Abstract
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}.
Highlights
Understanding the geometry of noncommutative space is believed to be crucial in order to approach a quantum theory of gravity
Other interesting approaches to matrix regularizations have appeared which focus slightly more on approximation properties, as well as connections to physics, and how geometry emerges from limits of matrix algebras
We study low dimensional representations of these algebras for arbitrary genus greater than one
Summary
Understanding the geometry of noncommutative space is believed to be crucial in order to approach a quantum theory of gravity. Both String theory (via the IKKT model [15]) and the matrix regularization of Membrane theory [9, 12], being candidates for describing quantum effects in gravity, contain noncommutative (matrix) analogues of 2-dimensional manifolds. [10, 12,13,14, 19]), but understanding the case of higher genus has turned out to be more difficult. There are several results that treat the case of higher genus and prove the existence
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