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

Details On a Poisson-Lie analogue of the classical dynamical Yang-Baxter equation for self-dual Lie algebras On a Poisson-Lie analogue of the classical dynamical Yang-Baxter equation for self-dual Lie algebras

2002-08-22

arxiv.org/abs/math/0208159v3

Lett.Math.Phys. 62 (2002) 51-62

Mathematics

Quantum Algebra

13 pages, LaTeX2e. Typos are corrected in v2, a note added in proof upon publication in LMP is included in v3

Scientific paper

We derive a generalization of the classical dynamical Yang-Baxter equation (CDYBE) on a self-dual Lie algebra $\cal G$ 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 $\cal G$. 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.

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