On the classification of quantum Poincar\'e groups

TL;DR

This paper classifies quantum Poincaré groups by solving polynomial equations for their parameters, establishing a one-to-one correspondence with quantum Minkowski spaces, and analyzing their R-matrices, extending classical symmetry structures into quantum realms.

Contribution

It provides a complete classification of quantum Poincaré groups and their associated quantum Minkowski spaces, including the analysis of fundamental R-matrices, based on polynomial parameter solutions.

Findings

Classified all quantum Poincaré groups without dilatations.

Established a one-to-one correspondence with quantum Minkowski spaces.

Determined possible R-matrices for fundamental representations.

Abstract

Using the general theory of [10] ( hep-th 9412058 ), quantum Poincar\'e groups (without dilatations) are described and investigated. The description contains a set of numerical parameters which satisfy certain polynomial equations. For most cases we solve them and give the classification of quantum Poincar\'e groups. Each of them corresponds to exactly one quantum Minkowski space. The Poincar\'e series of these objects are the same as in the classical case. We also classify possible $R$-matrices for the fundamental representation of the group.