# Questions on weakly symmetric algebras

A finite dimensional algebra $$A$$ over a field $$K$$ is called weakly symmetric in case $$soc(P)=top(P)$$ for every indecomposable projective module $$P$$ and it is called symmetric in case $$D(A) \cong A$$ as $$A$$-bimodules.

Questions:

1.In case $$A$$ is representation-finite, does weakly-symmetric imply symmetric?

1. Is there a construction of weakly symmetric algebras over the field $$F_2$$ with two elements that are not symmetric?

2. It is well known that every group algebra over a field is symmetric. Is there a monoid algebra over a field that is weakly-symmetric but not symmetric?

Question 1. has a positive answer for algebraically closed fields by a result of Kupisch ,see Folgerung 2 in https://www.sciencedirect.com/science/article/pii/0021869378901904 .

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