Quantum Berezinian for quantum affine superalgebra $U_q(\widehat{gl}_{M|N})$
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866908586055041024 |
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| author | Jing, Naihuan Zheng, Li Zhang, Jian |
| author_facet | Jing, Naihuan Zheng, Li Zhang, Jian |
| contents | We introduce the quantum Berezinian for the quantum affine superalgebra $\mathrm{U}_q(\widehat{\mathfrak{gl}}_{M|N})$ and show that the coefficients of the quantum Berezinian belong to the center of $\mathrm{U}_q(\widehat{\gl}_{M|N})$. We also construct another family of central elements which can be expressed in the quantum Berezinian by a Liouville-type theorem. Moreover, we prove analogues of the Jacobi identities, the Schur complementary theorem, the Sylvester theorem and the MacMahon Master theorem for the generator matrices of $\mathrm{U}_q(\widehat{\gl}_{M|N})$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_19385 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantum Berezinian for quantum affine superalgebra $U_q(\widehat{gl}_{M|N})$ Jing, Naihuan Zheng, Li Zhang, Jian Quantum Algebra Representation Theory Primary: 17B37, Secondary: 81R50 We introduce the quantum Berezinian for the quantum affine superalgebra $\mathrm{U}_q(\widehat{\mathfrak{gl}}_{M|N})$ and show that the coefficients of the quantum Berezinian belong to the center of $\mathrm{U}_q(\widehat{\gl}_{M|N})$. We also construct another family of central elements which can be expressed in the quantum Berezinian by a Liouville-type theorem. Moreover, we prove analogues of the Jacobi identities, the Schur complementary theorem, the Sylvester theorem and the MacMahon Master theorem for the generator matrices of $\mathrm{U}_q(\widehat{\gl}_{M|N})$. |
| title | Quantum Berezinian for quantum affine superalgebra $U_q(\widehat{gl}_{M|N})$ |
| topic | Quantum Algebra Representation Theory Primary: 17B37, Secondary: 81R50 |
| url | https://arxiv.org/abs/2412.19385 |