$q$-Analogue of the degree zero part of a rational Cherednik algebra
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866912088746622976 |
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| author | Feigin, Misha Vrabec, Martin |
| author_facet | Feigin, Misha Vrabec, Martin |
| contents | Inside the double affine Hecke algebra of type $GL_n$, which depends on two parameters $q$ and $τ$, we define a subalgebra $\mathbb{H}^{\mathfrak{gl}_n}$ that may be thought of as a $q$-analogue of the degree zero part of the corresponding rational Cherednik algebra. We prove that the algebra $\mathbb{H}^{\mathfrak{gl}_n}$ is a flat $τ$-deformation of the crossed product of the group algebra of the symmetric group with the image of the Drinfeld-Jimbo quantum group $U_q(\mathfrak{gl}_n)$ under the $q$-oscillator (Jordan-Schwinger) representation. We find all the defining relations and an explicit PBW basis for the algebra $\mathbb{H}^{\mathfrak{gl}_n}$. We describe its centre and establish a double centraliser property. As an application, we also obtain new integrable generalisations of Hamiltonians introduced by van Diejen. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_07543 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | $q$-Analogue of the degree zero part of a rational Cherednik algebra Feigin, Misha Vrabec, Martin Quantum Algebra Mathematical Physics Exactly Solvable and Integrable Systems Inside the double affine Hecke algebra of type $GL_n$, which depends on two parameters $q$ and $τ$, we define a subalgebra $\mathbb{H}^{\mathfrak{gl}_n}$ that may be thought of as a $q$-analogue of the degree zero part of the corresponding rational Cherednik algebra. We prove that the algebra $\mathbb{H}^{\mathfrak{gl}_n}$ is a flat $τ$-deformation of the crossed product of the group algebra of the symmetric group with the image of the Drinfeld-Jimbo quantum group $U_q(\mathfrak{gl}_n)$ under the $q$-oscillator (Jordan-Schwinger) representation. We find all the defining relations and an explicit PBW basis for the algebra $\mathbb{H}^{\mathfrak{gl}_n}$. We describe its centre and establish a double centraliser property. As an application, we also obtain new integrable generalisations of Hamiltonians introduced by van Diejen. |
| title | $q$-Analogue of the degree zero part of a rational Cherednik algebra |
| topic | Quantum Algebra Mathematical Physics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2311.07543 |