Affine Yangians as Limits of Quantum Toroidal Algebras
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arXiv
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| Format: | Preprint |
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2026
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| author | Bezerra, Luan Kashuba, Iryna Lin, Hongda |
| author_facet | Bezerra, Luan Kashuba, Iryna Lin, Hongda |
| contents | We establish a degeneration isomorphism between quantum toroidal algebras and untwisted affine Yangians, valid for all untwisted affine Kac-Moody Lie algebras. Specifically, we prove that the affine Yangian $Y_\hbar(\mathfrak{g})$ is isomorphic, as a $\mathbb{C}[\hbar]$-algebra, to the associated graded algebra of the quantum toroidal algebra $U_\hbar(\mathfrak{g}^{\mathrm{tor}})$ with respect to a canonical filtration. This result constitutes the affine analogue of Drinfeld's conjecture on the relationship between Yangians and quantum loop algebras, previously established in the finite-dimensional setting by Gautam--Toledano Laredo and by Guay--Ma. As principal applications of this isomorphism, we derive two fundamental structural properties of affine Yangians: a Poincaré--Birkhoff--Witt (PBW) basis for $Y_\hbar(\mathfrak{g})$ in all untwisted affine types, and the identification of its classical limit as the universal enveloping algebra $U(\mathfrak{g}[u])$ of the polynomial current Lie algebra. A key ingredient of independent interest is our construction of a PBW basis for $U_\hbar(\mathfrak{g}^{\mathrm{tor}})$ itself, which relies on a new torsion-freeness argument for the quantum toroidal algebra and the topological Nakayama lemma. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_12871 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Affine Yangians as Limits of Quantum Toroidal Algebras Bezerra, Luan Kashuba, Iryna Lin, Hongda Quantum Algebra Mathematical Physics Representation Theory 17B37, 17B65, 17B67 We establish a degeneration isomorphism between quantum toroidal algebras and untwisted affine Yangians, valid for all untwisted affine Kac-Moody Lie algebras. Specifically, we prove that the affine Yangian $Y_\hbar(\mathfrak{g})$ is isomorphic, as a $\mathbb{C}[\hbar]$-algebra, to the associated graded algebra of the quantum toroidal algebra $U_\hbar(\mathfrak{g}^{\mathrm{tor}})$ with respect to a canonical filtration. This result constitutes the affine analogue of Drinfeld's conjecture on the relationship between Yangians and quantum loop algebras, previously established in the finite-dimensional setting by Gautam--Toledano Laredo and by Guay--Ma. As principal applications of this isomorphism, we derive two fundamental structural properties of affine Yangians: a Poincaré--Birkhoff--Witt (PBW) basis for $Y_\hbar(\mathfrak{g})$ in all untwisted affine types, and the identification of its classical limit as the universal enveloping algebra $U(\mathfrak{g}[u])$ of the polynomial current Lie algebra. A key ingredient of independent interest is our construction of a PBW basis for $U_\hbar(\mathfrak{g}^{\mathrm{tor}})$ itself, which relies on a new torsion-freeness argument for the quantum toroidal algebra and the topological Nakayama lemma. |
| title | Affine Yangians as Limits of Quantum Toroidal Algebras |
| topic | Quantum Algebra Mathematical Physics Representation Theory 17B37, 17B65, 17B67 |
| url | https://arxiv.org/abs/2605.12871 |