A Hodge–de Rham Dirac operator on the quantum SU(2)

We describe how it is possible to define a Hodge–de Rham Dirac operator associated to a suitable Cartan-Killing metric form upon the exterior algebra over the quantum spheres [Formula: see text] equipped with a three-dimensional left covariant calculus.

Projective Modules Over Quantum Projective Line

Taking a groupoid C*-algebra approach to the study of the quantum complex projective spaces [Formula: see text] constructed from the multipullback quantum spheres introduced by Hajac and collaborators, we analyze the structure of the C*-algebra [Formula: see text] realized as a concrete groupoid C*-algebra, and find its [Formula: see text]-groups. Furthermore, after a complete classification of the unitary equivalence classes of projections or equivalently the isomorphism classes of finitely generated projective modules over the C*-algebra [Formula: see text], we identify those quantum principal [Formula: see text]-bundles introduced by Hajac and collaborators among the projections classified.

Inverse Semigroups and Sheu's Groupoid for Odd Dimensional Quantum Spheres

In this paper, we give a different proof of the fact that the odd dimensional quantum spheres are groupoid C*-algebras. We show that the C*-algebra is generated by an inverse semigroup T of partial isometries. We show that the groupoid 𝓖tight associated with the inverse semigroup T by Exel is exactly the same as the groupoid considered by Sheu.

HODGE DUALITY OPERATORS ON LEFT-COVARIANT EXTERIOR ALGEBRAS OVER TWO- AND THREE-DIMENSIONAL QUANTUM SPHERES

Using non-canonical braidings, we first introduce a notion of symmetric tensors and corresponding Hodge operators on a class of left-covariant 3d differential calculi over SU q(2), then we induce Hodge operators on the left-covariant 2d exterior algebra over the Podles quantum sphere.

EXAMPLES OF HODGE LAPLACIANS ON QUANTUM SPHERES

Using a non-canonical braiding over the three-dimensional left covariant calculus we present a family of Hodge operators on SU q(2) and on its homogeneous quantum space [Formula: see text].