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 
, and pick a basis 
of 
and the dual basis 
of 
. We define the map 
by 
. Now =\epsilon(a)](http://s0.wp.com/latex.php?latex=%5Cleft%5B%5Crho%28a%29%5Cright%5D%281%29%3D%5Cepsilon%28a%29&bg=e6e6e6&fg=333333&s=0 "\left[\rho(a)\right](1)=\epsilon(a)")
, so if we use the action of 
on 
before transferring to 
, we get 
. Be careful not to confuse the counit 
with the basis elements 
.
On the other hand, if we transfer first, we must calculate
\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}](http://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D%5Cleft%5B%5Cleft%5B%5Crho%5E%2A%5Cotimes%5Crho%5Cright%5D%28a%29%5Cright%5D%28%5Cepsilon%5Ei%5Cotimes+e_i%29%3D%5Cleft%5B%5Crho%5E%2A%5Cleft%28a_%7B%281%29%7D%5Cright%29%5Cotimes%5Crho%5Cleft%28a_%7B%282%29%7D%5Cright%29%5Cright%5D%28%5Cepsilon%5Ei%5Cotimes+e_i%29%5C%5C%3D%5Cleft%5B%5Crho%5Cleft%28S%5Cleft%28a_%7B%281%29%7D%5Cright%29%5Cright%29%5E%2A%5Cotimes%5Crho%5Cleft%28a_%7B%282%29%7D%5Cright%29%5Cright%5D%28%5Cepsilon%5Ei%5Cotimes+e_i%29%5C%5C%3D%5Cleft%5B%5Crho%5Cleft%28S%5Cleft%28a_%7B%281%29%7D%5Cright%29%5Cright%29%5E%2A%5Cright%5D%28%5Cepsilon%5Ei%29%5Cotimes%5Cleft%5B%5Crho%5Cleft%28a_%7B%282%29%7D%5Cright%29%5Cright%5D%28e_i%29%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\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 
and 
as matrices. We’ll just take on matrix indices on the right for our notation. Then we continue the calculation above:
\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}](http://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D%5Cleft%5B%5Crho%5Cleft%28S%5Cleft%28a_%7B%281%29%7D%5Cright%29%5Cright%29%5E%2A%5Cright%5D%28%5Cepsilon%5Ei%29%5Cotimes%5Cleft%5B%5Crho%5Cleft%28a_%7B%282%29%7D%5Cright%29%5Cright%5D%28e_i%29%3D%5Cepsilon%5Ej%5Crho%5Cleft%28S%5Cleft%28a_%7B%281%29%7D%5Cright%29%5Cright%29_j%5Ei%5Cotimes%5Crho%5Cleft%28a_%7B%282%29%7D%5Cright%29_i%5Eke_k%5C%5C%3D%5Cepsilon%5Ej%5Cotimes%5Crho%5Cleft%28S%5Cleft%28a_%7B%281%29%7D%5Cright%29%5Cright%29_j%5Ei%5Crho%5Cleft%28a_%7B%282%29%7D%5Cright%29_i%5Eke_k%5C%5C%3D%5Cepsilon%5Ej%5Cotimes%5Cleft%5B%5Crho%5Cleft%28%5Cmu%5Cleft%28%5Cleft%5BS%5Cotimes1_H%5Cright%5D%5Cleft%28%5CDelta%28a%29%5Cright%29%5Cright%29%5Cright%29%5Cright%5D%28e_j%29%5C%5C%3D%5Cepsilon%5Ej%5Cotimes%5Cleft%5B%5Crho%5Cleft%28%5Ciota%5Cleft%28%5Cepsilon%28a%29%5Cright%29%5Cright%29%5Cright%5D%28e_j%29%3D%5Cepsilon%5Ej%5Cotimes%5Cepsilon%28a%29e_j%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0 "\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 . Together with the evaluation map, it provides the duality on the category of representations of a Hopf algebra that we were looking for.
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November 14, 2008 -
Posted by John Armstrong |
Algebra, Category theory, Representation Theory
“map”???
Enough with the metaphors, chief.
We already know you’re somewhere between a hoax and a troll, scholar. Go find another bridge.
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