Shuffle algebras for quivers as quantum groups
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208_2024_2989_ReferencePDF.pdf
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Author(s)
Neguț, Andrei
Sala, Francesco
Schiffmann, Olivier
Date Issued
September 20, 2024
Journal
Mathematische Annalen
Publisher
Springer Berlin Heidelberg
Citation
Neguț, A., Sala, F. & Schiffmann, O. Shuffle algebras for quivers as quantum groups. Math. Ann. 391, 2981–3021 (2025).
Version
Author's final manuscript
Abstract
We define a quantum loop group U Q + associated to an arbitrary quiver Q = ( I , E ) and maximal set of deformation parameters, with generators indexed by I × Z and some explicit quadratic and cubic relations. We prove that U Q + is isomorphic to the (generic, small) shuffle algebra associated to the quiver Q and hence, by Neguț (Shuffle algebras for quivers and wheel conditions. arXiv:2102.11269 ), to the localized K-theoretic Hall algebra of Q. For the quiver with one vertex and g loops, this yields a presentation of the spherical Hall algebra of a (generic) smooth projective curve of genus g [invoking the results of Schiffmann and Vasserot (Math Ann 353(4):1399–1451, 2012)]. We extend the above results to the case of non-generic parameters satisfying a certain natural metric condition. As an application, we obtain a description by generators and relations of the subalgebra generated by absolutely cuspidal eigenforms of the Hall algebra of an arbitrary smooth projective curve [(invoking the results of Kapranov et al. (Sel Math (NS) 23(1):117–177, 2017)].
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00208-024-02989-4
