algebra vs Dual of a coalgebra

Question

Let $(A,m,u, \Delta, \varepsilon)$ be a bialgebra. Taking dual, $(A^\star, \Delta^\star,\varepsilon^\star)$ is a algebra. What is the relationship between the two algebras $(A, m, u)$ and $(A^\star, \Delta^\star,\varepsilon^\star)$? Thank you.

Answer 1

There is no apriori relation between these two algebras. In general they are different and non-isomorphic.

But in some cases they maybe isomorphic: take for example the group hopf algebra $kG$ of some finite abelian group $G$. Then, the dual of the coalgebra structure $\big((kG)^\star, \Delta^\star,\varepsilon^\star\big)$ is isomorphic to the original group algebra structure $(kG, m, u)$.

On the other hand, if $G$ is not abelian then $kG$ and $(kG)^*$ are in general non-isomorphic as algebras since the former is non-commutative while the later is commutative. You can see more details (and a full proof) of this point at: https://math.stackexchange.com/a/2056433/195021