Braided symmetries of $SU_{q,\phi}(2)$ and Podle\'s Spheres

TL;DR

This paper develops explicit braided symmetries for quantum spheres using a braided quantum Hopf algebra, advancing the understanding of their algebraic structures and symmetries.

Contribution

It introduces a braided quantum Hopf algebra and demonstrates that quantum spheres are braided Hopf modules over it, providing a systematic development of braided algebraic structures.

Findings

Explicit form of braided symmetries for quantum spheres

Introduction of a braided quantum Hopf algebra $U_{q, phi}$

Quantum spheres are shown to be braided Hopf modules over this algebra

Abstract

We present an explicit form of braided symmetries of the quantum spheres, by introducing a braided quantum Hopf algebra $\cU_{q, \phi}$ and demonstrating that they are braided Hopf modules over this braided Hopf algebra. To obtain this result, we systematically develop braided versions of structures like pairing between Hopf algebras, left module algebra as well as their compatibility with the $*$-structure.