(Co)bordism groups in quantum super PDE's. I: Quantum supermanifolds.

Following our previous works on noncommutative manifolds and noncommutative PDE's, we consider in these series of three papers, some further results on quantum supermanifolds and quantum super PDE's. In particular, in this first part we focus our attention on quantum supermanifolds. These structures globalize the notion of quantum superalgebras, obtaining noncommutative manifolds that are useful to give a fully covariant description of noncommutative geometric structures, hence of quantum physics. We study the geometry of quantum supermanifolds and characterize their (co)homological and (co)bordism properties. Covariant quantizations of super PDE's are given, obtaining examples of quantum supermanifolds justifying their definitions.