Hybrid algebras

TL;DR

This paper introduces hybrid algebras, a new class of symmetric algebras that unify Brauer graph and weighted surface algebras, and explores their structural properties and classifications.

Contribution

It defines hybrid algebras, shows they are blocks of weighted surface algebras up to socle deformation, and classifies the tree classes of certain tame symmetric algebras.

Findings

Hybrid algebras encompass Brauer graph and weighted surface algebras.

Hybrid algebras are blocks of weighted surface algebras up to socle deformation.

The tree class of Auslander-Reiten components is classified for certain tame symmetric algebras.

Abstract

We introduce a new class of symmetric algebras, which we call hybrid algebras. This class contains on one extreme Brauer graph algebras, and on the other extreme general weighted surface algebras. We show that hybrid algebras are precisely the blocks of idempotent algebras of weighted surface algebras, up to socle deformations. More generally, for tame symmetric algebras whose Gabriel quiver is 2-regular, we show that the tree class of an Auslander-Reiten component is Dynkin or Euclidean or one of the infinite tress $A_{\infty}$, $A_{\infty}^{\infty}$, or $D_{\infty}$.