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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Communicated by Y. Kawahigashi
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License ( https://creativecommons.org/licenses/by-nc/2.0 ), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Arnlind, J., Bordemann, M., Hofer, L. et al. Noncommutative Riemann Surfaces by Embeddings in \({\mathbb{R}^{3}}\) . Commun. Math. Phys. 288, 403–429 (2009). https://doi.org/10.1007/s00220-009-0766-8
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DOI: https://doi.org/10.1007/s00220-009-0766-8