• Goal: understand Yetter-Drinfeld module, Drinfeld double and then Andy Manion's comment paper

Let $H$ be a Hopf algebra, and $Rep(H)$ is the cat of finite dim rep. Let $(Y, \varphi)$ be such central element, then we have the following rep $$ \varphi: H \otimes Y \to Y \otimes H $$ As written this is a rep H morphism. We may restrict to $ \{1\} \otimes Y$, and get a linear map $$ \tau: Y \to Y \otimes H. $$ This has the potential to be a (right) co-action of $H$ on $Y$.

To check that this is indeed a co-action, we need to do splitting twice. either we split $Y$ twice, or the second split is on $H$.

Consider the following example: $H = \C G$ where $G$ is a finite group. Let $(Y, \varphi)$ be given as a $G$-graded $G$-rep, namely $Y = V = \oplus_g V_g$, and for $v_g \in V_g$, $h \in G$, we have $h \cdot v_g \in V_{h g h^{-1}}$. Note that any $G$ rep can be given a trivial $G$-gradation. Now, given another $G$-rep $W$, we can do $$ \varphi: W \otimes V \to V \otimes W, \quad w \otimes v_g \mapsto v_g \otimes g \cdot w. $$ Now, we need to check that this commute with $G$-action. OK, it does.

Now, we apply this to $W=H$, where $G$ acts on $\C G$ by left multiplication. Then we have $$ \tau: V \to V \otimes \C G, \quad v_g \to v_g \otimes g. $$ Now it is clear that this indeed is a co-action.

Now, we should check that the action is compatible with the co-action. We faces the funny condition: $$ \tau(h v_g) = h v_g \otimes h g h^{-1}. $$ OK, this indeed works. I don't know the general rule for co-product, but at least here, for the co-commutative coproduct, this works.

Let's try taking $H = O(G)$. Let $(Y, \varphi)$ be given as a $G$-graded $G$-rep. Then $O(G)$-action on $Y$ is pointwise multiplication. $O(G)$ has an interesting co-product, hence $Vec_G$ has an interesting monoidal structure, basically the grading multiply when they take product. Then, it is naturally true that, $W_1 \otimes W_2$ is not isomorphic to $W_2 \otimes W_1$: the grading won't match. Now comes $(Y, \varphi)$, say $Y = V = \oplus_g V_g$. Say $W = \C \delta_h$, a rank 1 skyscraper at position $h \in G$, then $$ (V \otimes W)_g = V_{gh^{-1}}, \quad (W \otimes V)_g = V_{h^{-1} g} $$ To have an isomorphism, we need to get for each $g$, an isomorphism $V_{gh^{-1}} \to V_{h^{-1} g}$, well, this is given by the action of $h^{-1}$ on $V$.

Now, we consider $1 \in H$, this is represented as the constant function $1 \in O(G)$. We have $1 = \sum_h \delta_h$ (constant function is the sum of delta functions). So, we just need to consider what is $\delta_h \otimes v_g \mapsto ?$. Now this will go to $ (h v_g) \otimes \delta_h \in V_{hgh^{-1}} \otimes H_h$. Put these all together, we have $$ \tau(v_g) = \sum_h (h v_g) \otimes \delta_h. $$ We first check that this is a co-action, we can do the comodule-split again, to get $$ \sum_{h_1, h_2} h_2 (h_1 v_g) \otimes \delta_{h_2} \otimes \delta_{h_1}. $$ Or, we can do the co-mult on $H$ to get $$ \tau(v_g) = \sum_{h_1, h_2} (h_1 h_2 v_g) \otimes (\delta_{h_1} \otimes \delta_{h_2}).$$