# The Unapologetic Mathematician

## Representations of Hopf Algebras II

Now that we have a coevaluation for vector spaces, let’s make sure that it intertwines the actions of a Hopf algebra. Then we can finish showing that the category of representations of a Hopf algebra has duals.

Take a representation $\rho:H\rightarrow\hom_\mathbb{F}(V,V)$, and pick a basis $\left\{e_i\right\}$ of $V$ and the dual basis $\left\{\epsilon^i\right\}$ of $V^*$. We define the map $\eta_\rho:\mathbf{1}\rightarrow V^*\otimes V$ by $\eta_\rho(1)=\epsilon^i\otimes e_i$. Now $\left[\rho(a)\right](1)=\epsilon(a)$, so if we use the action of $H$ on $\mathbf{1}$ before transferring to $V^*\otimes V$, we get $\epsilon(a)\epsilon^i\otimes e_i$. Be careful not to confuse the counit $\epsilon$ with the basis elements $\epsilon^i$.

On the other hand, if we transfer first, we must calculate

\begin{aligned}\left[\left[\rho^*\otimes\rho\right](a)\right](\epsilon^i\otimes e_i)=\left[\rho^*\left(a_{(1)}\right)\otimes\rho\left(a_{(2)}\right)\right](\epsilon^i\otimes e_i)\\=\left[\rho\left(S\left(a_{(1)}\right)\right)^*\otimes\rho\left(a_{(2)}\right)\right](\epsilon^i\otimes e_i)\\=\left[\rho\left(S\left(a_{(1)}\right)\right)^*\right](\epsilon^i)\otimes\left[\rho\left(a_{(2)}\right)\right](e_i)\end{aligned}

Now let’s use the fact that we’ve got this basis sitting around to expand out both $\rho\left(S\left(a_{(1)}\right)\right)$ and $\rho\left(a_{(2)}\right)$ as matrices. We’ll just take on matrix indices on the right for our notation. Then we continue the calculation above:

\begin{aligned}\left[\rho\left(S\left(a_{(1)}\right)\right)^*\right](\epsilon^i)\otimes\left[\rho\left(a_{(2)}\right)\right](e_i)=\epsilon^j\rho\left(S\left(a_{(1)}\right)\right)_j^i\otimes\rho\left(a_{(2)}\right)_i^ke_k\\=\epsilon^j\otimes\rho\left(S\left(a_{(1)}\right)\right)_j^i\rho\left(a_{(2)}\right)_i^ke_k\\=\epsilon^j\otimes\left[\rho\left(\mu\left(\left[S\otimes1_H\right]\left(\Delta(a)\right)\right)\right)\right](e_j)\\=\epsilon^j\otimes\left[\rho\left(\iota\left(\epsilon(a)\right)\right)\right](e_j)=\epsilon^j\otimes\epsilon(a)e_j\end{aligned}

And so the coevaluation map does indeed intertwine the two actions of $H$. Together with the evaluation map, it provides the duality on the category of representations of a Hopf algebra $H$ that we were looking for.

November 14, 2008