Some remarks on the {$q$}-Poincare algebra in R-matrix form

TL;DR

This paper reviews the construction of q-deformed Poincaré algebras using R-matrix formalism, highlighting their covariance, duality, and structural properties within a braided quantum group framework.

Contribution

It provides a comprehensive review of R-matrix formulas for q-Poincaré algebras in Minkowski and Euclidean spaces, including their duality and *-structure axioms.

Findings

Explicit R-matrix formulas for q-Poincaré groups

Duality between Euclidean and Minkowski cases

Discussion of *-structure and dilaton problem

Abstract

The braided approach to q-deformation (due to the author and collaborators) gives natural algebras $R_{21}u_1Ru_2=u_2R_{21}u_1R$ and $R_{21}x_1x_2=x_2x_1R$ for q-Minkowski and q-Euclidean spaces respectively. These algebras are covariant under a corresponding background `rotation' quantum group. Semidirect product by this according to the bosonisation procedure (also due to the author) gives the corresponding Poincar\'e quantum groups. We review the construction and collect the resulting R-matrix formulae for both Euclidean and Minkowski cases in both enveloping algebra and function algebra form, and the duality between them. Axioms for the Poincar\'e quantum group $*$-structure and the dilaton problem are discussed.