Bicrossproduct structure of $\kappa$-Poincare group and non-commutative geometry

TL;DR

This paper demonstrates that the $kappa$-deformed Poincaré algebra has a bicrossproduct Hopf algebra structure, leading to a covariant, non-commutative $kappa$-Minkowski space, with implications for Planck scale physics.

Contribution

It reveals the bicrossproduct Hopf algebra structure of the $kappa$-Poincaré algebra and its action on a deformed, non-commutative Minkowski space.

Findings

The algebra is a semidirect product of Lorentz and momentum sectors.

The coalgebra also has a semidirect structure with backreaction.

$kappa$-Minkowski space is necessarily deformed and non-commutative.

Abstract

We show that the $\kappa$-deformed Poincar\'e quantum algebra proposed for elementary particle physics has the structure of a Hopf agebra bicrossproduct $U(so(1,3))\cobicross T$. The algebra is a semidirect product of the classical Lorentz group $so(1,3)$ acting in a deformed way on the momentum sector $T$. The novel feature is that the coalgebra is also semidirect, with a backreaction of the momentum sector on the Lorentz rotations. Using this, we show that the $\kappa$-Poincar\'e acts covariantly on a $\kappa$-Minkowski space, which we introduce. It turns out necessarily to be deformed and non-commutative. We also connect this algebra with a previous approach to Planck scale physics.