Linearization of Poisson–Lie structures on the $2D$ Euclidean and $(1+1)$ Poincaré groups

The paper deals with linearization problem of Poisson-Lie structures on the $(1+1)$ Poincaré and $2D$ Euclidean groups. We construct the explicit form of linearizing coordinates of all these Poisson-Lie structures. For this, we calculate all Poisson-Lie structures on these two groups mentioned above, through the correspondence with Lie Bialgebra structures on their Lie algebras which we first determine.