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.

3$\begingroup$ by PBW, universal enveloping algebra of a Lie algebra is isomorphic to a symmetric coalgebra (in char zero at least). So the coalgebra structure depends only on dimension, and from the bialgebra structure you can reconstruct the Lie algebra you started with $\endgroup$– Grisha PapayanovSep 17 at 19:55
Yes: if $k$ is a field of characteristic 2, let $C$ be the coalgebra over $k$ spanned by 1, $x$, $y$, and $z$ with $x$ and $y$ primitive, $\Delta z = z \otimes 1 + 1 \otimes z + x \otimes y + y \otimes x$ — in the algebra structure, $z$ is going to equal $xy = yx$, so $\Delta z$ has to equal $(\Delta x)(\Delta y)$. Then put algebra structures on this as follows:
 $xy=yx$
 $y^2 = 0$
 either $x^2 = 0$ or $x^2 = y$
This gives two graded (with $\deg x = 1$, $\deg y = 2$) connected cocommutative bialgebras, and so by a theorem of Milnor and Moore, there is a unique antipode making such a bialgebra into a Hopf algebra.

$\begingroup$ I can believe it, but is it obvious those two algebras are nonisomorphic? $\endgroup$– LSpiceSep 17 at 21:57

2$\begingroup$ One of the algebras ($x^2=0$) has all nontrivial nilpotents of order two, while the other has a nilpotent of order 4 ($x^2 = y \neq 0$, $x^3=xy=z$, $x^4=y^2=0$). $\endgroup$ Sep 17 at 23:10

$\begingroup$ Right, one is $k[x,y]/(x^2, y^2)$ while the other is $k[x]/(x^4)$. $\endgroup$ Sep 18 at 16:37
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 $x_i=g_i1$. 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 $S_1(x_i)=g_i^{1}1=(x_i)+\cdots +(x_i)^{p1}$.
$A$ 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

$\begingroup$ Note that the Hopf algebras are not isomorphic because the space of primitives in the group Hopf algebra is trivial while it contains the $x_i$ in the other case. $\endgroup$– tj_Sep 17 at 23:58
