Cayley-Klein Lie Algebras and their Quantum Universal Enveloping Algebras

TL;DR

This paper introduces quantum deformations of Lie algebras associated with Cayley-Klein geometries, providing a unified algebraic framework for describing multiple geometric structures and their quantum groups.

Contribution

It presents a novel quantum deformation (Hopf algebra structure) for Lie algebras of Cayley-Klein geometries, extending the algebraic understanding of these geometric groups.

Findings

Quantum deformations of Cayley-Klein Lie algebras are constructed.

The Hopf algebra structure is expressed in a compact formalism.

The quasitriangularity of these quantum algebras is analyzed.

Abstract

The N-dimensional Cayley-Klein scheme allows the simultaneous description of $3^N$ geometries (symmetric orthogonal homogeneous spaces) by means of a set of Lie algebras depending on $N$ real parameters. We present here a quantum deformation of the Lie algebras generating the groups of motion of the two and three dimensional Cayley-Klein geometries. This deformation (Hopf algebra structure) is presented in a compact form by using a formalism developed for the case of (quasi) free Lie algebras. Their quasitriangularity (i.e., the most usual way to study the associativity of their dual objects, the quantum groups) is also discussed.