Restricting the bi-equivariant spectral triple on quantum SU(2) to the Podles spheres

TL;DR

This paper demonstrates how the spectral triple on quantum SU(2) can be restricted to Podles spheres, highlighting unique features of the equatorial sphere related to grading and real structures, and analyzing the algebraic properties involved.

Contribution

It establishes a connection between spectral triples on quantum SU(2) and Podles spheres, identifying conditions for equivariance and real structure compatibility.

Findings

Restriction of spectral triples preserves isospectrality

Equatorial Podles sphere admits a compatible grading and real structure

Failure of the real structure's commutant property relates to the R-matrix unitarity

Abstract

It is shown that the isospectral bi-equivariant spectral triple on quantum SU(2) and the isospectral equivariant spectral triples on the Podles spheres are related by restriction. In this approach, the equatorial Podles sphere is distinguished because only in this case the restricted spectral triple admits an equivariant grading operator together with a real structure (up to infinitesimals of arbitrary high order). The real structure is expressed by the Tomita operator on quantum SU(2) and it is shown that the failure of the real structure to satisfy the commutant property is related to the failure of the universal R-matrix operator to be unitary.