Generalized weighted surface algebras

TL;DR

This paper introduces generalized triangulation quivers extending surface triangulation concepts and studies associated weighted algebras, revealing their construction from surface triangulations with marked self-folded triangles and their mutation properties.

Contribution

It extends the notion of triangulation quivers to a generalized setting and explores the structure and mutation behavior of the associated weighted algebras.

Findings

Generalized triangulation quivers can be constructed from surface triangulations with self-folded triangles.

Weighted generalized triangulation algebras naturally arise from mutations of existing algebras.

The paper establishes foundational properties of these new algebraic structures.

Abstract

The weighted triangulation algebras associated to triangulation quivers and their socle deformations were recently introduced and studied in [15]-[20] and [2]. These algebras, based on surface triangulations and originated from the theory of cluster algebras, were also proved to be (with some minor exceptions) finite-dimensional tame symmetric and periodic algebras of period 4. In this paper, we introduce a new concept of a generalized triangulation quiver, extending the notion of a triangulation quiver. Inparticular, it is also shown that the generalized triangulation quivers can be constructed from triangulations of orientable surfaces with marked self-foldedtriangles. Moreover, motivated by the recent results of [28], we define and investigate so called weighted generalized triangulation algebras associated to generalized triangulation quivers, which naturally arise from mutations of weighted triangulation algebras.