Non-Commutative Geometry and Twisted Conformal Symmetry

TL;DR

This paper develops a framework for conformal symmetry in non-commutative geometries using twisted Hopf algebras, extending previous work on Poincaré symmetry to the full conformal algebra.

Contribution

It introduces a twisted conformal algebra as a Hopf algebra, enabling conformal invariance in non-commutative backgrounds, expanding the mathematical tools for quantum geometry.

Findings

Conformal symmetry can be compatible with non-commutative geometry.

The twisted co-product formalism is extended from Poincaré to conformal algebra.

Non-commutativity arises naturally from the twisting procedure.

Abstract

The twist-deformed conformal algebra is constructed as a Hopf algebra with twisted co-product. This allows for the definition of conformal symmetry in a non-commutative background geometry. The twisted co-product is reviewed for the Poincar\'e algebra and the construction is then extended to the full conformal algebra. It is demonstrated that conformal invariance need not be viewed as incompatible with non-commutative geometry; the non-commutativity of the coordinates appears as a consequence of the twisting, as has been shown in the literature in the case of the twisted Poincar\'e algebra.