Symplectic double groupoids over Poisson $(ax+b)$-groups

Abstract:

First, we classify all the multiplicative Poisson structures on the $(ax + b)$-group and determine their dual Poisson Lie groups. Next, we show the existence of symplectic groupoid over the Poisson $(ax + b)$-group. Finally, by the Hamilton-Jacobi method we construct nontrivial symplectic double groupoids and conclude that for each pair of nondegenerate multiplicative Poisson structures of the $(ax + b)$-group there exists a symplectic double groupoid.