I was trying to define a Frobenius Algebra structure over complex polynomials modulo $x^2$, but I am really struggling to do so. The algebra structure is rather evident, but I've tried many possible coproducts without success so far. My intuition tells me that the counit linear map will send every polynomial to its evaluation on $x=0$ (so the free coefficient), and I tried defining the coproduct as
$$\Delta(f(x)) = f(x) ⊗ 1(x) + 1(x) ⊗ f(x)$$
It does give me a coproduct structure, but it fails to work on the Frobenius structure, since it doesn't make the diagram below commute (at least the lower triangle doesn't commute).
Is there any known Frobenius structure for this problem?
