A quantum space of Euclidean lines

TL;DR

This paper constructs a differential groupoid with a coaction of the Quantum Euclidean Group, linking quantum group theory with the geometry of Euclidean lines through Poisson structures.

Contribution

It introduces a novel differential groupoid with quantum group coaction and identifies the dual Lie algebroid as a Poisson manifold of oriented lines in Euclidean space.

Findings

Established a differential groupoid with quantum group coaction.

Identified the dual Lie algebroid as a Poisson manifold of Euclidean lines.

Connected quantum group structures with classical geometric spaces.

Abstract

This article presents a differential groupoid with ``coaction'' of the groupoid underlying the Quantum Euclidean Group (i.e. its $C^*$-algebra is the $C^*$-algebra of this quantum group). The dual of the Lie algebroid is a Poisson manifold that can be identified with the space of oriented lines in Euclidean space equipped with a Poisson action of the Poisson-Lie Euclidean group.