$q$-Analogue of the degree zero part of a rational Cherednik algebra Article Swipe

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YOU? · · 2023 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2311.07543

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.

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Subalgebra Mathematics Loop algebra Zero (linguistics) Degree (music) Connection (principal bundle) Group (periodic table) Algebra over a field Pure mathematics Type (biology) Representation theory Filtered algebra Quantum affine algebra Cellular algebra Product (mathematics) Current algebra Algebra representation Physics Geometry Quantum mechanics Philosophy Biology Acoustics Ecology Linguistics

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Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2311.07543
PDF: https://arxiv.org/pdf/2311.07543
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DOI: https://doi.org/10.48550/arxiv.2311.07543

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Title: $q$-Analogue of the degree zero part of a rational Cherednik algebra

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Language: en

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Publication year: 2023

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Publication date: 2023-11-13

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Authors: Misha Feigin, Martin Vrabec

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Landing page: https://arxiv.org/abs/2311.07543

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Concepts: Subalgebra, Mathematics, Loop algebra, Zero (linguistics), Degree (music), Connection (principal bundle), Group (periodic table), Algebra over a field, Pure mathematics, Type (biology), Representation theory, Filtered algebra, Quantum affine algebra, Cellular algebra, Product (mathematics), Current algebra, Algebra representation, Physics, Geometry, Quantum mechanics, Philosophy, Biology, Acoustics, Ecology, Linguistics

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