# Higher Tetrahedral Algebras

## TL;DR

This paper introduces higher tetrahedral algebras, a new class of finite-dimensional tame symmetric algebras linked to tetrahedral quivers, and characterizes their periodicity based on singularity.

## Contribution

It defines higher tetrahedral algebras, explores their properties, and establishes a criterion for their periodicity related to non-singularity.

## Key findings

- Higher tetrahedral algebras are tame symmetric algebras associated with tetrahedral quivers.
- They are classified within algebras of generalised quaternion type but are distinct from weighted surface algebras.
- A higher tetrahedral algebra is periodic if and only if it is non-singular.

## Abstract

We introduce and study the higher tetrahedral algebras, an exotic family of finite-dimensional tame symmetric algebras over an algebraically closed field. The Gabriel quiver of such an algebra is the triangulation quiver associated to the coherent orientation of the tetrahedron. Surprisingly, these algebras occurred in the classification of all algebras of generalised quaternion type, but are not weighted surface algebras. We prove that a higher tetrahedral algebra is periodic if and only if it is non-singular.