Rieffel deformation of group coactions

TL;DR

This paper develops a method to deform group coactions using Rieffel deformation, resulting in quantum groups and deformed C*-algebras that retain complete information about the original structures, with applications to quantum Minkowski space.

Contribution

It introduces a way to construct deformed coactions of quantum groups on C*-algebras via Rieffel deformation, preserving essential information.

Findings

Deformed C*-algebra B(f) admits a continuous coaction of the quantum group G(f).

The deformation process retains complete information about the original coaction.

Application to Lorentz group yields a quantum Minkowski space.

Abstract

Let G be a locally compact group, H an abelian subgroup and let f be a continuous 2-cocycle on the dual group of H. Let B be a C*-algebra equipped with a continuous right coaction of G. Using Rieffel deformation, we can construct a quantum group G(f) and the deformed C*-algebra B(f). The aim of this paper is to show that B(f) is equipped with a continuous coaction of G(f). The transition from the original coaction to its deformed counterpart is nontrivial in the sense that the deformed one contains complete information about the undeformed one. In order to illustrate our construction we apply it to the action of the Lorentz group on the Minkowski space obtaining a C*-algebraic quantum Minkowski space.