Complex geometry of quantum cones

TL;DR

This paper explores quantum cones as fixed point algebras under cyclic group actions on quantum discs, revealing their smoothness, differential structures, and boundary-like properties in a non-commutative geometric setting.

Contribution

It demonstrates that quantum cone algebras are homologically smooth, admit rich differential calculi, and exhibit boundary characteristics, advancing the understanding of non-commutative geometric structures.

Findings

Quantum cones are homologically smooth.

Differential calculi on quantum cones resemble those on smooth complex curves.

Volume forms on quantum cones are exact, indicating boundary-like properties.

Abstract

The algebras obtained as fixed points of the action of the cyclic group $Z_N$ on the coordinate algebra of the quantum disc are studied. These can be understood as coordinate algebras of quantum or non-commutative cones. The following observations are made. First, contrary to the classical situation, the actions of $Z_N$ are free and the resulting algebras are homologically smooth. Second, the quantum cone algebras admit differential calculi that have all the characteristics of calculi on smooth complex curves. Third, the corresponding volume forms are exact, indicating that the constructed algebras describe manifolds with boundaries.