# Low dimensional matrix representations for noncommutative surfaces of arbitrary genus

## TL;DR

This paper explores finite-dimensional matrix representations of noncommutative deformations of compact surfaces of any genus, explicitly constructing low-dimensional irreducible representations and analyzing their structure.

## Contribution

It introduces a study of finite-dimensional representations for algebras related to noncommutative surfaces of arbitrary genus, including explicit constructions for genus greater than one.

## Key findings

- Explicit two and three-dimensional irreducible representations for higher genus surfaces.
- Representation existence depends on the polynomial's analytic structure.
- Graph representations help understand matrix element structures.

## 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 $\mathbb{R}^3$.