Virtual mutations of weighted surface algebras

TL;DR

This paper introduces and studies virtual mutations of weighted surface algebras, expanding the class of symmetric tame periodic algebras of period 4, and shows their derived equivalence but non-isomorphism to original weighted surface algebras.

Contribution

It defines virtual mutations of weighted surface algebras, demonstrating they form a new large class of symmetric tame periodic algebras of period 4.

Findings

All virtual mutated algebras are derived equivalent to weighted surface algebras.

These algebras are not isomorphic to the original weighted surface algebras.

The construction associates such algebras to triangulated surfaces via blow-ups and mutations.

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.