Representations of Hopf Algebras I
We’ve seen that the category of representations of a bialgebra is monoidal. What do we get for Hopf algebras? What does an antipode buy us? Duals! At least when we restrict to finite-dimensional representations.
Again, we base things on the underlying category of vector spaces. Given a representation 
, we want to find a representation 
. And it should commute with the natural transformations which make up the dual structure.
Easy enough! We just take the dual of each map to find 
. But no, this can’t work. Duality reverses the order of composition. We need an antiautomorphism 
to reverse the multiplication on 
. Then we can define 
.
The antiautomorphism we’ll use will be the antipode. Now to make these representations actual duals, we’ll need natural transformations 
and 
. This natural transformation 
is not to be confused with the counit of the Hopf algebra. Given a representation 
on the finite-dimensional vector space 
, we’ll just use the 
and 
that come from the duality on the category of finite-dimensional vector spaces.
Thus we find that 
is the pairing 
. Does this commute with the actions of ? On the one side, we calculate
\right](v\otimes\lambda)=\left[\rho\left(h_{(1)}\right)\otimes\rho^*\left(h_{(2)}\right)\right](v\otimes\lambda)\\=\left[\rho\left(h_{(1)}\right)\otimes\rho\left(S\left((h_{(2)}\right)\right)^*\right](v\otimes\lambda)\\=\left[\rho\left(h_{(1)}\right)\right](v)\otimes\left[\rho\left(S\left(h_{(2)}\right)\right)^*\right](\lambda)\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D%5Cleft%5B%5Cleft%5B%5Crho%5Cotimes%5Crho%5E%2A%5Cright%5D%28h%29%5Cright%5D%28v%5Cotimes%5Clambda%29%3D%5Cleft%5B%5Crho%5Cleft%28h_%7B%281%29%7D%5Cright%29%5Cotimes%5Crho%5E%2A%5Cleft%28h_%7B%282%29%7D%5Cright%29%5Cright%5D%28v%5Cotimes%5Clambda%29%5C%5C%3D%5Cleft%5B%5Crho%5Cleft%28h_%7B%281%29%7D%5Cright%29%5Cotimes%5Crho%5Cleft%28S%5Cleft%28%28h_%7B%282%29%7D%5Cright%29%5Cright%29%5E%2A%5Cright%5D%28v%5Cotimes%5Clambda%29%5C%5C%3D%5Cleft%5B%5Crho%5Cleft%28h_%7B%281%29%7D%5Cright%29%5Cright%5D%28v%29%5Cotimes%5Cleft%5B%5Crho%5Cleft%28S%5Cleft%28h_%7B%282%29%7D%5Cright%29%5Cright%29%5E%2A%5Cright%5D%28%5Clambda%29%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0&c=20201002)
Then we apply the evaluation to find
\right]\left(\left[\rho\left(h_{(1)}\right)\right](v)\right)=\lambda\left(\left[\rho\left(S\left(h_{(2)}\right)\right)\right]\left(\left[\rho\left(h_{(1)}\right)\right](v)\right)\right)\\=\lambda\left(\left[\rho\left(h_{(1)}S\left(h_{(2)}\right)\right)\right](v)\right)=\lambda\left(\left[\rho\left(\mu\left(h_{(1)}\otimes S\left(h_{(2)}\right)\right)\right)\right](v)\right)\\=\lambda\left(\left[\rho\left(\iota\left(\epsilon(h)\right)\right)\right](v)\right)=\epsilon(h)\lambda(v)\end{aligned}](https://s0.wp.com/latex.php?latex=%5Cbegin%7Baligned%7D%5Cleft%5B%5Cleft%5B%5Crho%5Cleft%28S%5Cleft%28h_%7B%282%29%7D%5Cright%29%5Cright%29%5E%2A%5Cright%5D%28%5Clambda%29%5Cright%5D%5Cleft%28%5Cleft%5B%5Crho%5Cleft%28h_%7B%281%29%7D%5Cright%29%5Cright%5D%28v%29%5Cright%29%3D%5Clambda%5Cleft%28%5Cleft%5B%5Crho%5Cleft%28S%5Cleft%28h_%7B%282%29%7D%5Cright%29%5Cright%29%5Cright%5D%5Cleft%28%5Cleft%5B%5Crho%5Cleft%28h_%7B%281%29%7D%5Cright%29%5Cright%5D%28v%29%5Cright%29%5Cright%29%5C%5C%3D%5Clambda%5Cleft%28%5Cleft%5B%5Crho%5Cleft%28h_%7B%281%29%7DS%5Cleft%28h_%7B%282%29%7D%5Cright%29%5Cright%29%5Cright%5D%28v%29%5Cright%29%3D%5Clambda%5Cleft%28%5Cleft%5B%5Crho%5Cleft%28%5Cmu%5Cleft%28h_%7B%281%29%7D%5Cotimes+S%5Cleft%28h_%7B%282%29%7D%5Cright%29%5Cright%29%5Cright%29%5Cright%5D%28v%29%5Cright%29%5C%5C%3D%5Clambda%5Cleft%28%5Cleft%5B%5Crho%5Cleft%28%5Ciota%5Cleft%28%5Cepsilon%28h%29%5Cright%29%5Cright%29%5Cright%5D%28v%29%5Cright%29%3D%5Cepsilon%28h%29%5Clambda%28v%29%5Cend%7Baligned%7D&bg=e6e6e6&fg=333333&s=0&c=20201002)
Which is the same as the result we’d get by applying the “unit” action after evaluating. Notice how we used the definition of the dual map, the fact that is a representation, and the antipodal property in obtaining this result.
This much works whether or not is a finite-dimensional vector space. The other direction, though, needs more work, especially since I waved my hands at it when I used 
as the motivating example of a category with duals. Tomorrow I’ll define this map.
November 12, 2008 -
Posted by John Armstrong |
Algebra, Category theory, Representation Theory
The Unapologetic Mathematician 
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