4D Quantum Affine Algebras and Space--Time q-Symmetries

TL;DR

This paper develops a quantum deformation framework for 4D affine Lie algebras related to space-time symmetries, extending classical geometries into the quantum domain with explicit algebraic constructions.

Contribution

It introduces a global q-deformation model for 4D affine Cayley--Klein geometries derived from 3D deformations, connecting classical and quantum symmetry algebras.

Findings

Quantum deformed space-time and space symmetry algebras are explicitly constructed.

Multiple q-deformations of Poincaré, Galilei, and Euclidean algebras are obtained.

Classical properties of Cayley--Klein systems are preserved in the quantum case.

Abstract

A global model of $q$-deformation for the quasi--orthogonal Lie algebras generating the groups of motions of the four--dimensional affine Cayley--Klein geometries is obtained starting from the three dimensional deformations. It is shown how the main algebraic classical properties of the CK systems can be implemented in the quantum case. Quantum deformed versions of either the space--time or space symmetry algebras (Poincar\'e (3+1), Galilei (3+1), 4D Euclidean as well as others) appear in this context as particular cases and several $q$-deformations for them are directly obtained.