Two-Parameter Quantum Groups and $R$-Matrices: Classical Types
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866912511248302080 |
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| author | Martin, Ian Tsymbaliuk, Alexander |
| author_facet | Martin, Ian Tsymbaliuk, Alexander |
| contents | We construct finite $R$-matrices for the first fundamental representation $V$ of two-parameter quantum groups $U_{r,s}(\mathfrak{g})$ for classical $\mathfrak{g}$, both through the decomposition of $V\otimes V$ into irreducibles $U_{r,s}(\mathfrak{g})$-submodules as well as by evaluating the universal $R$-matrix. The latter is crucially based on the construction of dual PBW-type bases of $U^{\pm}_{r,s}(\mathfrak{g})$ consisting of the ordered products of quantum root vectors defined via $(r,s)$-bracketings and combinatorics of standard Lyndon words. We further derive explicit formulas for affine $R$-matrices, both through the Yang-Baxterization technique of [Internat. J. Modern Phys. A 6 (1991), 3735-3779] and as the unique intertwiner between the tensor product of $V(u)$ and $V(v)$, viewed as modules over two-parameter quantum affine algebras $U_{r,s}(\widehat{\mathfrak{g}})$ for classical $\mathfrak{g}$. The latter generalizes the formulas of [Comm. Math. Phys. 102 (1986), 537-547] for one-parametric quantum affine algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_01450 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Two-Parameter Quantum Groups and $R$-Matrices: Classical Types Martin, Ian Tsymbaliuk, Alexander Representation Theory High Energy Physics - Theory Mathematical Physics Quantum Algebra Exactly Solvable and Integrable Systems We construct finite $R$-matrices for the first fundamental representation $V$ of two-parameter quantum groups $U_{r,s}(\mathfrak{g})$ for classical $\mathfrak{g}$, both through the decomposition of $V\otimes V$ into irreducibles $U_{r,s}(\mathfrak{g})$-submodules as well as by evaluating the universal $R$-matrix. The latter is crucially based on the construction of dual PBW-type bases of $U^{\pm}_{r,s}(\mathfrak{g})$ consisting of the ordered products of quantum root vectors defined via $(r,s)$-bracketings and combinatorics of standard Lyndon words. We further derive explicit formulas for affine $R$-matrices, both through the Yang-Baxterization technique of [Internat. J. Modern Phys. A 6 (1991), 3735-3779] and as the unique intertwiner between the tensor product of $V(u)$ and $V(v)$, viewed as modules over two-parameter quantum affine algebras $U_{r,s}(\widehat{\mathfrak{g}})$ for classical $\mathfrak{g}$. The latter generalizes the formulas of [Comm. Math. Phys. 102 (1986), 537-547] for one-parametric quantum affine algebras. |
| title | Two-Parameter Quantum Groups and $R$-Matrices: Classical Types |
| topic | Representation Theory High Energy Physics - Theory Mathematical Physics Quantum Algebra Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2407.01450 |