Virtual mutations of weighted surface algebras

authored by

Thorsten Holm, Andrzej Skowroński, Adam Skowyrski

Abstract

The finite-dimensional symmetric algebras over an algebraically closed field, based on surface triangulations, motivated by the theory of cluster algebras, have been extensively investigated and applied. In particular, the weighted surface algebras and their deformations were introduced and studied in [16]-[20], and it was shown that all these algebras, except few singular cases, are symmetric tame periodic algebras of period \(4\). In this article, using the general form of a weighted surface algebra from [19], we introduce and study so called virtual mutations of weighted surface algebras, which constitute a new large class of symmetric tame periodic algebras of period \(4\). We prove that all these algebras are derived equivalent but not isomorphic to weighted surface algebras. We associate such algebras to any triangulated surface, first taking blow-ups of a family of edges to \(2\)-triangle discs, and then virtual mutations of their weighted surface algebras. The results of this paper form an essential step towards a classification of all tame symmetric periodic algebras.

Organisation(s)

Institute of Algebra, Number Theory and Discrete Mathematics

External Organisation(s)

Nicolaus Copernicus University

Type

Article

Journal

Journal of Algebra

Volume

619

Pages

822-859

No. of pages

38

ISSN

0021-8693

Publication date

01.04.2023

Publication status

Published

Peer reviewed

Yes

ASJC Scopus subject areas

Algebra and Number Theory

Electronic version(s)

http://arxiv.org/abs/2006.13075v2 (Access: Open)
https://doi.org/10.1016/j.jalgebra.2022.11.026 (Access: Closed)