QUANTUM GAUGE-THEORIES

We consider a gauge theory by taking real quantum groups of nondegener ate bilinear form as a symmetry. The construction of this quantum gaug e theory is developed in order to fit with the Hopf algebra structure. In this framework, we show that an appropriate definition of the infi nitesimal gauge variations and the axioms of the Hopf algebra structur e of the symmetry group lead to the closure of the infinitesimal gauge transformations without any assumption on the commutation rules of th e gauge parameters, the connection and the curvature. An adequate defi nition of the quantum trace is given leading to the quantum Killing fo rm. This is used to construct an invariant quantum Yang-Mills Lagrangi an.