  The purpose of the current paper is twofold: to provide a conceptual link
between the quantization framework based on Lie integration of algebroids
proposed by N.P. Landsman in the book "Mathematical Topics between Classical
and Quantum Mechanics" (1998) and the dynamical formulation of the Einstein's
equation, and to clarify how from the relevant groupoid the Poisson bracket
between constraints of the hamiltonian formulation emerges as an algebroid
bracket. To do so, we adapt the groupoid proposed by C. Blohmann, M. Fernandes
and A. Weinstein in the paper "Groupoid symmetry and constraints in General
Relativity" (2010) by using a different, in our view simpler and more natural,
diffeological structure. As a result, we get a different algebroid associated
to it, such that the algebroid bundle can be understood as a configuration
space of the dynamical formulation of General Relativity, the bracket structure
between its sections being of the desired form. We point out the conceptual
advantages of this perspective, as well as some interesting issues that require
further investigation. We also discuss some of the difficulties that need to be
overcome to apply our approach to quantize the Einstein's theory in this sense.
This is an extract from the Master's thesis written under the supervision of
Prof. Klaas Landsman.
