TITLE: Different Bialgebra/Hopf algebra structures on coalgebras
QUESTION [7 upvotes]: Given a coalgebra $C$, can there exist more than one algebra structure on $C$ giving it the structure of a bialgebra? I will also ask the same question for Hopf algebras.

REPLY [7 votes]: By dualizing, you are looking for an algebra $A$ such that there are two different coproducts $\Delta_1, \Delta_2: A \to A\otimes A$ that make $A$ into a bialgebra resp. Hopf algebra.
As an example let $p$ be prime, $k$ a field of char. $p$ and let $A=k[x_1,...,x_n]/(x_1^p,...,x_n^p)$ be the truncated polynomial algebra.
$A$ is the group algebra of the elementary abelian group $(\mathbb{Z}/p)^n=\langle g_1,...,g_n\rangle$ via $x_i=g_i-1$. Thus $A$ is a Hopf with coproduct
$\Delta_1(x_i)=x_i\otimes 1 + 1\otimes x_i + x_i\otimes x_i$ and antipode $S_1(x_i)=g_i^{-1}-1=(-x_i)+\cdots +(-x_i)^{p-1}$.
$A$ is also the restricted enveloping algebra of $k^n$ considered as trivial restricted $p$-Lie algebra. As such $A$ is a Hopf algebra with coproduct $\Delta_2(x_i)=x_i \otimes 1 + 1\otimes x_i$ and antipode $S_2(x_i)=-x_i$.
These Hopf algebras where used by Avrunin and Scott in their proof of Carlson's conjecture on the rank variety in group cohomology. For a reference see section 4 of Carlson, Iyengar: Hopf algebra structures and tensor products for group algebras