Questions on weakly symmetric algebras

Question

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?

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

3. 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 .

Answer 1

According to QPA the algebra $K<x,y>/(x^2,y^3,(x+y)^3)$ over a field with 2 elements is weakly-symmetric but not symmetric.

Answer 2

After answering a related question that was just posed on this site (Associated graded algebras and symmetric Frobenius algebras) I noticed that the answer also provides an answer to two of the three questions here. So I'm copying the relevant parts of the answer.

There are weakly symmetric algebras of finite representation type over $\mathbb{F}_2$ that are not symmetric. An example can be given as follows. Let $\mathbb{F}_4=\{0,1,\omega,\bar\omega\}$ be the field of four elements, with bar denoting the non-trivial field automorphism, and let $A$ be the four dimensional algebra over $\mathbb{F}_2$ consisting of matrices $\left(\begin{smallmatrix}a&0&0&0\\b&\bar a&0&0\\0&0&\bar a&0\\0&0&\bar b&a\end{smallmatrix}\right)$ with entries in $\mathbb{F}_4$. This algebra has just one simple module, whose dimension is two, so it is weakly symmetric. To see that it is not symmetric, the property of being symmetric is invariant under field extension, and once the field is extended from $\mathbb{F}_2$ to $\mathbb{F}_4$ there are now two one-dimensional simples instead of one two-dimensional simple, and the Nakayama permutation after field extension becomes non-trivial. Note that this example also shows that the property of being weakly symmetric is not invariant under field extension.

To summarise, 1. No, 2. Yes, 3. I don't know much about monoid algebras... (cue music).