### Abstract

A dynamical r-matrix is associated with every self-dual Lie algebra A which is graded by finite-dimensional subspaces as A = ⊕_{n∈Z}A_{n}, where A_{n} is dual to A_{-n} with respect to the invariant scalar product on A, and A_{0} admits a nonempty open subset A_{0} for which ad κ is invertible on A_{n} if n ≠ 0 and κ ∈ Ǎ_{0}. Examples are furnished by taking A to be an affine Lie algebra obtained from the central extension of a twisted loop algebra ℓ(G, μ) of a finite-dimensional self-dual Lie algebra G. These r-matrices, R: Ǎ_{0} → End(A), yield generalizations of the basic trigonometric dynamical r-matrices that, according to Etingof and Varchenko, are associated with the Coxeter automorphisms of the simple Lie algebras, and are related to Felder's elliptic r-matrices by evaluation homomorphisms of ℓ(G, μ) into G. The spectral-parameter-dependent dynamical r-matrix that corresponds analogously to an arbitrary scalar-product-preserving finite order automorphism of a self-dual Lie algebra is here calculated explicitly.

Original language | English |
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Pages (from-to) | 622-642 |

Number of pages | 21 |

Journal | Nuclear Physics B |

Volume | 621 |

Issue number | 3 |

DOIs | |

Publication status | Published - Jan 21 2002 |

### Keywords

- Classical Yang-Baxter equation
- Dynamical r-matrix
- Self-dual Lie algebra

### ASJC Scopus subject areas

- Nuclear and High Energy Physics

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## Cite this

*Nuclear Physics B*,

*621*(3), 622-642. https://doi.org/10.1016/s0550-3213(01)00609-5