Are there examples of finite-dimensional complex non-semisimple non-commutative symmetric Frobenius algebras? 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 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?
 A: 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.
A: I just realize that my question is actually rather trivial the way I posed it: There exist non-commutative semisimple examples (with the $2\times 2$ matrix algebra being the smallest example). There also exist commutative non-semisimple examples, the smallest one given by
$$e_0\cdot e_0 = e_0,\quad e_0\cdot e_1=e_1,\\
e_1\cdot e_0=e_1,\quad \mu(e_1)=1\;,$$
where $\mu$ is the linear form generating the symmetric non-degenerate pairing and all other entries $0$.
All one needs to do is consider direct sums or tensor products of a non-commutative (but semisimple) and a non-semisimple (but commutative) example, and it will be both non-commutative and non-semisimple. The smallest such example is of dimension $6$.
I guess a more sensible but a bit artificial question to ask would have been whether there are examples that do not come from the tensor product or direct sum of a commutative and a semi-simple example.
