A note on coalgebra gauge theory

Abstract

The formulation of classical gauge field theory in terms of thetheory of fiber bundles is well known. The gauge field isdescribed by a connection on a certain fiber bundle with a gaugegroup as structure group. Recently the description of space timeat the Planck scale has been one of the fundamental problems oftheoretical physics. There is an approach to this subject basedon the concept of q deformations and quantum groups. It isbased on the generalization of the standard notion of the theoryof fibre bundles and corresponding gauge field theory. Such ageneralization has been given recently by S. Majid and the authorof the paper. It has been expressed in terms of Hopf algebras andtheir actions and coactions on algebras. In the present noteanother generalization of quantum principal bundles is brieflydescribed and summarized. Some of the results are based on thecollaboration of the author with Majid. This report containsessential ideas concerning the notion of so called ψbundles. Details are published elsewhere. A few examples aregiven. The main notion of ψ bundles is based on theentwining structure of coalgebra and algebra. The formalismpresented in the paper should be important for the furtherdevelopment of braided gauge theory.\par {For the entirecollection see MR \Cite{Budzynski97:Quantum:Polish}[98f:00028].}