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)?
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> Question 2: Is there a way to calculate the Frobenius dimension using the GAP-package QPA?
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On nearly Frobenius algebras
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.


