Dirac Operators on Quantum Weighted Projective Spaces

TL;DR

This paper constructs and analyzes Dirac operators on quantum weighted projective spaces, providing spectral triples that extend classical geometric concepts into the quantum algebra setting.

Contribution

It introduces two new spectral triples on quantum weighted projective spaces, one odd and one even, based on coinvariant spinors and quantum group structures.

Findings

Construction of 2-summable spectral triples for each index pair (k,l)

Development of an odd spectral triple using coinvariant spinors

Development of an even spectral triple on quantum weighted projective spaces

Abstract

The quantum weighted projective algebras $\mathbb{C}[\mathbb{WP}_{k,l,q}]$ are coinvariant subalgebras of the quantum group algebra $\mathbb{C}[SU_{q,2}]$. For each pair of indices $k,l$, two $2$-summable spectral triples will be constructed. The first one is an odd spectral triple based on coinvariant spinors on $\mathbb{C}[SU_{q,2}]$. The second one is an even spectral triple.