A Frobenius algebra is a tuple $(A,\mu,\delta,\eta,\varepsilon)$, where $A$ is a vector space over some field, $(A,\mu,\eta)$ a unital associative algebra, and $(A,\delta,\varepsilon)$ a counital coassociative coalgebra, satisfying the Frobenius relation: $$(\mu\otimes\mathrm{id})\circ(\mathrm{id}\otimes\delta)=\delta\circ\mu=(\mathrm{id}\otimes\mu)\circ(\delta\otimes\mathrm{id})\,.$$ In this setting, $A$ is automatically finite-dimensional since the pairing $\varepsilon\circ\mu$ is non-degenerate.
However, by giving up the (co)unitality, there is still hope to find infinite-dimensional ones (while I could not find them yet). My question is:
- Are there such, non-trivial (i.e. having a non-vanishing (co)product) Frobenius algebras?
- In particular, does the symmetric algebra $S(V)$ over some vector space $V$ admit a structure of a non-trivial Frobenius algebra?
Thank you!
