Twisted Classical Poincar\'{e} Algebras

TL;DR

This paper explores the twisting of the classical Poincaré algebra's Hopf structure, providing explicit formulas for the 4D case, and demonstrates how certain quantum deformations are special cases of this twisting approach.

Contribution

It introduces explicit twisting formulas for the Poincaré algebra's Hopf structure and connects recent quantum deformations to this twisting framework.

Findings

Explicit formulas for twisted coproducts in 4D Poincaré algebra

Universal R-matrices are triangular in the twisted setting

Quantum deformation by Chaichian and Demiczev is a twisted classical algebra

Abstract

We consider the twisting of Hopf structure for classical enveloping algebra $U(\hat{g})$, where $\hat{g}$ is the inhomogenous rotations algebra, with explicite formulae given for $D=4$ Poincar\'{e} algebra $(\hat{g}={\cal P}_4).$ The comultiplications of twisted $U^F({\cal P}_4)$ are obtained by conjugating primitive classical coproducts by $F\in U(\hat{c})\otimes U(\hat{c}),$ where $\hat{c}$ denotes any Abelian subalgebra of ${\cal P}_4$, and the universal $R-$matrices for $U^F({\cal P}_4)$ are triangular. As an example we show that the quantum deformation of Poincar\'{e} algebra recently proposed by Chaichian and Demiczev is a twisted classical Poincar\'{e} algebra. The interpretation of twisted Poincar\'{e} algebra as describing relativistic symmetries with clustered 2-particle states is proposed.