BRST cohomology in classical mechanics

The classical (non-quantum) cohomology of the Becchi-Rouet-Stora-Tyutin (BRST) symmetry in phase space is defined and worked out. No group action for the gauge transformations is assumed. The results cover, therefore, the general case of an open algebra and are valid off-shell. Each cohomology class contains all BRST invariant functions with fixed ghost number (an integer) which differ from each other by a BRST variation. If the ghost number is negative there is only the trivial class whose elements are equivalent to zero. If the ghost number is positive or zero there is a bijective correspondence between the BRST classes and those of the exterior derivative along the gauge orbits. These gauge orbits lie in the phase space surface on which the gauge generators are constrained to vanish. The BRST invariant functions of ghost numberp are then related to closedp-forms along the orbits. The addition of a BRST variation corresponds to the addition of an exact form. Some comments about the quantum case are included. The physical meaning of the classes with ghost number greater than zero is not discussed.Chercheur qualifié du Fonds National de la Recherche Scientifique (Belgium)     

Finite dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra

Let be a complex simple Lie algebra. We show that whent is not a root of 1 all finite dimensional representations of the quantum analogU _t are completely reducible, and we classify the irreducible ones in terms of highest weights. In particular, they can be seen as deformations of the representations of the (classical)U .