Braided Hopf algebras and gauge transformations. II: \(\ast\)-structures and examples (Q6155598)

In this paper the authors pursue the study of the notions developed in their previous work [``Braided Hopf algebras and gauge transformations'', Preprint.\url{ arXiv:2203.13811}]. Their focus is on \(K\)-equivariant Hopf-Galois extensions, where \((K,R)\) is a triangular Hopf algebra, and their braided Lie algebras of gauge symmetries. They present here a systematic analysis of \(*\)-structures on braided Hopf algebras associated with quasitriangular Hopf algebras and study their compatibility with actions on \(*\)-algebras. In the triangular case, they further consider braided Lie \(*\)-algebras and their representations on \(*\)-algebras. Examples of infinite dimensional braided Lie and Hopf algebras of infinitesimal gauge transformations of bundles on noncommutative spheres are presented; these are given by two Hopf-Galois extensions, the \textit{instanton bundle} and the \textit{orthogonal bundle}, of the algebra \(\mathcal O(S^4_{\theta})\) of the noncommutative 4-sphere of \textit{A. Connes} and \textit{G. Landi} [Commun. Math. Phys. 221, No. 1, 141--159 (2001; Zbl 0997.81045)] associated to an abelian twist.