Constructing nearly Frobenius algebras

TL;DR

This paper explores the properties and constructions of nearly Frobenius algebras, generalizing Frobenius algebras, and characterizes their structures in relation to quivers and specific algebra families.

Contribution

It introduces the concept of nearly Frobenius algebras, studies their structural properties, and provides a classification for certain quiver-related algebras with an algorithm for counting structures.

Findings

Nearly Frobenius algebras do not have traces or self-duality.

Constructed nearly Frobenius structures are preserved under direct sums, tensor products, and quotients.

Only quivers of type A_n with no relations admit non-trivial nearly Frobenius structures.

Abstract

In the first part we study nearly Frobenius algebras. The concept of nearly Frobenius algebras is a generalization of the concept of Frobenius algebras. Nearly Frobenius algebras do not have traces, nor they are self-dual. We prove that the known constructions: direct sums, tensor, quotient of nearly Frobenius algebras admit natural nearly Frobenius structures. In the second part we study algebras associated to some families of quivers and the nearly Frobenius structures that they admit. As a main theorem, we prove that an indecomposable algebra associated to a bound quiver $(Q,I)$ with no monomial relations admits a non trivial nearly Frobenius structure if and only if the quiver is $\overrightarrow{\mb{A}_n}$ and I=0. We also present an algorithm that determines the number of independent nearly Frobenius structures for Gentle algebras without oriented cycles.