I am working on a relatively easy example with the graphical calculus. I am very familiar with this while I am still very uncomfortable with Sweedlers notation and translating graphs into algebraic formulas. But let me present my example:
I have a finite dimensional Hopf algebra $H$, an $H$-comodule C with coaction $\delta: C \rightarrow C \otimes H$, an $H$-module M with action $\rho: H \otimes M \rightarrow M$, and an vector space $V$ which is not important at this time.
Then I want to show that the following map is an isomorphism:
I will argue that this map here is the inverse:
This happens in one direction (the other is similar) like this:
Where I used the well-known fact that to act two times is the same as act with the multiplication (same for coaction). In the second step I used the following property of the antipode:
So far, so good. Now I don't want to give graphs but formulas. I expect this to be relatively easy formulas as the graphs aren't complicated either.
My first step was to define the maps:
The first one is given by $$(\rho \otimes \mathrm{id}_{V \otimes C}) \circ ASS \circ ( \mathrm{id}_{M \otimes V} \otimes \delta)$$
and the second one is given by:
$$(\rho \otimes \mathrm{id}_{V \otimes C}) \circ ASS \circ (\mathrm{id}_{M \otimes V \otimes C} \otimes S) \circ ( \mathrm{id}_{M \otimes V} \otimes \delta)$$
in both ASS is just the associator for switching things around, I guess we can just leave those, because they are isomorphisms anyway.
Now I get confused:
\begin{align} (\rho \otimes \mathrm{id}_{V \otimes C}) \circ ASS \circ ( \mathrm{id}_{M \otimes V} &\otimes \delta)(m \otimes v \otimes c) \\ &=(\rho \otimes \mathrm{id}_{V \otimes C}) \circ ASS(m \otimes v \otimes c_{(0)} \otimes c_{(-1)}) \\ &=(\rho \otimes \mathrm{id}_{V \otimes C})(c_{(-1)} \otimes m \otimes v \otimes c_{(0)}) \\ &= m_{(0)} \otimes v \otimes c_{(0)} \end{align}
Now the second equation: \begin{align} (\rho \otimes \mathrm{id}_{V \otimes C}) \circ ASS \circ (\mathrm{id}_{M \otimes V \otimes C} &\otimes S) \circ ( \mathrm{id}_{M \otimes V} \otimes \delta)(m_{(0)} \otimes v \otimes c_{(0)}) \\ &=(\rho \otimes \mathrm{id}_{V \otimes C}) \circ ASS \circ (\mathrm{id}_{M \otimes V \otimes C} \otimes S)(m_{0} \otimes v \otimes c_{(0)(0)} \otimes c_{(-1)}) \\ &=(\rho \otimes \mathrm{id}_{V \otimes C}) \circ ASS (m_{0} \otimes v \otimes c_{(0)(0)} \otimes S(c_{(-1)}) \\ &= (\rho \otimes \mathrm{id}_{V \otimes C})(S(c_{(-1)}) \otimes m_{0} \otimes v \otimes c_{(0)(0)}) \end{align}
yeah but well, in this last line I have no chance to continue or to use any rule. Can anybody help me fix this?
Thanks in advance!!