example of algebras and coalgebras which cannot be turned into Hopf algebras

Let $H={𝕄}_n(k)$ be a matrix algebra over a field $k$ with standard multiplication and assume that $n>1$. Assume that $H$ can be turned into a Hopf algebra. In particular, there is $\epsilon :H\to k$ such that $\epsilon$ is a morphism of algebras. It can be shown that matrix algebra is simple, i.e. if $I\subseteq H$ is a two-sided ideal, then $I=0$ or $I=H$. Thus we have that $\ker \epsilon =0$ (because $\epsilon (1)=1$). Contradiction, because ${\dim }_{k}H>1={\dim }_{k}k$.

Now consider $H={𝕄}^c(n,k)$ a vector space of all $n\times n$ matrices over $k$. We introduce coalgebra structure on $H$. Let $E_{ij}$ be a matrix in $H$ with $1$ in $(i,j)$ place and $0$ everywhere else. Of course $\{E_{ij}\}$ forms a basis of $H$ and it is sufficient to define comultiplication and counit on it. Define

$$\mathrm{\Delta }(E_{ij})=\sum _{p=1}^n{E}_{ip}\otimes E_{pj};$$

$$\epsilon (E_{ij})={\delta }_{ij},$$

where ${\delta }_{ij}$ denotes Kronecker delta. It can be easily checked, that $({𝕄}^c(n,k),\mathrm{\Delta },\epsilon )$ is a coalgebra known as the matrix coalgebra. Also, is well known that the dual algebra ${𝕄}^c{(n,k)}^{*}$ is isomorphic to the standard matrix algebra.

Now assume that matrix coalgebra $H={𝕄}^c(n,k)$ (where $n>1$) can be turned into a Hopf algebra. Since $H$ is finite dimensional, then we can take dual Hopf algebra $H^{*}$. But the underlaying algebra structure of $H^{*}$ is isomorphic to a matrix algebra (as we remarked earlier), which we’ve already shown to be impossible. Thus matrix coalgebra cannot be turned into a Hopf algebra.