification

Primary: 17B37 17B67 22E65 Secondary: 14A20 

Notes

Acknowledgements

The authors are grateful to Mattia Talpo for many enlightening conversations about root stacks and their categories of coherent sheaves, to Jyun-Ao Lin for helpful explanations about his work and to the referee for a careful reading and useful comments. They also thank Andrea Appel and Mikhail Kapranov for helpful discussions and comments; and David Hernandez for pointing out his paper [25]. Last but not least, they thank Tatsuki Kuwagaki for writing the Appendix B. The author of the Appendix B thanks Francesco Sala for his intriguing talk at GTM seminar at IPMU, which motivated this note. He also thanks Francesco and Olivier Schiffmann for their kindness to include this note as an appendix of their work. The work of Tatsuki Kuwagaki was supported by World Premier International Research Center Initiative (WPI), MEXT, Japan and JSPS KAKENHI Grant Number JP18K13405.

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Authors and Affiliations

  1. 1.Kavli IPMU (WPI), UTIASThe University of TokyoKashiwa, ChibaJapan
  2. 2.Département de MathématiquesUniversité de Paris-Sud Paris-SaclayOrsay CedexFrance

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