Weighted surface algebras: general version. Corrigendum

TL;DR

This paper corrects and clarifies the definition of weighted surface algebras, expanding their class by allowing arrows outside the Gabriel quiver and ensuring they remain symmetric and properly dimensioned.

Contribution

It provides the corrected and complete definition of weighted surface algebras, addressing previous omissions and modifications needed for symmetry and dimension consistency.

Findings

Corrected the zero relations for the algebra class.

Expanded the class to include arrows outside the Gabriel quiver.

Ensured the algebras are symmetric and dimensionally accurate.

Abstract

In [arXiv:1902.04063] we generalize the original definition of weighted surface algebras in [arXiv:1703.02346] by allowing the possibility that arrows might not be part of the Gabriel quiver, which gives a much larger class of algebras. This means that the zero relations need modification, to make sure that the algebras are symmetric, and of the appropriate dimension. We found recently that we had missed one necessary modification for the zero relations. Here we give the correct definition, and revise the parts of [arXiv:1902.04063] which are affected by this modification.