Location
MC 5479
Candidate
Thiago Oliveira Ferreira | Applied Mathematics, University of Waterloo
Title
Towards a phase space for polyhedra using Lie 2-groups
Abstract
It has been known that the coadjoint orbits of a Lie group carry a natural symplectic structure, studied extensively by Kirillov in his “orbit method”. It is also known that non-planar polygons have a natural phase space, first introduced by Kapovich and Millson (1996), wherein each edge is represented by a sphere (which, being a coadjoint orbit of SU(2), is naturally symplectic), and a symplectic reduction by global rotations is performed. This polygon phase space can be extended to polyhedra using Minkowski’s Theorem (1897). However, in the latter case, the lack of information about edge lengths and the impossibility of coherently gluing 2d surfaces decorated with non-abelian group elements — as shown by Eckmann and Hilton (1962) — point to the need for higher algebra structures, such as 2-vector spaces, Lie 2-groups and Lie 2-algebras. Inspired by that, we are studying whether it is possible to extend the Kirillov–Kostant–Souriau symplectic structure to the coadjoint orbits of strict Lie 2-groups and to build the analogue of the Kapovich–Millson phase space for polyhedra without Mikowski's theorem.