Recall that a finite dimensional associative algebra $A$ over a field $k$ is called a symmetric Frobenius algebra (sometimes called "closed" Frobenius algebra) if its equipped with a symmetric non degenerate bilinear form $< , >: A \otimes A \to k$ satisfying $<ab,c>=<a,bc>$.

We can obtain a coproduct $v: A \to A \otimes A$ through the following composition of maps:

$A \cong A^* \to (A \otimes A)^* \cong A^* \otimes A^* \cong A \otimes A$

where the first map is the isomorphism of $A$-bimodules between $A$ and its dual $A^*$ defined by the pairing, the second is the dual of the product in $A$, the third follows from the finite dimensionality of $A$, and the last is again obtained from isomorphism induced by the pairing. One can check that that the compatibility of the product in $A$ with $<, >$ implies that $v$ is a map of $A$-bimodules.

This yields the notion of a "Frobenius bialgebra" (sometimes called an "open" Frobenius algebra), which is essentially an associative algebra $A$ equipped with a coassociative coproduct which is a map of $A$-bimodules. A closed Frobenius algebra is equivalent to an open Frobenius algebra with unit and counit.

There is a homotopy notion of a closed Frobenius algebra - I think originally due to Kontsevich- known as a cyclic $A_{\infty}$-algebra, which is by definition a finite dimensional $A_{\infty}$-algebra $(A, m_k:A^{\otimes k} \to A)$ equipped a non degenerate bilinear form $< , >: A \otimes A \to k$ cyclically compatible with the $m_k$'s in the sense that $<m_k(a_1,...,a_k),a_{k+1}>= \pm <m_k(a_{k+1},a_1,...,a_{k-1},a_k>$. In particular, this cyclic compatibility condition implies that the isomorphism $A \to A^*$ is a map of $A_{\infty}$-bimodules over $A$.

My question is the following: Is there an explicit notion of a Frobenius $A_{\infty}$-bialgebra? In particular, given an a cyclic $A_{\infty}$-algebra $A$ we can obtain an $A_{\infty}$-coalgebra structure on $A$ by dualizing the structure maps and using the isomorphism $A \cong A^*$ induced by the pairing as we did above for the strict case. What are the explicit compatibilities between the $A_{\infty}$-algebra and $A_{\infty}$-coalgebra maps characterizing such structure?