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Given a commutative Frobenius algebra, finite dimensional over a field $k$. We assume that the algebra is connected and in fact given by quiver and relations. Let $S$ be the unique simple modules of such an algebra and $A^e= A \otimes_K A$ the enveloping algebra.

I have not much experience on those algebras (so there is a chance that this question is stupid) but I wonder if one can characterise them using certain sequences. Let $V$ be the set of all maps $\mathbb{N} \rightarrow \mathbb{N}$ and $X=X_K$ be the set of commutative Frobenius algebras over a fixed field $K$. We can assume first that the field is finite for simplicity.

Let $\psi: X \rightarrow V$ be the map $\psi(A)=f_A$ with $f_A (n) =dim ( \Omega_A^n(S))$.

Question 1: Is $\psi$ injective? What is the image?

For example the algebra $K[X]/(X^p)$ gives the sequence $(p-1,1,p-1,1,p-1,1,....)$.

Let $\phi: X \rightarrow V$ be the map $\psi(A)=g_A$ with $g_A (n) =dim ( \Omega_{A^e}^n(A))$.

For example the algebra $K[X]/(X^p)$ gives the sequence $(p^2-p,p,p^2-p,p,p^2-p,p,p^2-p,....)$.

Question 2: Is $\phi$ injective? What is the image?

Question 3 is how one might modify the maps in order to be able to characterise commutative Frobenius algebras via sequences, if possible at all. Maybe take the map $\zeta : X \rightarrow V \times V$ with $ \zeta(A)=(f_A,g_A)$ in case injectivity fails in 1 or 2.

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