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.