Can a (finite dimenaional) $\mathbb{K}$-algebra $A$ be equipped with more than one Frobenius structure $\lambda:A \to \mathbb{K}$? Of course we identify two structures $\lambda$ and $\lambda'$ if they differ by a scalar multiple.


If it can what is a good example? If we restrict to filtered Frobenius algebras can this help with uniqueness?



An algebra with more than one Frobenius algebra structure
If $A$ is a Frobenius $K$-algebra and $\lambda\colon A\to K$ is a Frobenius form, then the Frobenius forms are the mappings of the form $a\mapsto \lambda(ua)$ with $u$ a unit of $A$.


One way to see that is being Frobenius means $A\_A\cong \hom\_K({}\_AA,K)$ (where $M\_A$ means $M$ is a right $A$-module and ${}\_AM$ means $M$ is a left $A$-module). A Frobenius form is precisely the image of $1$ under such an isomorphism. Since the group of units of $A$, acting via left multiplication, is the automorphism group of $A$ we see that any isomorphism takes $1$ to $\lambda u$ for some unit $u$.


For example, in @JaSch's answer $\lambda\_2=\lambda\_1(-i(a+ib))$ so the unit $u$ is $-i$.


