Bidynamical Poisson Groupoids

• Published in: arXiv.org
• Main Author: Pujol, Romaric
• Format: Paper
• Language: English
• Published: Ithaca Cornell University Library, arXiv.org 11.10.2006
• Subjects: Decomposition
• ISSN: 2331-8422

We give relations between dynamical Poisson groupoids, classical dynamical Yang--Baxter equations and Lie quasi-bialgebras. We show that there is a correspondance between the class of bidynamical Lie quasi-bialgebras and the class of bidynamical Poisson groupoids. We give an explicit, analytical and canonical equivariant solution of the classical dynamical Yang--Baxter equation (classical dynamical \(\ell\)-matrices) when there exists a reductive decomposition \(\g=\l\oplus\m\), and show that any other equivariant solution is formally gauge equivalent to the canonical one. We also describe the dual of the associated Poisson groupoid, and obtain the characterization that a dynamical Poisson groupoid has a dynamical dual if and only if there exists a reductive decomposition \(\g=\l\oplus\m\).