The Ding-Frenkel Isomorphism Theorem for two-parameter quantum affine algebra $U_{r,s}\mathcal(\widehat{\mathfrak{so}_{2n+1}})$
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| Format: | Preprint |
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2024
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| _version_ | 1866916001920057344 |
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| author | Hu, Naihong Xu, Xiao Zhuang, Rushu |
| author_facet | Hu, Naihong Xu, Xiao Zhuang, Rushu |
| contents | From the theory of finite-dimensional weight modules, we get the basic braided $R$-matrix $\widehat R$ of $U_{r, s}(\mathfrak{so}_{2n+1})$. For its FRT presentation $U(\widehat R)$, we achieve two word-formation methods of quantum Lyndon bases (whose bracketing rules are regulated by the $RLL$-formalism) and elucidate their distribution rule within the triangular $L$-matrix. Consequently, we contribute an algebraic proof for establishing an isomorphism between the Drinfeld-Jimbo presentation and the FRT presentation. In the affine setting, we first derive two spectral parameter-dependent $R$-matrices through the Yang-Baxterization. Next, we select the only one that satisfies the intertwining property with respect to the minimal affinization. Accordingly, we obtain the $RLL$ realization of $U_{r, s}(\widehat{\mathfrak{so}_{2n+1}})$ through the Gauss decompositions of the generating matrices. Finally, we contribute an algebraic proof to the Ding-Frenkel Isomorphism Theorem between the Drinfeld realization and the $RLL$ realization. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2405_06587 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Ding-Frenkel Isomorphism Theorem for two-parameter quantum affine algebra $U_{r,s}\mathcal(\widehat{\mathfrak{so}_{2n+1}})$ Hu, Naihong Xu, Xiao Zhuang, Rushu Quantum Algebra From the theory of finite-dimensional weight modules, we get the basic braided $R$-matrix $\widehat R$ of $U_{r, s}(\mathfrak{so}_{2n+1})$. For its FRT presentation $U(\widehat R)$, we achieve two word-formation methods of quantum Lyndon bases (whose bracketing rules are regulated by the $RLL$-formalism) and elucidate their distribution rule within the triangular $L$-matrix. Consequently, we contribute an algebraic proof for establishing an isomorphism between the Drinfeld-Jimbo presentation and the FRT presentation. In the affine setting, we first derive two spectral parameter-dependent $R$-matrices through the Yang-Baxterization. Next, we select the only one that satisfies the intertwining property with respect to the minimal affinization. Accordingly, we obtain the $RLL$ realization of $U_{r, s}(\widehat{\mathfrak{so}_{2n+1}})$ through the Gauss decompositions of the generating matrices. Finally, we contribute an algebraic proof to the Ding-Frenkel Isomorphism Theorem between the Drinfeld realization and the $RLL$ realization. |
| title | The Ding-Frenkel Isomorphism Theorem for two-parameter quantum affine algebra $U_{r,s}\mathcal(\widehat{\mathfrak{so}_{2n+1}})$ |
| topic | Quantum Algebra |
| url | https://arxiv.org/abs/2405.06587 |