Biserial algebras via subalgebras and the path algebra of D_4

TL;DR

This paper introduces two new criteria to identify biserial algebras, based on subalgebra properties and module conditions, generalizing Nakayama criteria and relating to the path algebra of D_4.

Contribution

It provides novel criteria for recognizing biserial algebras through subalgebra and module conditions, expanding the understanding of their structure.

Findings

An algebra is biserial iff all subalgebras supported on at most 4 vertices are biserial.

Certain modules with properties similar to the (1,1,1,1) dimension vector in D_4 are forbidden in biserial algebras.

The criteria generalize existing Nakayama algebra conditions.

Abstract

We give two new criteria for a basic algebra to be biserial. The first one states that an algebra is biserial iff all subalgebras of the form eAe where e is supported by at most 4 vertices are biserial. The second one gives some condition on modules that must not exist for a biserial algebra. These modules have properties similar to the module with dimension vector (1,1,1,1) for the path algebra of the quiver D_4. Both criteria generalize criteria for an algebra to be Nakayama. They rely on the description of a basic biserial algebra in terms of quiver and relations given by R. Vila-Freyer and W. Crawley- Boevey.