Are there any examples of finite-dimensional complex non-semisimple non-commutative symmetric Frobenius algebras? Or can one show that none exist?


I went through [this](https://arxiv.org/pdf/0910.0932.pdf) list of all complex associative algebras up to dimension $4$ and couldn't find any non-commutative non-semisimple ones that can be equipped with a corresponding linear form to make them into a symmetric Frobenius algebra. Are there examples in higher dimensions?



Are there examples of finite-dimensional complex non-semisimple non-commutative symmetric Frobenius algebras?
Given any finite dimensional algebra $A$, consider the linear dual $\hat{A}= \hom(A, k)$ as an $A$-$A$-bimodule. Then $R = A \oplus \hat{A}$ may be equipped with an algebra structure as follows:


$$(a, x) \cdot (b,y) = (ab, x \cdot b + a \cdot y)$$


for $a,b \in A$ and $x,y \in \hat{A}$. The algebra $R$ has a natural symmetric Frobenius algebra structure. So every finite dimensional (possibly non-commutative) algebra embeds into a symmetric Frobenius algebra.


