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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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