Kinematical Gravitational Charge Algebra

12 pagesIn its first order formulation in terms of connection and coframes, the phase space of general relativity consists in a pair of conjugate fields, the flux 2-form (co-triad) and the Ashtekar connection, living on the canonical spatial hypersurface. It is supplemented with the Gauss constraints, the vector and scalar Hamiltonian constraints, respectively generating local SU$(2)$ gauge transformations, spatial diffeomorprhisms and time diffeomorphisms. Putting aside the scalar constraint and focusing on kinematical structures, we show that the Gauss constraint and the space diffeomorphism constraint can both be written as conservation laws for a set of boundary charges. We prove that these kinematical charges generates a local Poincar\'e ISU$(2)$ symmetry algebra. This comes in support of the recent proposal of Poincar\'e charge networks for discretized general relativity presented in arXiv:1906.07876