b-Structures on Lie groups and Poisson reduction
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 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 transfor...
| Authors: | , , |
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| Format: | article |
| Publication Date: | 2022 |
| Country: | España |
| Institution: | Universitat Politècnica de Catalunya (UPC) |
| Repository: | UPCommons. Portal del coneixement obert de la UPC |
| Language: | English |
| OAI Identifier: | oai:upcommons.upc.edu:2117/371005 |
| Online Access: | https://hdl.handle.net/2117/371005 https://dx.doi.org/10.1016/j.geomphys.2022.104471 |
| Access Level: | Open access |
| Keyword: | Symplectic geometry Poisson reduction Lie groups b-Symplectic manifolds Galilean transformations Cotangent models Minimal coupling Geometria simplèctica Classificació AMS::53 Differential geometry::53D Symplectic geometry, contact geometry Àrees temàtiques de la UPC::Matemàtiques i estadística::Geometria |
| Summary: | 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 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 ¿ together with its reduction theory. Namely, we extend the minimal coupling procedure to ¿ 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 ¿ (where is the Lie algebra of H) and the canonical b-symplectic structure on ¿ , where is viewed as a one-dimensional b-manifold having as critical hypersurface (in the sense of b-manifolds) the identity element. |
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