$b$-Structures on Lie groups and Poisson reduction

Abstract

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