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In Classification of commutative Frobenius algebras , Jeremy Rickard showed that there are infinitely many commutative (local without loss of generality) Frobenius algebras of vector space dimension 14 over an infinite field.

Question 1: What is the smallest integer $l$ such that there are infinitely many commutative Frobenius algebras up to isomorphism of dimension $l$ (does this depend on the field?)?

In case this is field independent it might be interesting to find all such algebras of a given dimension $\leq l-1$ to do some tests (as there are some open problems on such algebras). This motivates the next question:

Question 2: Is there a quick way, using QPA, to obtain all finite dimensional local quiver algebras whose quiver has $r \geq 2$ loops of a given (small) vector space dimension over a finite field (with lets say 2 or 3 elements)?

This might be used to give a classification of Frobenius algebras over the field with 2 elements for small dimensions.

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    $\begingroup$ In analogous problems for other algebraic structures, it often depends on the field. For instance for nilpotent Lie algebras, in dimension $\le 5$ the classification is finite and uniform in characteristic $\neq 2,3$ for all fields, while in dimension $6$, it's still finite for algebraically closed fields, and some in the list split, over subfields $K$, as a family indexed by $K^*/{K^*}^2$. Then in dimension $\ge 7$ there are families indexed by the field itself. So your question has two distinct aspects, both of interest, one being the algebraically closed case (in large enough char.). $\endgroup$ – YCor Apr 16 '20 at 19:28

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