# Weighted Surface Algebras

## TL;DR

This paper introduces weighted surface algebras associated with triangulated surfaces, demonstrating they are symmetric tame periodic algebras of period 4, advancing the classification of periodic symmetric tame algebras.

## Contribution

It defines weighted surface algebras for triangulated surfaces and proves their properties, including symmetry, tameness, and periodicity, contributing to the classification of such algebras.

## Key findings

- All weighted surface algebras, except singular tetrahedral ones, are symmetric tame periodic algebras of period 4.
- Socle deformations of these algebras also share the same properties.
- Orbit closures contain classes of tame symmetric algebras related to dihedral and semidihedral types.

## Abstract

A finite-dimensional algebra $A$ over an algebraically closed field $K$ is called periodic if it is periodic under the action of the syzygy operator in the category of $A-A-$ bimodules. The periodic algebras are self-injective and occur naturally in the study of tame blocks of group algebras, actions of finite groups on spheres, hypersurface singularities of finite Cohen-Macaulay type, and Jacobian algebras of quivers with potentials. Recently, the tame periodic algebras of polynomial growth have been classified and it is natural to attempt to classify all tame periodic algebras. We introduce the weighted surface algebras of triangulated surfaces with arbitrarily oriented triangles and describe their basic properties. In particular, we prove that all these algebras, except the singular tetrahedral algebras, are symmetric tame periodic algebras of period $4$. Moreover, we describe the socle deformations of the weighted surface algebras and prove that all these algebras are symmetric tame periodic algebras of period $4$. The main results of the paper form an important step towards a classification of all periodic symmetric tame algebras of non-polynomial growth, and lead to a complete description of all algebras of generalized quaternion type. Further, the orbit closures of the weighted surface algebras (and their socle deformations) in the affine varieties of associative $K$-algebra structures contain wide classes of tame symmetric algebras related to algebras of dihedral and semidihedral types, which occur in the study of blocks of group algebras with dihedral and semidihedral defect groups.