Quasitriangular structure and twisting of the 2+1 bicrossproduct model

TL;DR

This paper demonstrates that a 2+1 dimensional bicrossproduct quantum Poincare group is related to the quantum double via a twist, connecting different quantum spacetime models through a limiting process from q-deformation.

Contribution

It establishes a twist relation between bicrossproduct and quantum double quantum groups in 2+1 dimensions, linking different quantum spacetime structures.

Findings

Derived the twist relation via a scaling limit as q approaches 1.

Connected bicrossproduct and quantum double models through Drinfeld twist.

Recovered the twist at the Lie bialgebra level.

Abstract

We show that the bicrossproduct model $C[SU_2^*]{\blacktriangleright\!\!\triangleleft} U(su_2)$ quantum Poincare group in 2+1 dimensions acting on the quantum spacetime $[x_i,t]=\imath\lambda x_i$ is related by a Drinfeld and module-algebra twist to the quantum double $U(su_2)\ltimes C[SU_2]$ acting on the quantum spacetime $[x_\mu,x_\nu]=\imath\lambda\epsilon_{\mu\nu\rho}x_\rho$. We obtain this twist by taking a scaling limit as $q\to 1$ of the $q$-deformed version of the above where it corresponds to a previous theory of $q$-deformed Wick rotation from $q$-Euclidean to $q$-Minkowski space. We also recover the twist result at the Lie bialgebra level.