Shuffle algebras for quivers and wheel conditions
Name
10.1515_crelle-2022-0074.pdf
Description
Published version
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623.78 KB
Format
Adobe PDF
Checksum (MD5)
2def8f7460b8e33cc240a1d5b11692b1
Author(s)
NeguĹŁ, Andrei
Date Issued
December 1, 2022
Journal
Journal fĂźr die reine und angewandte Mathematik (Crelles Journal)
Publisher
Walter de Gruyter GmbH
Citation
NeguĹŁ, Andrei. "Shuffle algebras for quivers and wheel conditions" Journal fĂźr die reine und angewandte Mathematik (Crelles Journal), vol. 2023, no. 795, 2023, pp. 139-182.
Version
Final published version
Abstract
We show that the shuffle algebra associated to a doubled quiver (determined by 3-variable wheel conditions) is generated by elements of minimal degree. Together with results of VaragnoloâVasserot and Yu Zhao, this implies that the aforementioned shuffle algebra is isomorphic to the localized đž-theoretic Hall algebra associated to the quiver by Grojnowski, SchiffmannâVasserot and YangâZhao. With small modifications, our theorems also hold under certain specializations of the equivariant parameters, which will allow us in joint work with Sala and Schiffmann to give a generators-and-relations description of the Hall algebra of any curve over a finite field (which is a shuffle algebra due to KapranovâSchiffmannâVasserot). When the quiver has no edge loops or multiple edges, we show that the shuffle algebra, localized đž-theoretic Hall algebra, and the positive half of the corresponding quantum loop group are all isomorphic; we also obtain the non-degeneracy of the Hopf pairing on the latter quantum loop group.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1515/crelle-2022-0074
