Noncommutative Riemann Surfaces by Embeddings in R^3

We introduce C-Algebras of compact Riemann surfaces $${\Sigma}$$ as non-commutative analogues of the Poisson algebra of smooth functions on $${\Sigma}$$ . Representations of these algebras give rise to sequences of matrix-algebras for which matrix-commutators converge to Poisson-brackets as N → ∞. For a particular class of surfaces, interpolating between spheres and tori, we completely characterize (even for the intermediate singular surface) all finite dimensional representations of the corresponding C-algebras

Fuzzy Torus via q-Parafermion

We note that the recently introduced fuzzy torus can be regarded as a q-deformed parafermion. Based on this picture, classification of the Hermitian representations of the fuzzy torus is carried out. The result involves Fock-type representations and new finite dimensional representations for q being a root of unity as well as already known finite dimensional ones.Comment: 12pages, no figur