A Hodge - De Rham Dirac operator on the quantum ${\rm SU}(2)$

TL;DR

This work constructs a Hodge–de Rham Dirac operator on the quantum group SU_q(2) using a three-dimensional left covariant calculus, addressing the lack of a canonical exterior calculus in noncommutative spaces. It develops a quantum analogue of the Cartan– Killing metric to define a Hodge duality, then defines and analyzes a quantum Dirac operator 𝔇_(q) on a finite left module, showing its spectrum is a genuine quantum deformation of the classical SU(2) Hodge–de Rham spectrum. The results provide explicit eigenvalues and eigen-spinors, illustrating how degeneracies shift with quantization (via the quantum numbers J,N) and offering a concrete framework for quantum spectral geometry on SU_q(2). These constructions pave the way for extending Dirac-type operators to other quantum homogeneous spaces and different calculi within the SU_q(2) family.

Abstract

We describe how it is possible to describe irreducible actions of the Hodge - de Rham Dirac operator upon the exterior algebra over the quantum spheres ${\rm SU}_q(2)$ equipped with a three dimensional left covariant calculus.