Braided Hopf algebras and gauge transformations

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 θ .Date: March 2022. 1 7. Braided Hopf algebra of gauge transformations 38 7.1. Quantum principal bundle over quantum homogeneous space 40 8. Braided Hopf algebra gauge symmetry from twist deformation 42 8.1. Gauge transformations of twisted principal bundles 43 Appendix A. Proof of Proposition 4.14 52 References 54

Levi–Civita Connections on Quantum Spheres

“…Thus we do not require A-linearity in the argument ∂ of ∇ ∂ . This is in contrast with the braided geometry framework [4,6] where for a braided commutative algebra the braided Lie algebra of its braided derivations is a module over the algebra and such a A-linearity on the connection can be stated. Braided commutativity of a Hopf algebra is a feature of its being cotriangular (and not just coquasitriangular).…”

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

“…The main objects investigated in this paper are K-equivariant Hopf-Galois extensions, for (K, R) a triangular Hopf algebra, and their braided Lie algebras of gauge symmetries. We briefly recall from [5] the main notions and results that are needed.…”

“…In our previous paper [5] we looked at the problem from the infinitesimal view-point by considering algebra derivations, rather than algebra morphisms. We then considered the case of H-Hopf-Galois extensions which are equivariant under a triangular Hopf algebra (K, R).…”

“…The construction is shown to be compatible with the theory of Drinfeld twists, and thus suitable for the study of noncommutative principal bundles that are obtained via twist deformation (quantization) of classical ones. We refer to [5] for details and for a discussion of different approaches and of the literature on the subject.…”

“…In the present paper we illustrate the general theory, developed in [5], with the computation of the braided Lie algebras of infinitesimal gauge transformations of two important examples of noncommutative principal bundles. These are given by two Hopf-Galois extensions of the algebra O(S 4 θ ) of the noncommutative 4-sphere S 4 θ of [8].…”