Calculi, Hodge operators and Laplacians on a quantum Hopf fibration

TL;DR

This paper develops Laplacian operators and Hodge dualities on quantum groups SUq(2) and the quantum sphere S2q, including gauged Laplacians on line bundle sections, advancing noncommutative geometry tools.

Contribution

It introduces explicit Laplacian and Hodge duality constructions on quantum groups and homogeneous spaces, including gauged Laplacians on quantum line bundles.

Findings

Defined Laplacians on SUq(2) and S2q with specific differential calculi

Constructed Hodge dualities for exterior algebras on these quantum spaces

Analyzed gauged Laplacians on sections of quantum line bundles

Abstract

We describe Laplacian operators on the quantum group SUq (2) equipped with the four dimensional bicovariant differential calculus of Woronowicz as well as on the quantum homogeneous space S2q with the restricted left covariant three dimensional differential calculus. This is done by giving a family of Hodge dualities on both the exterior algebras of SUq (2) and S2q . We also study gauged Laplacian operators acting on sections of line bundles over the quantum sphere.