Classical $N$-Reflection Equation and Gaudin Models
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arXiv
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| Format: | Preprint |
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2018
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| _version_ | 1866910917359304704 |
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| author | Caudrelier, Vincent Crampe, Nicolas |
| author_facet | Caudrelier, Vincent Crampe, Nicolas |
| contents | We introduce the notion of $N$-reflection equation which provides a large generalization of the usual classical reflection equation describing integrable boundary conditions. The latter is recovered as a special example of the $N=2$ case. The basic theory is established and illustrated with several examples of solutions of the $N$-reflection equation associated to the rational and trigonometric $r$-matrices. A central result is the construction of a Poisson algebra associated to a non skew-symmetric $r$-matrix whose form is specified by a solution of the $N$-reflection equation. Generating functions of quantities in involution can be identified within this Poisson algebra. As an application, we construct new classical Gaudin-type Hamiltonians, particular cases of which are Gaudin Hamiltonians of $BC_L$-type . |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1803_09931 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Classical $N$-Reflection Equation and Gaudin Models Caudrelier, Vincent Crampe, Nicolas Mathematical Physics Exactly Solvable and Integrable Systems We introduce the notion of $N$-reflection equation which provides a large generalization of the usual classical reflection equation describing integrable boundary conditions. The latter is recovered as a special example of the $N=2$ case. The basic theory is established and illustrated with several examples of solutions of the $N$-reflection equation associated to the rational and trigonometric $r$-matrices. A central result is the construction of a Poisson algebra associated to a non skew-symmetric $r$-matrix whose form is specified by a solution of the $N$-reflection equation. Generating functions of quantities in involution can be identified within this Poisson algebra. As an application, we construct new classical Gaudin-type Hamiltonians, particular cases of which are Gaudin Hamiltonians of $BC_L$-type . |
| title | Classical $N$-Reflection Equation and Gaudin Models |
| topic | Mathematical Physics Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/1803.09931 |