| Résume | We construct a notion of quantum surface which enjoys the main nice properties of the smooth version of Connes-Rieffel noncommutative torus. In particular, we give, on a surface of any genus, a family of noncommutative Fréchet algebra structures that closes on the space of smooth functions the surface. The constructed field of Fréchet algebras deforms the commutative one where the space of smooth functions is endowed with the pointwise product. We will discuss also the Hilbert representation theory of those surfaces in connection with the representations of Fuschian arithmetic groups in the discrete series of SL(2,R). |
