5
$\begingroup$

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 .

$\endgroup$
0

1 Answer 1

1
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.