On Double Groups and the Poincar\'e group

Abstract

In [22], Crane and Sheppard considered the structure of the Poincare group as a 2-Group, and derived important information about its representations in a 2-Category suited for representations of non-compact 2-groups, following a lead of [23]. In this paper, starting from the position that the most natural structure to describe cobordisms with corners, as in the recently published work of A. Voronov [27], is the cubical approach to higher category theory of Ehreshman, we explore some possibilities given by double groups to build TQFTs. Our main theorem is an extension of the work of [4], where we prove a theorem on the structure of maximally exclusive double groups. This result gives a presentation of the Poincar\'e group where the distinction between boosts, rotations and translations is part of the structure, from which a TQFT could be build with space and spacetime transformations kept separate. This article drafts a program that will hopefully yield new state sum models of physical interest.