Quantum algebra of multiparameter Manin matrices
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866911947279040512 |
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| author | Jing, Naihuan Liu, Yinlong Zhang, Jian |
| author_facet | Jing, Naihuan Liu, Yinlong Zhang, Jian |
| contents | Multiparametric quantum semigroups $\mathrm{M}_{\hat{q}, \hat{p}}(n)$ are generalization of the one-parameter general linear semigroups $\mathrm{M}_q(n)$, where $\hat{q}=(q_{ij})$ and $\hat{p}=(p_{ij})$ are $2n^2$ parameters satisfying certain conditions. In this paper, we study the algebra of multiparametric Manin matrices using the R-matrix method. The systematic approach enables us to obtain several classical identities such as Muir identities, Newton's identities, Capelli-type identities, Cauchy-Binet's identity both for determinant and permanent as well as a rigorous proof of the MacMahon master equation for the quantum algebra of multiparametric Manin matrices. Some of the generalized identities are also generalized to multiparameter $q$-Yangians. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_12608 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quantum algebra of multiparameter Manin matrices Jing, Naihuan Liu, Yinlong Zhang, Jian Quantum Algebra Combinatorics Rings and Algebras Primary: 05E10, Secondary: 17B37, 58A17, 15A75, 15B33, 15A15 Multiparametric quantum semigroups $\mathrm{M}_{\hat{q}, \hat{p}}(n)$ are generalization of the one-parameter general linear semigroups $\mathrm{M}_q(n)$, where $\hat{q}=(q_{ij})$ and $\hat{p}=(p_{ij})$ are $2n^2$ parameters satisfying certain conditions. In this paper, we study the algebra of multiparametric Manin matrices using the R-matrix method. The systematic approach enables us to obtain several classical identities such as Muir identities, Newton's identities, Capelli-type identities, Cauchy-Binet's identity both for determinant and permanent as well as a rigorous proof of the MacMahon master equation for the quantum algebra of multiparametric Manin matrices. Some of the generalized identities are also generalized to multiparameter $q$-Yangians. |
| title | Quantum algebra of multiparameter Manin matrices |
| topic | Quantum Algebra Combinatorics Rings and Algebras Primary: 05E10, Secondary: 17B37, 58A17, 15A75, 15B33, 15A15 |
| url | https://arxiv.org/abs/2303.12608 |