The Unapologetic Mathematician

Mathematics for the interested outsider

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:


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 Posted by | Algebra, Category theory, Representation Theory | 3 Comments



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