  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 $\mathbb{R}^3$.
