=============================================================

Let $n>1$ be a positive integer with $p|n$. Let $G=\langle g\rangle$ be the cyclic group of order $n$. Then $kG$ has a 2-dimensional indecomposable right-right Yetter-Drinfeld module $V$. $V$ has a basis $\{v_1, v_2\}$ such that the right $kG$-action and $kG$-coaction are defined by $$v_1\cdot g=v_1,\ v_2\cdot g=v_1+v_2,\ \rho(v)=v\otimes g,\ v\in V.$$ Then one can form a Nichols algebra $\mathcal{B}(V)$ and the corresponding pointed Hopf algebra $\mathcal{B}(V)\# kG$. $\mathcal{B}(V)\# kG$ is a finite dimensional graded Hopf algebra, which is generated as an algebra by three elements $g$, $a$ and $b$ (see [@Cib09]).

When $p=2$, the generators $g$, $a$ and $b$ of $\mathcal{B}(V)\# kG$ are subject to the relations: $$\begin{aligned}
& g^n=1,\ g^{-1}ag=a,\ g^{-1}bg=a+b,&\end{aligned}$$ $$\begin{aligned}
& a^2=0, \ b^4=0, \ baba=abab, \ {b^2}a=ab^2+aba. &\end{aligned}$$ When $p>2$, the generators $g$, $a$ and $b$ of $\mathcal{B}(V)\# kG$ are subject to the relations: $$\begin{aligned}
& g^n=1, \ g^{-1}ag=a,\  g^{-1}bg=a+b, &\end{aligned}$$ $$\begin{aligned}
& a^p=0, \ b^p=0, \ ba=ab+{\frac{1}{2}}a^2. &\end{aligned}$$ The coalgebra structure and the antipode of $\mathcal{B}(V)\# kG$ are determined by $$\begin{aligned}
& \bigtriangleup(g)=g\otimes g, \ \bigtriangleup(a)={a\otimes
1}+{g\otimes a}, \ \bigtriangleup(b)={b\otimes 1}+{g\otimes b};&\end{aligned}$$ $$\begin{aligned}
 & \varepsilon(g)=1,\ \varepsilon(a)=\varepsilon(b)=0;&\end{aligned}$$ $$\begin{aligned}
S(g)=g^{-1}, \ S(a)=-g^{-1}a, \ S(b)=-g^{-1}b.\end{aligned}$$ Note that $kG$ is the coradical of $\mathcal{B}(V)\# kG$ and $kG$ 