Braided Hopf algebras and gauge transformations II: $*$-structures and examples

TL;DR

This paper explores noncommutative principal bundles with triangular Hopf algebra symmetry, providing explicit examples of braided Lie and Hopf algebras of gauge transformations on noncommutative spheres, and analyzing compatible *-structures.

Contribution

It introduces explicit examples of braided gauge transformation algebras on noncommutative spheres and systematically studies compatible *-structures, including quasitriangular cases.

Findings

Explicit examples of braided Lie and Hopf algebras of gauge transformations.

Systematic analysis of compatible *-structures.

Implementation of braiding via the triangular structure.

Abstract

We consider noncommutative principal bundles which are equivariant under a triangular Hopf algebra. We present explicit examples of infinite dimensional braided Lie and Hopf algebras of infinitesimal gauge transformations of bundles on noncommutative spheres. The braiding of these algebras is implemented by the triangular structure of the symmetry Hopf algebra. We present a systematic analysis of compatible $*$-structures, encompassing the quasitriangular case.