Marked non-orientable surfaces and cluster categories via symmetric representations

TL;DR

This paper explores the representation theory of non-orientable surfaces by developing a symmetric representation framework that links geometric curves to algebraic objects, advancing the categorification of quasi-cluster algebras.

Contribution

It introduces a symmetric representation approach to non-orientable surfaces, connecting curves to indecomposable symmetric objects and establishing a new symmetric extension concept.

Findings

Correspondence between curves on non-orientable surfaces and symmetric objects

Definition of symmetric extension and its relation to arcs and quasi-arcs

Quasi-triangulations correspond to symmetric cluster tilting objects

Abstract

We initiate the investigation of representation theory of non-orientable surfaces. As a first step towards finding an additive categorification of Dupont and Palesi's quasi-cluster algebras associated marked non-orientable surfaces, we study a certain modification on the objects of the cluster category associated to the orientable double covers in the unpunctured case. More precisely, we consider symmetric representation theory studied by Derksen-Weyman and Boos-Cerulli Irelli, and lift it to the cluster category. This gives a way to consider `indecomposable orbits of objects' under a contravariant duality functor. Hence, we can assign curves on a non-orientable surface $(\mathbb{S}, \mathbb{M})$ to indecomposable symmetric objects. Moreover, we define a new notion of symmetric extension, and show that the arcs and quasi-arcs on $(\mathbb{S}, \mathbb{M})$ correspond to the indecomposable symmetric objects without symmetric self-extension. Consequently, we show that quasi-triangulations of $(\mathbb{S}, \mathbb{M})$ correspond to a symmetric analogue of cluster tilting objects.