On nearly Frobenius algebras

Question

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?

Answer 1

I realized that I didn't look at the definition careful enough. So I have erased my first answer, since it didn't say anything useful.

The requirements actually says that

$(1)\quad \Delta(ab) = \Delta(a)b$

and

$(2)\quad \Delta(ab) = a\Delta(b).$

In other words, the linear homomorphism $\Delta\colon A \to A\otimes_k A$ should be a homomorphism of $A$-$A$-bimodules. Therefore the Frobenius dimension of an algebra is nothing else than the dimension of the space of homomorphisms

$\operatorname{Hom}_{A\otimes_k A^{\operatorname{op}}}(A,A\otimes_k A).$

This can be easily computed in QPA by the following line of commands:

Aenv := EnvelopingAlgebra( A );
P := DirectSumOfQPAModules( IndecProjectiveModules( Aenv ) );
M := AlgebraAsModuleOverEnvelopingAlgebra( A );
Length( HomOverAlgebra( M, P ) );

The last number is the Frobenius dimension of the algebra $A$.

So, Frobenius dimension = $\dim_k \operatorname{Hom}_{A\otimes_k A^{\operatorname{op}}}( A, A\otimes_k A )$.

The QPA-team.