The aim of the paper is to attach a noncommutative cluster-like structure to each marked surface Σ. This is a noncommutative algebra AΣ generated by “noncommutative geodesics” between marked points subject to certain triangle relations and noncommutative analogues of Ptolemy–Plücker relations. It turns out that the algebra AΣ exhibits a noncommutative Laurent Phenomenon with respect to any triangulation of Σ which confirms its “cluster nature”. As a surprising byproduct, we obtain a new topological invariant of Σ which is a free or a 1-relator group easily computable in terms of any triangulation of Σ. Another application is the proof of Laurentness and positivity of certain discrete noncommutative integrable systems.
All Science Journal Classification (ASJC) codes
- Laurent Phenomenon
- Noncommutative cluster-like structures
- Noncommutative triangulations