The Cangemi-Jackiw manifold in high dimensions and symplectic structure

The notion of Poincaré gauge manifold (G), proposed in the context of an (1 + 1) gravitational theory by Cangemi and Jackiw ( D. Cangemi and R. Jackiw, Ann. Phys. (N.Y.) 225 (1993) 229), is explored from a geometrical point of view. First G is defined for arbitrary dimensions and in the sequence a symplectic structure is attached to T ∗G. Treating then the case of 5-dimensions, a (4, 1) de-Sitter space, applications are presented studing representations of the Poincaré group in association with kinetic theory and the Weyl operators in phase space. The central extension in the Aghassi-RomanSantilli group (Aghassi, Roman and Santilli, Phys. Rev. D 1 (1970) 2753) is derived as a subgroup of linear transformations in G with 6 dimensions.