Shuffle Products, Degenerate Affine Hecke Algebras, and Quantum Toda Lattice
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912921800409088 |
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| author | Kalmykov, Artem |
| author_facet | Kalmykov, Artem |
| contents | We revisit an identification of the quantum Toda lattice for $\mathrm{GL}_N$ and the truncated shifted Yangian of $\mathfrak{sl}_2$, as well as related constructions, from a purely algebraic point of view, bypassing the topological medium of the homology of the affine Grassmannian. For instance, we interpret the Gerasimov-Kharchev-Lebedev-Oblezin homomorphism into the algebra of difference operators via a finite analog of the Miura transform. This algebraic identification is deduced by relating degenerate affine Hecke algebras to the simplest example of a rational Feigin-Odesskii shuffle product. As a bonus, we obtain a presentation of the latter via a mirabolic version of the Kostant-Whittaker reduction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_09520 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Shuffle Products, Degenerate Affine Hecke Algebras, and Quantum Toda Lattice Kalmykov, Artem Representation Theory Quantum Algebra We revisit an identification of the quantum Toda lattice for $\mathrm{GL}_N$ and the truncated shifted Yangian of $\mathfrak{sl}_2$, as well as related constructions, from a purely algebraic point of view, bypassing the topological medium of the homology of the affine Grassmannian. For instance, we interpret the Gerasimov-Kharchev-Lebedev-Oblezin homomorphism into the algebra of difference operators via a finite analog of the Miura transform. This algebraic identification is deduced by relating degenerate affine Hecke algebras to the simplest example of a rational Feigin-Odesskii shuffle product. As a bonus, we obtain a presentation of the latter via a mirabolic version of the Kostant-Whittaker reduction. |
| title | Shuffle Products, Degenerate Affine Hecke Algebras, and Quantum Toda Lattice |
| topic | Representation Theory Quantum Algebra |
| url | https://arxiv.org/abs/2505.09520 |