Let $A$ be a quiver algebra over a field $k$ with multiplication $m$. By https://arxiv.org/pdf/1705.10222.pdf definition 6, $A$ is called nearly Frobenius in case there exists a (non-zero? Was this forgotten in the definition?) linear map $\Delta : A \rightarrow A \otimes_k A$ such that the following holds:

(1) $\Delta m = (1 \otimes m) (\Delta \otimes 1)$

(2) $\Delta m = (m \otimes 1) (1 \otimes \Delta)$.

The Frobenius dimension is defined as the vector space dimension of the space of such $\Delta$ (so $A$ is nearly Frobenius iff the Frobenius dimension is non-zero?).

This a generalisation of Frobenius algebras.

Question 1: For Frobenius algebras there are many equivalent characterisations, for example $A \cong D(A)$, is there also a characterisation of nearly Frobenius algebras by some isomorphism of modules (or any other non-obvious characterisation)?

Question 2: Is there a way to calculate the Frobenius dimension using the GAP-package QPA?