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 Varagnolo–Vasserot and Yu Zhao, this implies that the aforementioned shuffle algebra is isomorphic to the localized 𝐾-theoretic Hall algebra associated to the quiver by Grojnowski, Schiffmann–Vasserot and Yang–Zhao. 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 Kapranov–Schiffmann–Vasserot). When the quiver has no edge loops or multiple edges, we show that the shuffle algebra, localized 𝐾-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.
Где-то есть люди, для которых теорема верна
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1760264
Award Identifier / Grant number: DMS-1845034
Funding statement: I gratefully acknowledge NSF grants DMS-1760264 and DMS-1845034, as well as support from the Alfred P. Sloan Foundation and the MIT Research Support Committee.
Acknowledgements
I would like to thank Boris Feigin, Francesco Sala, Olivier Schiffmann, Tudor Pădurariu, Alexander Tsymbaliuk (with special thanks for the substantial feedback), Michela Varagnolo, Éric Vasserot, and Yu Zhao for many interesting conversations about 𝐾-theoretic Hall algebras, shuffle algebras, and much more.
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