b-Structures on Lie groups and Poisson reduction

Motivated by the group of Galilean transformations and subgroup which fix time zero, we introduce notion a $b$-Lie as pair $(G,H)$ where $G$ is Lie $H$ codimension-one subgroup. Such allows us to give theoretical framework for space-time initial can be seen boundary. In this framework, develop basics theory study associated canonical $b$-symplectic structure on $b$-cotangent bundle $^b {T}^\ast G$ together with its reduction theory. Namely, extend minimal coupling procedure $^bT^*G/H$ prove that Poisson under cotangent lifted action left translations described in terms $\mathfrak{h}^\ast$ (where $\mathfrak{h}$ algebra $H$) {T}^\ast(G/H)$, $G/H$ viewed one-dimensional $b$-manifold having critical hypersurface (in sense $b$-manifolds) identity element.