### Abstract

We derive a generalization of the classical dynamical Yang-Baxter equation (CDYBE) on a self-dual Lie algebra G by replacing the cotangent bundle T*G in a geometric interpretation of this equation by its Poisson-Lie (PL) analogue associated with a factorizable constant r-matrix on G. The resulting PL-CDYBE, with variables in the Lie group G equipped with the Semenov-Tian-Shansky Poisson bracket based on the constant r-matrix, coincides with an equation that appeared in an earlier study of PL symmetries in the WZNW model. In addition to its new group theoretic interpretation, we present a self-contained analysis of those solutions of the PL-CDYBE that were found in the WZNW context and characterize them by means of a uniqueness result under a certain analyticity assumption.

Original language | English |
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Pages (from-to) | 51-62 |

Number of pages | 12 |

Journal | Letters in Mathematical Physics |

Volume | 62 |

Issue number | 1 |

DOIs | |

Publication status | Published - Oct 2002 |

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### Keywords

- Classical dynamical Yang-Baxter equation
- Poisson-Lie groups and groupoids
- Self-dual Lie algebra, Wess-Zumino-Novikov-Witten model

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

### Cite this

**On a poisson-lie analogue of the classical dynamical Yang-Baxter equation for self-dual Lie algebras.** / Fehér, L.; Marshall, I.

Research output: Contribution to journal › Article

*Letters in Mathematical Physics*, vol. 62, no. 1, pp. 51-62. https://doi.org/10.1023/A:1021681826447

}

TY - JOUR

T1 - On a poisson-lie analogue of the classical dynamical Yang-Baxter equation for self-dual Lie algebras

AU - Fehér, L.

AU - Marshall, I.

PY - 2002/10

Y1 - 2002/10

N2 - We derive a generalization of the classical dynamical Yang-Baxter equation (CDYBE) on a self-dual Lie algebra G by replacing the cotangent bundle T*G in a geometric interpretation of this equation by its Poisson-Lie (PL) analogue associated with a factorizable constant r-matrix on G. The resulting PL-CDYBE, with variables in the Lie group G equipped with the Semenov-Tian-Shansky Poisson bracket based on the constant r-matrix, coincides with an equation that appeared in an earlier study of PL symmetries in the WZNW model. In addition to its new group theoretic interpretation, we present a self-contained analysis of those solutions of the PL-CDYBE that were found in the WZNW context and characterize them by means of a uniqueness result under a certain analyticity assumption.

AB - We derive a generalization of the classical dynamical Yang-Baxter equation (CDYBE) on a self-dual Lie algebra G by replacing the cotangent bundle T*G in a geometric interpretation of this equation by its Poisson-Lie (PL) analogue associated with a factorizable constant r-matrix on G. The resulting PL-CDYBE, with variables in the Lie group G equipped with the Semenov-Tian-Shansky Poisson bracket based on the constant r-matrix, coincides with an equation that appeared in an earlier study of PL symmetries in the WZNW model. In addition to its new group theoretic interpretation, we present a self-contained analysis of those solutions of the PL-CDYBE that were found in the WZNW context and characterize them by means of a uniqueness result under a certain analyticity assumption.

KW - Classical dynamical Yang-Baxter equation

KW - Poisson-Lie groups and groupoids

KW - Self-dual Lie algebra, Wess-Zumino-Novikov-Witten model

UR - http://www.scopus.com/inward/record.url?scp=0041366830&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0041366830&partnerID=8YFLogxK

U2 - 10.1023/A:1021681826447

DO - 10.1023/A:1021681826447

M3 - Article

AN - SCOPUS:0041366830

VL - 62

SP - 51

EP - 62

JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

SN - 0377-9017

IS - 1

ER -