  The diffeomorphism symmetry of general relativity leads in the canonical
formulation to constraints, which encode the dynamics of the theory. These
constraints satisfy a complicated algebra, known as Dirac's hypersurface
deformation algebra. This algebra has been a long standing challenge for
quantization. One reason is that discretizations, on which many quantum gravity
approaches rely, generically break diffeomorphism symmetry. In this work we
find a representation for the Dirac constraint algebra of hypersurface
deformations in a formulation of discrete 3D gravity and for the flat as well
as homogeneously curved sector of discrete 4D gravity. In these cases
diffeomorphism symmetry can be preserved. Furthermore we present different
versions of the hypersurface deformation algebra for the boundary of a simplex
in arbitrary dimensions.
