Nearly Frobenius dimension on Frobenius algebras

TL;DR

This paper explores the Frobenius space structure in Frobenius algebras over fields and extends the concept to nearly Frobenius algebras over commutative rings, linking to solutions of the Yang-Baxter equation.

Contribution

It proves the Frobenius space is generated by the coproduct and introduces nearly Frobenius algebras with new characterizations and applications.

Findings

Frobenius dimension equals the algebra's dimension over a field

Constructs solutions to the Yang-Baxter equation from Frobenius space elements

Provides characterizations of nearly Frobenius algebras

Abstract

This article is divided into two parts. In the first part we work over a field $\mathbb{k}$ and prove that the Frobenius space associated to a Frobenius algebra is generated as left A-module by the Frobenius coproduct. In particular, we prove that the Frobenius dimension coincides with the dimension of the algebra. In the second part we work with a commutative ring $k$. We introduce the concept of nearly Frobenius algebras in this context and construct solutions of the Yang-Baxter equation starting from elements in the Frobenius space. Also, we give a list of equivalent characterizations of nearly Frobenius algebras.