The universe is not a Lie, but actually an Hopf, algebra

Abstract

In this paper I would like to show how the Deformed Special Relativity family of models - developed to approach spacetime quantization - can actually be applied to the description of classical cosmology. I use the bicrossproduct basis of $\kappa$-Poincar\'{e} algebra to describe photon propagation in deSitter classical General Relativity. I show the Hopf algebraic aspects of deSitter model, and give an explicit physical interpretation of $\kappa$-Poincar\'{e} Hopf algebraic features in spacetime. Such an approach allows to unravel some not yet known General Relativistic relations of deSitter cosmology, and reinterpret many features of Quantum Gravity phenomenology as classical properties of maximally symmetric spacetime models. In the last section of the paper I give a first example on how to apply this mathematical framework to more realistic (non maximally symmetric) spacetime models, such as $\Lambda\text{CDM}$ and matter-dominated universe.