Are there attempts to classify commutative finite dimensional Frobenius algebras? They appear often in mathematics, such as in algebraic geometry and the famous category equivalence between commutative Frobenius algebras and 2-dimension topolocial quantum field theories. However, I have not yet seen attempts to classify this class of algebras (up to isomorphism of k-algebras). Recall that a commutative Frobenius algebra is a finite dimensional algebra $A$ with $A \cong D(A)$ or equivalently simple socle in case it is local.
Here are two questions related to such a classification (we can assume that commutative Frobenius algebra are connected):
Question 1: Is a commutative Frobenius algebra "field-independent"? This means that in its presentation $KQ/I$ by quiver and relations, there exists such $I$ which only contains the field element 1 or -1 so that a given commutative Frobenius algebra is defined over all fields.
In case question 1 has a positive answer, this would mean that a classification is independet of the field (maybe excluding characteristic 2).
Question 2: For a given integer $d$, are the only finitely many $d$-dimensional commutative Frobenius algebras of vector space dimension $d$?
(here we say that two algebras are isomorphic in case they are isomorphic as $K$-algebras)
A positive answer to question 2 would be surprising, but I think it should be true for $d \leq 5$ at least.