The Lorentz symmetry and the space and time translational symmetry are fundamental symmetries of nature. Crystals are themanifestation of the continuous space translational symmetry being spontaneously broken into a discrete one.We argue that, following the space translational symmetry, the continuous Lorentz symmetry should also be broken into a discrete one, which further implies that the con...

The Lorentz symmetry and the space and time translational symmetry are fundamental symmetries of nature. Crystals are themanifestation of the continuous space translational symmetry being spontaneously broken into a discrete one.We argue that, following the space translational symmetry, the continuous Lorentz symmetry should also be broken into a discrete one, which further implies that the con...

In this paper we introduce the notion of coalgebra symmetry for discrete systems. With concept prove that all radially symmetric systems in standard form are quasi-integrable and variational quasi-radially Poincar\'e--Lyapunov--Nekhoroshev maps order $N-2$, where $N$ degrees freedom system. We also discuss integrability properties several vector which generalisations well-known one degree integ...

Using the differential calculus on discrete group, we study the general relativity in the space-time which is the product of a four dimensional manifold by a two-point space. We generalize the usual concept of frame and connection in our space-time, and from the generalized torsion free condition we obtain an action of a scalar field coupled to Einstein gravity, which may be related to the Jord...