见 Sorin Dăscălescu《Hopf Algebras: An Introduction》第20页
Exercise 1.1.5 Let $C$ be a $k$-space with basis $\{s, c\}$. We define $\Delta: C \rightarrow$ $C \otimes C$ and $\varepsilon: C \rightarrow k$ by
\[
\begin{aligned}
\Delta(s) & =s \otimes c+c \otimes s \\
\Delta(c) & =c \otimes c-s \otimes s \\
\varepsilon(s) & =0 \\
\varepsilon(c) & =1
\end{aligned}
\]
Show that $(C, \Delta, \varepsilon)$ is a coalgebra.
SolutionWe have
\[
(I \otimes \Delta) \Delta(s)=(\Delta \otimes I) \Delta(s)=s \otimes c \otimes c+c \otimes s \otimes c+c \otimes c \otimes s-s \otimes s \otimes s
\]
and
\[
(I \otimes \Delta) \Delta(c)=(\Delta \otimes I) \Delta(c)=c \otimes c \otimes c-s \otimes s \otimes c-s \otimes c \otimes s-c \otimes s \otimes s
\]
The counit property is obvious. | 
