Remarks on simple interpolation between Jordanian twists

TL;DR

This paper introduces a one-parameter family of Jordanian twists that generalize existing twists, all leading to the same noncommutative spacetime structure, with detailed constructions and a unifying diagram.

Contribution

It proposes a simple generalization of Jordanian twists, showing they produce identical deformations and providing comprehensive constructions and a unifying diagram.

Findings

All twists produce the same $oldsymbol{ ext{kappa}-}$Minkowski relations.

Explicit constructions of star product, coproduct, and twist are provided.

A diagram relates all possible algebraic constructions.

Abstract

In this paper, we propose a simple generalization of the locally r-symmetric Jordanian twist, resulting in the one-parameter family of Jordanian twists. All the proposed twists differ by the coboundary twists and produce the same Jordanian deformation of the corresponding Lie algebra. They all provide the $\kappa$-Minkowski spacetime commutation relations. Constructions from noncommutative coordinates to the star product and coproduct, and from the star product to the coproduct and the twist are presented. The corresponding twist in the Hopf algebroid approach is given. Our results are presented symbolically by a diagram relating all of the possible constructions.