$\kappa$-deformations of D=3 conformal versus deformations of D=4 AdS symmetries

TL;DR

This paper explores classical $o(3,2)$ $r$-matrices and their quantum deformations related to D=3 conformal and D=4 AdS symmetries, using Drinfeld twist and basis transformations, highlighting the role of the AdS radius.

Contribution

It introduces a unified approach to quantum deformations of $o(3,2)$, connecting D=3 conformal and D=4 AdS symmetries through basis changes and deformation methods.

Findings

Quantum $o(3,2)$ algebra expressed in Hopf algebra form.

Deformation parameters relate to AdS radius and conformal mass.

Bilinear Casimir expressed in deformed basis.

Abstract

We describe the classical $o(3,2)$ $r$-matrices as generating the quantum deformations of either D=3 conformal algebra with mass-like deformation parameters or D=4 $AdS$ algebra with dimensionless deformation parameters. We describe the quantization of classical $o(3,2)$ $r$-matrices via Drinfeld twist method which locates the deformation in the coalgebra sector. Further we obtain the quantum $o(3,2)$ algebra in a convenient Hopf algebra form by considering suitable deformation maps from classical to deformed $o(3,2)$ algebra basis. It appears that if we pass from $\kappa$-deformed D =3 conformal algebra basis to the deformed D=4 $AdS$ generators basis the role of dimensionfull parameter is taken over by the $AdS$ radius $R$. We provide also the bilinear $o(3,2)$ Casimir which we express using the deformed D=3 conformal basis.