# Nearly Frobenius structures in some families of algebras

## TL;DR

This paper investigates nearly Frobenius structures across various finite-dimensional algebras, establishing conditions for their existence in radical square zero, string, and toupie algebras, and extending results to quotients of path algebras.

## Contribution

It provides new criteria for the presence of nearly Frobenius structures in specific algebra families and generalizes these conditions to quotients of path algebras.

## Key findings

- Radical square zero algebras with paths of length two are nearly Frobenius.
- Non-gentle string algebras have at least one non-trivial nearly Frobenius structure.
- Monomial relations in toupie algebras imply non-trivial nearly Frobenius structures.

## Abstract

In this article we continue with the study started in [1] of nearly Frobenius structures in some representative families of finite dimensional algebras, as the radical square zero algebras, string algebras and the toupie algebras. We prove that the radical square zero algebras with at least one path of length two are nearly Frobenius. As for the string algebras, in the ones that are not gentle, we can afirm that there is at least one non-trivial nearly Frobenius structure. Finally, in the case of the toupie algebras, we prove that the existence of monomial relations is a suficient condition to have non-trivial nearly Frobenius structure. Using the technics developed for the previous families of algebras we prove suficient conditions for the existence of non-trivial Frobenius structures in quotients of path algebras in general.