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Frobenius algebras are finite-dimensional algebras together with a compatible inner product. Commutative Frobenius algebras have attracted recent interest because they're equivalent to 2D oriented TQFTs.
3
votes
A non-example of a graded Frobenius algebra
A pretty trivial example is an algebra with the zero product. Less trivial is $k[x,y]/(x,y)^2$ where $x$ has degree $1$ and $y$ has degree $2$. One can make up various similar examples, e.g., $k[x,y,z …
2
votes
Reference on the classification of (low rank) Gorenstein rings over $\mathbb{C}$
This paper might possibly be relevant:
http://www.ams.org/mathscinet-getitem?mr=2922602
It's not quite exactly what you are asking for. Instead of Gorenstein algebras of low rank, they are consideri …