In this note, we initiate a study of the finite-dimensional representation theory of a class of algebras that correspond to noncommutative deformations of compact surfaces of arbitrary genus. Low dimensional representations are investigated in detail and graph representations are used in order to understand the structure of non-zero matrix elements. In particular, for arbitrary genus greater than one, we explicitly construct classes of irreducible two and three dimensional representations. The existence of representations crucially depends on the analytic structure of the polynomial defining the surface as a level set in $\mathbb{R}^3$.

中文翻译：

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任意属非对易曲面的低维矩阵表示

在这篇笔记中，我们开始研究一类代数的有限维表示理论，这些代数对应于任意属的紧密表面的非对易变形。详细研究了低维表示，并使用图表示来理解非零矩阵元素的结构。特别是，对于大于 1 的任意属，我们明确地构造了不可约的二维和三维表示的类。表示的存在关键取决于多项式的解析结构，该多项式将表面定义为 $\mathbb{R}^3$ 中的水平集。