### Abstract

Recent developments in the theory of quantum integrable particle systems in one-dimension with inverse square interactions are reviewed. First the Yangian symmetry is introduced and the energy spectra of the related spin models are discussed. The character of the su(n)_{1} WZNW theory is shown to be closely related with the Rogers-Szegö polynomial. Second, the infinite dimensional representation for solutions of the Yang-Baxter equation and the reflection equation is given. Based on the representation, the Dunkl operators associated with the classical root systems are constructed. The Macdonald polynomial and its generalization are discussed in connection with the eigenstates for the trigonometric case. Finally, some results on short-range interacting systems are mentioned.

Original language | English |
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Pages (from-to) | 3569-3594 |

Number of pages | 26 |

Journal | Journal of Mathematical Physics |

Volume | 44 |

Issue number | 8 |

DOIs | |

Publication status | Published - Aug 1 2003 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

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## Cite this

*Journal of Mathematical Physics*,

*44*(8), 3569-3594. https://doi.org/10.1063/1.1588743