• 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