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.



Different Bialgebra/Hopf algebra structures on coalgebras
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.


