Spectral geometry of $\kappa$-Minkowski space

TL;DR

This paper explores the spectral geometry of the $k$-Minkowski space, a noncommutative space modeled by Lie algebra coordinates, using quantum groups and noncommutative geometry tools to analyze its spectral properties and Dirac operators.

Contribution

It introduces a spectral triple for $k$-Minkowski space, connecting the cyclic integral to the Dixmier trace and constructing Dirac operators with symmetry properties.

Findings

Constructed a representation of $k$-Minkowski algebra on a Hilbert space.

Established a spectral triple whose Dixmier trace recovers the cyclic integral.

Developed two Dirac operators, one linked to the spectral triple and another equivariant under quantum Euclidean group.

Abstract

After recalling Snyder's idea of using vector fields over a smooth manifold as `coordinates on a noncommutative space', we discuss a two dimensional toy-model whose `dual' noncommutative coordinates form a Lie algebra: this is the well known $\kappa$-Minkowski space. We show how to improve Snyder's idea using the tools of quantum groups and noncommutative geometry. We find a natural representation of the coordinate algebra of $\kappa$-Minkowski as linear operators on an Hilbert space study its `spectral properties' and discuss how to obtain a Dirac operator for this space. We describe two Dirac operators. The first is associated with a spectral triple. We prove that the cyclic integral of M. Dimitrijevic et al. can be obtained as Dixmier trace associated to this triple. The second Dirac operator is equivariant for the action of the quantum Euclidean group, but it has unbounded commutators with the algebra.