Braided Momentum Structure of the q-Poincare Group

TL;DR

This paper reveals that the $q$-Poincaré group has a semidirect product structure involving a braided quantum group on $q$-Minkowski space, introducing braid statistics into the quantum group framework.

Contribution

It demonstrates the braided-quantum group structure of the $q$-Poincaré group and its decomposition, extending quantum group theory with braid statistics.

Findings

The $q$-Poincaré group is a semidirect product of a braided quantum group and $SO_q(1,3)$.

The braided quantum group $B$ has a braided-coproduct on 4-momentum.

Existence of braided vectors and covectors for general R-matrices.

Abstract

The $q$-Poincar\'e group of \cite{SWW:inh} is shown to have the structure of a semidirect product and coproduct $B\cocross \widetilde{SO_q(1,3)}$ where $B$ is a braided-quantum group structure on the $q$-Minkowski space of 4-momentum with braided-coproduct $\und\Delta \vecp=\vecp\tens 1+1\tens \vecp$. Here the necessary $B$ is not a usual kind of quantum group, but one with braid statistics. Similar braided-vectors and covectors $V(R')$, $V^*(R')$ exist for a general R-matrix. The abstract structure of the $q$-Lorentz group is also studied.