Braided Hopf algebras and gauge transformations

TL;DR

This paper explores the structure of infinitesimal gauge transformations within noncommutative geometry using braided Hopf and Lie algebras, including their deformations and specific quantum examples.

Contribution

It introduces a framework for understanding gauge transformations as braided Lie algebras and analyzes their deformations via Drinfeld twists, with explicit quantum examples.

Findings

Braided Lie algebra of gauge transformations for quantum bundles described.

Drinfeld twist deformations of braided Hopf and Lie algebras studied.

Explicit examples on quantum spheres provided.

Abstract

We study infinitesimal gauge transformations of an equivariant noncommutative principal bundle as a braided Lie algebra of derivations. For this, we analyse general $K$-braided Hopf and Lie algebras, for $K$ a (quasi)triangular Hopf algebra of symmetries, and study their representations as braided derivations. We then study Drinfeld twist deformations of braided Hopf algebras and of Lie algebras of infinitesimal gauge transformations. We give examples coming from deformations of abelian and Jordanian type. In particular we explicitly describe the braided Lie algebra of gauge transformations of the instanton bundle and of the orthogonal bundle on the quantum sphere $S^4_\theta$.