The necessity of (co)unit in nearly Frobenius algebra

TL;DR

This paper investigates nearly Frobenius algebras, proving that under certain conditions they are actually Frobenius algebras, which has implications for 2D topological quantum field theories and link homology constructions.

Contribution

It establishes that nearly Frobenius algebras with surjective multiplication and injective comultiplication over a principal ideal domain are Frobenius algebras, clarifying their structure.

Findings

Nearly Frobenius algebras with specified properties are Frobenius algebras.

Results have implications for 2D-TQFT and link homology.

Provides algebraic criteria for Frobenius structure.

Abstract

In this article, we concern the concept of nearly Frobenius algebra, which corresponds to most 2D-TQFT of which each cobordism admits no critical points of index 0 or 2. We prove that any nearly Frobenius algebra over a principal ideal domain with surjective multiplication and injective comultiplication is indeed a Frobenius algebra. The motivation of this study mainly emanates from the investigation of potential constructions of link homology.