is an $H$-bimodule, with left and right $H$-action $$\lan h\leftact h^*\rightact k, l\ran =\lan h^*, klh\ran,$$ or $${\label{eq:4.0}}
h\leftact h^*\rightact k=\lan h_{(1)}^*,k\ran \lan h_{(3)}^*,h\ran h_{(2)}^*.$$ Using [(\[eq:1.41\])]{}, we find a weak smash product structure $(H,H^*,R)$, with $R:\ H^*\ot H\to H\ot H^*$ given by $$\begin{aligned}
R(h^*\ot h)&=&
\hbox{$\sum_i$} \lan h^*, h_{(3)}h_iS^{-1}(h_{(1)})\ran h_{(2)}\ot h_i^*\nonumber\\
&=& \hbox{$\sum_i$} \lan S^{-1}(h_{(1)})\leftact h^*\rightact h_{(3)}, h_i\ran h_{(2)}\ot h_i^*\nonumber\\
&=& h_{(2)} \ot \Bigl(S^{-1}(h_{(1)})\leftact h^*\rightact h_{(3)}\Bigr).
{\label{eq:4.1}}\end{aligned}$$

From [Section \[se:1.4\]]{}, we know that $H\#_R H^*$, which we will also denote by $H\bowtie H^*$, is an associative algebra with preunit $1\# \varepsilon$. Using [(\[eq:smash\])]{}, we compute the multiplication rule on $H\bowtie H^*$. $$\begin{aligned}
&&\hspace*{-2cm} (h\bowtie h^*)(k\bowtie k^*)=
\hbox{$\sum_i$} \lan h^*, k_{(3)}h_iS^{-1}(k_{(1)})\ran hk_{(2)}
\bowtie h_i^**k^*\nonumber\\
&=& hk_{(2)}\bowtie (S^{-1}(k_{(1)})\leftact h^*\rightact k_{(3)})*k^*
{\label{eq:4.2}}\\
&=& hk_{(2)} \bowtie \lan h_{(1)}^*, k_{(3)}\ran \lan h_{(3)}^*,S^{-1}(k_{(1)})\ran
h_{(2)}^**k^*{\label{eq:4.3}}.\end{aligned}$$ We have a projection $p:\ H\bowtie H^*\to H\bowtie H^*$, $$p(h\bowtie h^*)=(1\bowtie \varepsilon)(h\bowtie h^*)=(h\bowtie h^*)(1\bowtie \varepsilon)= (h\bowtie h^*)(1\bowtie \varepsilon)^2,$$ and $D(H)=\ol{H\bowtie H^*}= (H\bowtie H^*)/{{\rm Ker}\,}p$ is a $k$-algebra with unit $[1\bowtie \varepsilon]$, which we call the [*Drinfeld double*]{} of $H$. $D(H)$ is also isomorphic to $\ul{H\bowtie H^*}={{\rm Im}\,}(p)$, which is a $k$-algebra with unit $(1\bowtie \varepsilon)^2$. Observe that the multiplication rule [(\[eq:4.3\])]{} is the same as in [@Bohm; @Nenciu]. We show that 