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Noncommutative Riemann Surfaces by Embeddings in $${mathbb{R}^{3}}$$
Authors:Joakim Arnlind   Martin Bordemann   Laurent Hofer   Jens Hoppe  Hidehiko Shimada
Affiliation:1.Institut des Hautes études Scientifiques,Bures-sur-Yvette,France;2.Max Planck Institute for Gravitational Physics,Golm,Germany;3.Laboratoire de MIA, 4, rue des Frères Lumière,Université de Haute-Alsace,Mulhouse,France;4.Université du Luxembourg,Luxembourg City,Luxembourg;5.Department of Mathematics,KTH,Stockholm,Sweden
Abstract: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.
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