The circle quantum group and the infinite root stack of a curve
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Francesco Sala [1] ; Olivier Schiffmann [2]
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[1] University of Tokyo
University of Tokyo
Japón
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[2] Université de Paris-Sud Paris-Saclay, Francia
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- Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 25, Nº. 5, 2019
- Idioma: inglés
- DOI: 10.1007/s00029-019-0521-8
- Enlaces
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Resumen
In the present paper, we give a definition of the quantum group Uυ(sl(S1)) of the circle S1:=R/Z, and its fundamental representation. Such a definition is motivated by a realization of a quantum group Uυ(sl(S1Q)) associated to the rational circle S1Q:=Q/Z as a direct limit of Uυ(slˆ(n))’s, where the order is given by divisibility of positive integers. The quantum group Uυ(sl(S1Q)) 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 gX, of Uυ(sl(S1Q)). Moreover, we show that Uυ(slˆ(+∞)) and Uυ(slˆ(∞)) are subalgebras of Uυ(sl(S1Q)). As proved by T. Kuwagaki in the appendix, the quantum group Uυ(sl(S1)) naturally arises as well in the mirror dual picture, as a Hall algebra of constructible sheaves on the circle S1.
