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 T⁎bG together with its reduction theory. Namely, we extend the minimal coupling procedure to T⁎bG/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 h⁎ (where h is the Lie algebra of H) and the canonical b-symplectic structure on T⁎b(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.