TY - JOUR

T1 - Virtual mutations of weighted surface algebras

AU - Holm, Thorsten

AU - Skowroński, Andrzej

AU - Skowyrski, Adam

PY - 2023/4/1

Y1 - 2023/4/1

N2 - 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.

AB - 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.

KW - Derived equivalence

KW - Mutation

KW - Periodic algebra

KW - Symmetric algebra

KW - Tame algebra

KW - Weighted surface algebra

UR - http://www.scopus.com/inward/record.url?scp=85146069378&partnerID=8YFLogxK

U2 - 10.1016/j.jalgebra.2022.11.026

DO - 10.1016/j.jalgebra.2022.11.026

M3 - Article

VL - 619

SP - 822

EP - 859

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

ER -