Quantum Euclidean Spaces with Noncommutative Derivatives

TL;DR

This paper explores quantum Euclidean spaces with noncommutative derivatives, developing a symbol calculus and a local index formula, advancing understanding of noncommutative geometry structures.

Contribution

It introduces a framework for pseudo-differential operators with noncommuting derivatives and derives a simplified local index formula for quantum Euclidean spaces.

Findings

Established a symbol calculus for noncommutative derivatives

Derived a local index formula similar to the classical case

Provided examples of semi-finite spectral triples with non-flat geometry

Abstract

Quantum Euclidean spaces, as Moyal deformations of Euclidean spaces, are the model examples of noncompact noncommutative manifold. In this paper, we study the quantum Euclidean space equipped with partial derivatives satisfying canonical commutation relation (CCR). This gives an example of semi-finite spectral triple with non-flat geometric structure. We develop an abstract symbol calculus for the pseudo-differential operators with noncommuting derivatives. We also obtain a simplified local index formula (even case) that is similar to the commutative setting.