TITLE: Compatibility conditions for Yetter-Drinfeld modules
QUESTION [6 upvotes]: In the paper, page 28, Definition 4.2.1, the compatibility condition for a Yetter-Drinfeld module over $H$ is 
$$
h_{(1)} v_{(-1)} \otimes h_{(2)}.v_{(0)} = (h_{(1)}.v)_{(-1)}h_{(2)} \otimes (h_{(1)}.v)_{(0)}, v \in V, h \in H.
$$ 
On the other hand, in the article, the compatibility condition for a Yetter-Drinfeld module over $H$ is 
$$
\delta(h.v) = h_{(1)} v_{(-1)} S(h_{(3)}) \otimes h_{(2)}.v_{(0)}, v \in V, h \in H.
$$ 
Are the two conditions equivalent? Thank you very much.

REPLY [3 votes]: Assume that the second equation holds. Then 
\begin{align*}
\delta(h_{1}.v)=(h_{1}. v)_{-1}\otimes(h_{1}.v)_{0}=h_{1,1}v_{-1}Sh_{1,3}\otimes h_{1,2}. v_{0}
\end{align*}
and therefore
$$
(h_{1}.v)_{-1}h_{2}\otimes(h_{1}.v)_{0}=h_{1,1}v_{-1}Sh_{1,3}h_{2}\otimes h_{1,2}.v_{0}=h_{1}v_{-1}\otimes h_{2}.v_{0}.
$$
Conversely, assume that the first equation holds. Then
\begin{align*}
        (m\otimes &id)(h_{11}v_{-1}\otimes Sh_2\otimes(h_{12}.v_0) )\\  
        &=(m\otimes id)\left( (h_{11}.v)_{-1}h_{12}\otimes Sh_2\otimes (h_{11}. v)_0 \right)\\
        &=(h_1.v)_{-1}h_2Sh_3\otimes (h_1.v)_0\\
        &=(h.v)_{-1}\otimes (h.v)_0.
\end{align*}