is a Hopf subalgebra of $\mathcal{B}(V)\# kG$.

Furthermore, one may construct filtered pointed Hopf algebras as “lifting" of $\mathcal{B}(V)\# kG$, that is those whose associated graded algebra is $\mathcal{B}(V)\# kG$. In the case of $p>2$, Cibils, Lauve and Witherspoon gave some examples of liftings of $\mathcal{B}(V)\# kG$, which can be described as follows.

Assume $p>2$, and let $\lambda, \mu\in k$. The Hopf algebra $H(\lambda, \mu)$ is generated, as an algebra, by $g$, $a$ and $b$ with the relations $$\begin{aligned}
& g^n=1,\  g^{-1}ag=a, \ g^{-1}bg=a+b, &\end{aligned}$$ $$\begin{aligned}
 & a^p=\lambda(1-g^p), \ b^p=\mu(1-g^p),\
ba=ab+{\frac{1}{2}}a^2.&\end{aligned}$$ The coalgebra structure and the antipode of $H(\lambda, \mu)$ are determined by the same equations as $\mathcal{B}(V)\# kG$. Note that $kG$ is the coradical of $H(\lambda, \mu)$ and $kG$ is a Hopf subalgebra of $H(\lambda, \mu)$. Moreover, when $\lambda=\mu=0$, $H(0,0)=\mathcal{B}(V)\# kG$.

\[2.1\] When $p=2$, in $\mathcal{B}(V)\# kG$ we have

$(1)$ $bg^i=ig^ia+g^ib$, $i\geqslant 0$. In particular, $g^2$ is central in $\mathcal{B}(V)\# kG$.

$(2)$ $\mathcal{B}(V)\# kG$ is a symmetric Hopf algebra.

\(1) It can be proved by induction on $i$ from the relation $g^{-1}bg=a+b$.

\(2) Let $H=\mathcal{B}(V)\# kG$. Then the set $\{g^iabab^3|0\leqslant i\leqslant n-1\}$ are linearly independent in $H$ by [@Cib09 Theorem 3.1 and Corollary 3.4]. Let $t=(\sum\limits_{0\leqslant i\leqslant n-1}g^i)abab^3$. Then 