# The Unapologetic Mathematician

## 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 $\rho:H\rightarrow\hom_\mathbb{F}(V,V)$, we want to find a representation $\rho^*:H\rightarrow\hom_\mathbb{F}(V^*,V^*)$. 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 $\rho(h)^*:V^*\rightarrow V^*$. But no, this can’t work. Duality reverses the order of composition. We need an antiautomorphism $S$ to reverse the multiplication on $H$. Then we can define $\rho^*(h)=\rho(S(h))^*$.

The antiautomorphism we’ll use will be the antipode. Now to make these representations actual duals, we’ll need natural transformations $\eta_\rho:\mathbf{1}\rightarrow\rho^*\otimes\rho$ and $\epsilon_\rho:\rho\otimes\rho^*\rightarrow\mathbf{1}$. This natural transformation $\epsilon$ is not to be confused with the counit of the Hopf algebra. Given a representation $\rho$ on the finite-dimensional vector space $V$, we’ll just use the $\eta_V$ and $\epsilon_V$ that come from the duality on the category of finite-dimensional vector spaces.

Thus we find that $\epsilon_\rho$ is the pairing $v\otimes\lambda\mapsto\lambda(v)$. Does this commute with the actions of $H$? On the one side, we calculate

$\begin{aligned}\left[\left[\rho\otimes\rho^*\right](h)\right](v\otimes\lambda)=\left[\rho\left(h_{(1)}\right)\otimes\rho^*\left(h_{(2)}\right)\right](v\otimes\lambda)\\=\left[\rho\left(h_{(1)}\right)\otimes\rho\left(S\left((h_{(2)}\right)\right)^*\right](v\otimes\lambda)\\=\left[\rho\left(h_{(1)}\right)\right](v)\otimes\left[\rho\left(S\left(h_{(2)}\right)\right)^*\right](\lambda)\end{aligned}$

Then we apply the evaluation to find

$\begin{aligned}\left[\left[\rho\left(S\left(h_{(2)}\right)\right)^*\right](\lambda)\right]\left(\left[\rho\left(h_{(1)}\right)\right](v)\right)=\lambda\left(\left[\rho\left(S\left(h_{(2)}\right)\right)\right]\left(\left[\rho\left(h_{(1)}\right)\right](v)\right)\right)\\=\lambda\left(\left[\rho\left(h_{(1)}S\left(h_{(2)}\right)\right)\right](v)\right)=\lambda\left(\left[\rho\left(\mu\left(h_{(1)}\otimes S\left(h_{(2)}\right)\right)\right)\right](v)\right)\\=\lambda\left(\left[\rho\left(\iota\left(\epsilon(h)\right)\right)\right](v)\right)=\epsilon(h)\lambda(v)\end{aligned}$

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 $\rho$ is a representation, and the antipodal property in obtaining this result.

This much works whether or not $V$ is a finite-dimensional vector space. The other direction, though, needs more work, especially since I waved my hands at it when I used $\mathbf{FinVect}$ as the motivating example of a category with duals. Tomorrow I’ll define this map.

## 3 Comments	»

1. [...] 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. [...]

Pingback by | November 14, 2008 | Reply

2. [...] Category of Representations of a Hopf Algebra It took us two posts, but we showed that the category of representations of a Hopf algebra has duals. This is on [...]

Pingback by | November 18, 2008 | Reply

3. [...] algebra. Thus the category of representations of is monoidal — symmetric, even — and has duals. Let’s consider these structures a bit more [...]

Pingback by | November 21, 2008 | Reply

« Previous | Next »

## About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 21, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8988432288169861, "perplexity_flag": "head"}