Shuffle algebras for quivers as quantum groups
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866909331806486528 |
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| author | Neguţ, Andrei Sala, Francesco Schiffmann, Olivier |
| author_facet | Neguţ, Andrei Sala, Francesco Schiffmann, Olivier |
| contents | We define a quantum loop group $\mathbf{U}^+_Q$ associated to an arbitrary quiver $Q=(I,E)$ and maximal set of deformation parameters, with generators indexed by $I \times \mathbb{Z}$ and some explicit quadratic and cubic relations. We prove that $\mathbf{U}^+_Q$ is isomorphic to the (generic, small) shuffle algebra associated to the quiver $Q$ and hence, by [Neg21a], 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 [SV12]). 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 [KSV17]). |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2111_00249 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Shuffle algebras for quivers as quantum groups Neguţ, Andrei Sala, Francesco Schiffmann, Olivier Representation Theory Algebraic Geometry Quantum Algebra We define a quantum loop group $\mathbf{U}^+_Q$ associated to an arbitrary quiver $Q=(I,E)$ and maximal set of deformation parameters, with generators indexed by $I \times \mathbb{Z}$ and some explicit quadratic and cubic relations. We prove that $\mathbf{U}^+_Q$ is isomorphic to the (generic, small) shuffle algebra associated to the quiver $Q$ and hence, by [Neg21a], 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 [SV12]). 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 [KSV17]). |
| title | Shuffle algebras for quivers as quantum groups |
| topic | Representation Theory Algebraic Geometry Quantum Algebra |
| url | https://arxiv.org/abs/2111.00249 |