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\begin{document}
\title{Lifting of Nichols Algebras of Type
$A_2$ and Pointed Hopf Algebras of order $p^4$}
\author{Nicol\'as Andruskiewitsch\\
Universidad de C\'ordoba\\
C\'ordoba, Argentina\and
Hans-J\"urgen Schneider\\
Universit\"at M\"unchen\\
M\"unchen, Germany}
\date{}
\maketitle

\begin{abstract}\noindent
We compute all finite-dimensional Hopf algebras whose coradical is the
group algebra of an abelian group $\Gamma$ and such that
$\gr(A)\cong \BBBB(V) \# \k[\Gamma]$, $V$ a 2-dimensional
Yetter-Drinfeld module over $\Gamma$ of type $A_2$. We  classify
pointed Hopf algebras of  order $ p^4$ up to isomorphism and show that
the number of quasi-isomorphism types is finite.
\end{abstract}

\section{Introduction}\selabel{0}
We shall work over an algebraically closed field $\k$  of
characteristic 0. Let $p$ be a prime number. Several classification
results are known for Hopf algebras of order $p^{N}$.  Semisimple
Hopf algebras of order $p^{N}$ have always a non-trivial central
group-like element; so  they can be obtained recursively  by an
extension process. The explicit description of all isomorphism types
for $p^{2}$ and $p^{3}$ is actually known \cite{Ma}.

We deal in this paper with {\it pointed} Hopf algebras of dimension
$p^{N}$. A Hopf algebra with coradical $\k[\Bbb Z/(2)]$ has dimension
$2^{N}$ and for each $N$ there is exactly one such Hopf algebra up to
isomorphisms \cite[Th. 4.2.1]{N}. Pointed Hopf algebras of order  8,
16 and 32 are respectively classified in \cite{W}, \cite{CDR},
\cite{Gr}.

>From now on we assume that $p$ is odd. The only known non-semisimple
Hopf algebras of dimension $p^{2}$ are pointed \cite{AS1} and it is easy to
see that they are Taft algebras. Pointed non-semisimple Hopf algebras of
dimension $p^3$ are classified in \cite{AS2}, \cite{CD}, \cite{SvO}.
Infinite families of pointed non-semisimple Hopf algebras of
dimension $p^4$ were constructed in \cite{AS2}, \cite{BDG},
\cite{G}. In this paper we complete the classification of all
pointed non-semisimple Hopf algebras of dimension $p^4$.

If $A$ is a pointed Hopf algebra of dimension $p^4$, then we know
by \cite[\S 7]{AS3} $\gr(A)$, the graded Hopf algebra associated to
the coradical filtration. We have to describe all possible liftings
$A$ of $\gr(A)$. There are two main cases: $\gr(A) \cong \BBBB(V)
\# \k[\Gamma]$, and $V$ is a Yetter-Drinfeld module over $\Gamma$
of  (1) type $A_1$ or $A_1 \times A_1$, or (2) type $A_2$ over the
cyclic group of order $p$. The liftings of type a product of copies
of $A_1$ are determined in \cite{AS2}. We describe explicitly all
the isomorphism types in \seref{3}. In \seref{4} of this paper we
compute all liftings of type $A_2$ over any finite abelian group
$\Gamma$, if the corresponding root of 1 is odd and greater than 3.
We conclude that a pointed Hopf algebra of dimension $p^{4}$ with
coradical of dimension $p$ is necessarily coradically graded,
and we obtain the classification of pointed Hopf algebras of dimension $p^4$
in \thref{5.1}.

In the appendix we show that although the number of isomorphism types of
pointed Hopf
algebras of dimension $p^4$ is infinite, the number of
quasi-isomor\-phism types is in
fact finite.

\section{Preliminaries}\selabel{1}
We use standard notation for
Hopf algebras. Our references for the theory of Hopf algebras are
\cite{Mo}, \cite{Sw}.

 We shall follow the method developed in  \cite{AS2}, \cite{AS3}. Let us
briefly recall the main ideas. Given a pointed Hopf algebra $A$ with
coradical $A_{0} = \k[\Gamma]$, the graded coalgebra associated to the
coalgebra filtration is in fact a graded Hopf algebra which we denote by
$\gr(A)$. In fact, $\gr(A)$ is  a {\it coradically graded}
Hopf algebra, and can be obtained as a bosonization $R\# \k[\Gamma]$, where
$R$ is a braided Hopf algebra in the category of Yetter-Drinfeld modules
over $\Gamma$. Now we know from \cite[\S 7]{AS3} the classification of
all possible coradically graded Hopf algebras of order $p^{4}$:

\begin{theorem}\thlabel{2.1}
Let $p$ be an odd prime number. Let $A$ be a pointed
non-semisimple Hopf algebras of order $p^4$  with coradical $\k[\Gamma]$.
Then the associated graded algebra $\gr(A)$ is isomorphic to
$R\# \k [\Gamma]$, where\\
\hspace*{8mm} (A). $R$ is a quantum line if $\Gamma$ has order $p^3$;\\
\hspace*{8mm} (B). $R$ is a quantum line or plane if $\Gamma$ has order
$p^2$;\\
\hspace*{8mm} (C). $R = \BBBB(V)$, where $V$ has Dynkin diagram $A_{2}$
if $\Gamma$
has order $p$, where $p = 3$ or  $p-1$ is divisible by 3.
\end{theorem}

To obtain our result we have to determine all possible Hopf algebras $A$
with $\gr(A)$ as in (A), (B) or (C) of Theorem 2.1. For (A) and (B), the
description follows from  \cite[Theorem 2]{AS2}; see \seref{3}.  We treat
the case (C) in \seref{4}.

In principle it is easy to describe all possible isomorphisms
between liftings of $\BBBB(V) \# \k[\Gamma]$ when the
Yetter-Drinfel'd module $V$ is of finite Cartan type (once the defining
relations are known). Let $(a_{ij})$
be a Cartan matrix of finite type of size $\theta \times \theta$,
$\Gamma$ a finite abelian group, $g_{1}, \dots, g_{\theta} \in
\Gamma$, $\chi_{1}, \dots, \chi_{\theta} \in \widehat{\Gamma}$ such
that for all $i,j$
 $$\chi_{i} (g_{j}) \chi_j(g_i) = \chi_i(g_i)^{a_{ij}}.$$
We define $q_i = \chi_i(g_i)$ and assume that the orders $N_i$ of
all $q_i$ are odd and $>1$. Let $V$ be a Yetter-Drinfel'd module over
$\k[\Gamma]$ with a basis $x_1, \ldots, x_{\theta}$ such that
$$h\cdot x_i = \chi_i(h) x_i \quad {\rm and}\quad \delta(x_i) = g_i
\otimes x_i$$ for all $h \in \Gamma$ and all $i$. Let $A$ be a
pointed Hopf algebra with $\gr(A) \cong \BBBB(V) \# \k[\Gamma]$.
We choose elements $a_i$ in the first term of the coradical
filtration of $A$ such that the image of $x_{i} \# 1$ in $\gr(A)$ is
the canonical image of $a_i$ in $A_1/A_0$. Then $A$ is generated as
an algebra by $\Gamma$ and the elements $a_i$. We consider another
such Yetter-Drinfel'd module $W$ over $\k[\widetilde{\Gamma}]$ of Cartan type
$(\widetilde{a}_{ij})$ of size $\widetilde{\theta} \times
\widetilde{\theta}$ with elements $h_j \in \widetilde{\Gamma}$, characters
$\eta_j$ and corresponding basis elements $w_j$. Let $B$ be a
lifting of $\BBBB(W) \# \k[\widetilde{\Gamma}]$ with corresponding elements
$b_j$.

\begin{lemma}\lelabel{2.2}
Assume for all $i \ne j$, $g_i \ne g_j$ or
$\chi_i \ne \chi_j$. (In particular this holds if for all $i \ne j$,
the order of $q_i$ does not divide $2 - a_{ij}$). Let $\Phi : A \to
B$ be a Hopf algebra isomorphism. Then $\theta =
\widetilde{\theta}$, $q_i^{a_{ij}} = q_i^{\widetilde{a}_{ij}}$ for
all $i,j$ and there are a group isomorphism $\phi :\ \Gamma \to
\widetilde{\Gamma}$, a permutation $\sigma$ and non-zero scalars $\alpha_i$ such
that $$\Phi \vert \Gamma = \phi, \quad \Phi(a_i) = \alpha_i
b_{\sigma(i)}, \quad \phi(g_i) = h_{\sigma(i)}, \quad \chi_i =
\eta_{\sigma(i)}\phi, \quad 1 \leq i \leq \theta.$$
\end{lemma}

\begin{proof}
$\Phi$ induces a Hopf algebra isomorphism $\gr(A) \to
\gr(B)$. By \cite[Proposition 6.3]{AS3} the assertion holds on the
graded level since by our assumption the Yetter-Drinfel'd module $V$
is a direct sum of pairwise non-isomorphic one-dimen\-sional
submodules $\k x_i$. Thus for all $i$, $\Phi(a_i) -
\alpha_ib_{\sigma(i)}$ is an element in the group algebra which must
be zero since the group is acting on it via the non-trivial
character $\eta_{\sigma(i)}$.
\end{proof}

Conversely, data $\phi,\sigma$ and $\alpha_i$ as in \leref{2.2}
define a Hopf algebra isomorphism $\Phi$ by $\Phi \vert \Gamma =
\phi$ and $\Phi(a_i) = \alpha_ib_{\sigma(i)}$ if all the relations
between the $a_i$ are respected. This gives a condition on the
scalars $\alpha_i$ which has to be worked out in each case.

In the case of quantum linear spaces, that is if $a_{ij} = 0$ for
all $ i \ne j$, this is very easy. By \cite[Section 5]{AS2}, a
lifting $A$ of a quantum linear space is given by scalars $\mu_i \in
\{0,1\}$ for all $i$, $\lambda_{ij} \in \k$ for all $i < j$ such
that $\mu_i = 0$ if $g_i^{N_i} = 1$ or $\chi_i^{N_i} \ne 1$, and
$\lambda_{ij} = 0$ if $g_ig_j = 1$ or $\chi_i\chi_j \ne 1$. The
relations we have to check are $$a_i^{N_i} = \mu_i(1 - g_i^{N_i})
\quad {\rm for ~ all} ~ i,\quad a_ja_i =\chi_i(g_j)a_ia_j
+ \lambda_{ij}(1 - g_ig_j) \quad {\rm for ~ all} ~ i < j.$$
If $B$ is another such lifting with data $\widetilde{\mu}_i,
\widetilde{\lambda}_{ij}$, then $A$ is isomorphic to $B$ if and only
if there are $\sigma$ and $\alpha_i$ as in \leref{2.2} satisfying for
all $i$ and $i < j$, $$\alpha_i^{N_i} \widetilde{\mu}_{\sigma(i)} =
\mu_i;$$  $$\alpha_i\alpha_j\widetilde{\lambda}_{\sigma(i)\sigma(j)}
= \lambda_{ij} ~{\rm if}~ \sigma (i) <
\sigma(j); ~
-\chi_i(g_j)\alpha_i\alpha_j\widetilde{\lambda}_{\sigma(j)\sigma(i)}
= \lambda_{ij} ~{\rm if} ~ \sigma(i) >
\sigma(j).$$

\section{The braided Hopf algebra $R$ is a quantum linear space}\selabel{3}
Let $p$ be an odd prime number. In order to classify pointed Hopf algebras
of order
$p^4$ we describe all liftings of quantum linear lines over a group
$\Gamma$ of order
$p^3$ and of quantum lines or planes over $\Gamma$ of order $p^2$. The
following list
contains all such Hopf algebras, and the remarks at the end of the last
section show
that Hopf algebras in two different cases of this list are non-isomorphic.
We leave the details of the proof to the reader.
\vspace*{3mm}\\
{\sl 2.1 Index $p$}\\
Here $\Gamma$ is a group of order $p^{3}$.
Since $R$ is a quantum line, there exist $g\in \Gamma$ central and a
character $\chi \in \widehat{\Gamma}$ such that $ \chi(g)$ has order
$p$.  Note that $\Gamma$ is necessarily abelian: indeed, the center
of a non-abelian group of order $p^{3}$ is contained in its
commutator, so $\chi(g) = 1$ for any $g\in \Gamma$ central, $\chi
\in \widehat{\Gamma}$. There are three abelian groups of order
$p^{3}$ which we write as    $\langle u_1, \ldots, u_r\rangle$, $1
\le r \leq 3$. Then $A$ can be presented by generators $u_{i}$,
$1\le i\le r$, $a$,  and relations
\begin{equation}\eqlabel{3.1}
\begin{array}{l}
u_{i}u_{j} = u_{j}u_{i}~~{\rm for~all}~~1 \leq i,j \leq r;\\
u_{1}^{p^3} = 1~{\rm  if}~
r = 1;~~ u_1^{p^2} = 1,~ u_2^p = 1 ~{\rm  if}~r = 2;\\
u_i^p = 1 ~~{\rm for~all}~~i ~{\rm  if}~r =3.\\
a^{p} = \mu(1- g^{p}), {\rm ~with}~\mu~{\rm either}~ 0{\rm~or~}1;\\
hah^{-1} = \chi(h)a,~~h\in \Gamma.\end{array}
\end{equation}
If $\mu = 1$, then $g^p \neq 1$ and $\chi^p = 1$. The value of $\mu$ is invariant under isomorphism.
The Hopf algebra structure of $A$ is determined by
$$\Delta(h) =
h\otimes h, \qquad \Delta(a) = a\otimes 1 + g\otimes a,\quad  h\in
\Gamma.$$
Up to isomorphism there are various subcases according to the order
of $g$.
\vspace*{3mm}\\
If $\Gamma = \langle u_1 \rangle$, we have three cases where $g =
u_1^{p^s}$ and $\chi(u_1) = \xi$, $\xi$ a root of unity of order
$p^{s+1}$, $0 \leq s \leq 2$. In each case, if $\eta$ is another
root of unity of the same order $p^{s+1}$, then the corresponding
Hopf algebras are isomorphic if and only if there is an integer $i$
with $\xi = \eta^i$ and $i \equiv 1 \mod {p^{3 - s}}$ (hence $\xi = \eta$ in case $s = 0,1$).
\vspace*{3mm}\\
If $\Gamma = \langle u_1,u_2 \rangle$, we take $g = u_1^p$ and
$\chi(u_1) = \xi$, $\chi(u_2) = 1$, $\xi$ a root of unity of order
$p^2$ (we get an isomorphic Hopf algebra only if we replace $\xi$
by $\xi^i$, $i \equiv 1 \mod p$); or $g = u_2$ and $\chi(u_1) =
\rho$, $\chi(u_2) =\xi$, $\xi$ a primitive $p$-th root of unity
and $\rho=1$ or an a priori given root of unity of order $p^2$;
or $g = u_1$ and $\chi(u_1) = \xi,~
\chi(u_2) = 1$, $\xi$ a $p$-th root of unity. In the last two cases
the Hopf algebras are pairwise non-isomorphic.
\vspace*{3mm}\\
If $\Gamma = \langle u_1,u_2,u_3 \rangle$, we take $g = u_1$ and
$\chi(u_1) = \xi$, $\xi$ a primitive $p$-th root of unity, $\chi(u_2) =1,
\chi(u_3) = 1$. These Hopf algebras are pairwise non-isomorphic.
\vspace*{3mm}\\
{\sl 2.2 Index $p^{2}$, $R$ a quantum line}\\
There exist $g\in \Gamma$ and a
character $\chi \in \widehat{\Gamma}$ such that $\chi(g)$ has order
$p^{2}$.  Necessarily, $\Gamma$ is isomorphic to $\Z/(p^{2})$. Then there
exists a primitive $p^2$-th root of unity $\xi\in \k$ such
that $A$ can be presented by generators $g$, $a$ and relations
\begin{equation}\eqlabel{3.4}
\begin{array}{l}
g^{p^{2}} = 1,\\
a^{p^{2}} = 0;\\
gag^{-1} = \xi a.
\end{array}\end{equation}
The Hopf algebra structure
of $A$ is determined by  $$\Delta(g) = g\otimes g, \qquad \Delta(a) =
a\otimes 1 + g\otimes a.$$
That is, $A$ is isomorphic to a Taft algebra of order $p^{4}$.
For different roots of unity the Hopf algebras are non-isomorphic.
\vspace*{3mm}\\
{\sl 2.3 Index $p^{2}$, $R$ a quantum plane and $\Gamma\simeq
\Z/(p)\times \Z/(p)$}
\vspace*{3mm}\\
Then there exist $g_{1},
g_{2}\in \Gamma$ and  characters $\chi_{1}, \chi_{2} \in
\widehat{\Gamma}$ such that $\chi_{1}(g_{1})$ and
$\chi_{2}(g_{2})$ have order $p$, $\chi_{1} (g_{2}) \chi_{2} (g_{1}) = 1$.
$A$ can be presented by generators $u_{1}$, $u_{2}$, $a_{1}$, $a_{2}$ and
relations
\begin{equation}\eqlabel{3.7}
\begin{array}{l}
u_{i}^{p} = 1,~~i = 1,2,~~u_1u_2 = u_2u_1;\\
a_{i}^{p} = 0,~~i = 1,2;\\
ha_{i}h^{-1} = \chi_{i}(h)a_{i},~~h\in \Gamma,~~i =1,2;\\
a_{1}a_{2} - \chi_{2}(g_{1}) a_{2}a_{1} =
\lambda(1 - g_{1}g_{2}),~~  {\rm with}~\lambda~{\rm either}~ 0
~{\rm or}~ 1.
\end{array}\end{equation}
If $\lambda = 1$, then $\chi_{1}\chi_{2} = 1$ and $g_1g_2 \ne 1$. The value of $\lambda$ is invariant under isomorphism. The
Hopf algebra
structure of $A$ is determined by
$$\Delta(u_{i}) = u_{i}\otimes
u_{i}, \qquad \Delta(a_{i}) = a_{i}\otimes 1 + g_{i}\otimes
a_{i},\quad i = 1,2.$$
There are two subcases depending on whether or not $g_1,g_2$ are
linearly independent over $\Z /(p)$. In the first subcase we take
$g_i = u_i$ and $\chi_i(u_i) = \xi_i$, $\xi_i$ primitive $p$-th roots
of unity, $1 \leq i \leq 2$ and $\chi_1(u_2) = \rho$, $\rho$ a $p$-th
root of unity. Note that necessarily $\chi_2(u_1) =
\rho^{-1}$. If we interchange $\xi_1,\xi_2$ and replace $\rho$ by
$\rho^{-1}$, then we get isomorphic Hopf algebras, and this is the
only possibility to get an isomorphism.
\vspace*{3mm}\\
In the second subcase we take $g_1 = u_1, g_2 = u_1^l$, $l$ an integer
not divisible by $p$, $\chi_1(u_1) = \xi, \chi_2(u_1) = \xi^{-l},
\chi_1(u_2) = 1, \chi_2(u_2) = \xi^{\varepsilon}$, $\xi$ a primitive
$p$-th root of unity and $\varepsilon = 0,1$. The only way to get
isomorphic Hopf algebras is to replace $l$ by $l^{-1}$ mod $p$ and
$\xi$ by $\xi^{-l^2}$.
\vspace*{3mm}\\
{\it 2.4 Index $p^{2}$, $R$ a quantum plane and $\Gamma\simeq
\Z /(p^{2})$}
\vspace*{3mm}\\
Then there exist $g_{1}, g_{2}\in \Gamma$ and  characters $\chi_{1},
\chi_{2} \in \widehat{\Gamma}$ such that $\chi_{1}(g_{1})$ and
$\chi_{2}(g_{2})$ have order $p$, $\chi_{1} (g_{2}) \chi_{2} (g_{1})
= 1$.  There are further possible subcases according to the orders
of $g_{1}$, $g_{2}$,  $\chi_{1}$, and $\chi_{2}$.  It is easy to see that
either ord$(g_i) = p$ and ord$(\chi_i) = p^2$, $i = 1,2$ (first subcase) or
ord$(g_i) = p^2$ and ord$(\chi_i) = p$, $i =1,2$ (second, third and fourth subcases according to the different choices of $\mu_1$ and $\mu_2$).
\vspace*{3mm}\\
{\it First subcase.} Let $\xi$ be a primitive
$p^{2}$-th root of unity and integers $b, c_{1}, c_{2}$ integers prime to $p$
such that $p\vert (bc_{1} + c_{2})$. $A$ is generated by
$h$, $x_{1}$, $x_{2}$ with relations
\begin{equation}\eqlabel{3.11}\begin{array}{l}
h^{p^{2}} = 1\\
hx_{1} = \xi x_{1}h,~~hx_{2} = \xi^{b}x_{2}h;\\
x_{1}^{p} = 0,~~x_{2}^{p} = 0;\\
x_{2}x_{1} - \xi^{pbc_{1}} x_{1}x_{2} = \lambda
\left(1 - h^{p(c_{1} + c_{2})} \right), {\rm ~where~} \lambda
{\rm~is~} 0 {\rm~or~} 1\\
{\rm (and~ it~ could~ be~ 1 ~only~ if~} b \equiv -1 \mod {p^{2}}
{\rm~ and~} c_1 +c_2 \not\equiv 0 \mod p).
\end{array}\end{equation}
carries a Hopf algebra structure given by
$$\Delta(h) =h\otimes h, \qquad
\Delta(x_{i}) = x_{i} \otimes 1 + h^{pc_{i}}\otimes x_{i}, \quad
1\leq i \leq 2. $$
The only ways to get isomorphic Hopf algebras
(except congruence mod $p$ for $c_1,c_2$ and mod $p^2$ for $b$) are
replacing $c_1,c_2,b,\xi$ by $sc_1, sc_2,b, \eta$ where $s$ is an
integer prime to $p$ and $\eta^s = \xi$; or by $sc_2, sc_1, d, \eta$
where $s$ again is prime to $p$, $d$ is an integer with
$bd \equiv 1 \mod {p^2}$ and $\eta^s = \xi^b$.
\vspace*{3mm}\\
{\it Second subcase.}
Let $q$ be a primitive
$p$-th root of unity and $a, d_{1}, d_{2}$ integers prime to $p$
such that $p\vert (ad_{1} + d_{2})$. $A$ is generated by
$h$, $x_{1}$, $x_{2}$ with relations
\begin{equation}\eqlabel{3.15}\begin{array}{l}
h^{p^{2}} = 1;\\
hx_{1} = q^{d_{1}} x_{1}h,~~hx_{2} = q^{d_{2}}x_{2}h;\\
x_{1}^{p} = 0,~~x_{2}^{p} = 0;\\
x_{2}x_{1} - q^{d_{2}} x_{1}x_{2} = \lambda \left(1 -
h^{1 + a} \right), {\rm~where~}\lambda~{\rm is~} 0 {\rm~or~} 1\\
{\rm (and~ it ~could~ be~ 1~
only~ if~} d_{1} + d_{2} \equiv  0\mod p~{\rm and~} 1 + a \not \equiv 0
\mod p).
\end{array}\end{equation}
carries a Hopf algebra structure given by $$\Delta(h) = h\otimes h , \quad
\Delta(x_{1}) = x_{1} \otimes 1 + h\otimes x_{1}, \quad
\Delta(x_{2}) = x_{2} \otimes 1 + h^{a}\otimes x_{2}. $$ The only
way to get isomorphic Hopf algebras (except congruence $\mod p$ for
$d_1,d_2$ and $\mod p^2$ for $a$) is to replace $d_1,d_2,a$ by $d_2a,
d_1a,b$ where $ab \equiv 1 \mod p^2$.
\vspace*{3mm}\\
{\it Third subcase.}\\
Let $q$ be a primitive $p$-th root of unity and
$a, d_{1}, d_{2}$ integers prime to $p$
such that $p\vert (ad_{1} + d_{2})$. $A$ is generated by
$h$, $x_{1}$, $x_{2}$ with relations
\begin{equation}\eqlabel{3.19}\begin{array}{l}
h^{p^{2}} = 1;\\
hx_{1} = q^{d_{1}} x_{1}h,~~hx_{2} = q^{d_{2}}x_{2}h;\\
x_{1}^{p} = 1 - h^{p},~~x_{2}^{p} = 0;\\
x_{2}x_{1} - q^{d_{2}} x_{1}x_{2} = \lambda \left(1 -
h^{1 + a} \right), {\rm~where~}\lambda~{\rm is~} 0 {\rm~or~} 1\\
{\rm (and~ it~ could~ be~ 1~
only~ if~} d_{1} + d_{2} \equiv  0 \mod p{\rm~and~}1 + a \not \equiv 0 \mod p).
\end{array}\end{equation}
carries a Hopf algebra structure given by $$\Delta(h) = h\otimes h , \quad
\Delta(x_{1}) = x_{1} \otimes 1 + h\otimes x_{1}, \quad
\Delta(x_{2}) = x_{2} \otimes 1 + h^{a}\otimes x_{2}. $$
Here the only way to get isomorphic Hopf algebras is to replace
$d_1,d_2$ by mod $p$ congruent elements and $a$ by a mod $p^2$
congruent element.
\vspace*{3mm}\\
In the first three subcases, $\lambda$ is invariant under isomorphism. 
\vspace*{3mm}\\
{\it Fourth subcase.}\\
It is only here that we get an infinite family of non-isomorphic Hopf algebras.\\
Let $q$ be a primitive $p$-th root of unity and
$a, d_{1}, d_{2}$ integers prime to $p$
such that $p\vert (ad_{1} + d_{2})$. $A$ is generated by
$h$, $x_{1}$, $x_{2}$ with relations
\begin{equation}\eqlabel{3.24}\begin{array}{l}
h^{p^{2}} = 1;\\
hx_{1} = q^{d_{1}} x_{1}h,~~hx_{2} = q^{d_{2}}x_{2}h;\\
x_{1}^{p} = 1 - h^{p},~~x_{2}^{p} = 1 - h^{pa};\\
x_{2}x_{1} - q^{d_{2}} x_{1}x_{2} = \lambda \left(1 -
h^{1 + a} \right), {\rm where~}\lambda\in \k\\
{\rm (and~ it~ could~ be~ non~ zero~
only~ if~} d_{1} + d_{2} \equiv  0 \mod p {\rm ~and}\\
1 + a \not \equiv 0 \mod p).
\end{array}\end{equation}
carries a Hopf algebra structure given by $$\Delta(h) = h\otimes h , \quad
\Delta(x_{1}) = x_{1}
\otimes 1 + h\otimes x_{1}, \quad \Delta(x_{2}) = x_{2} \otimes 1 +
h^{a}\otimes x_{2}. $$
Hopf algebras defined by $d_1,d_2,a,\lambda$ and
$\widetilde{d}_1,\widetilde{d}_2, \widetilde{a},\widetilde{\lambda}$ are
isomorphic if
and only if there exists a scalar $\alpha \in \k$ such that $\alpha^p = 1$
and (1)
$\lambda = \alpha \widetilde{\lambda}$, $\widetilde{d}_i \equiv d_i \mod
p$, $1 \leq i
\leq 2$, $\widetilde{a} \equiv a \mod p^2$, or (2) $\lambda = - \alpha
\widetilde{\lambda}$, $\widetilde{d}_2 \equiv d_1a \mod p$,
$\widetilde{d}_1 \equiv
d_2a \mod p$ and $a\widetilde{a} \equiv 1 \mod p^2$.

\section{Nichols algebras of type $A_{2}$ and their liftings}\selabel{4}
We fix a finite abelian group $\Gamma$, elements $g_1,g_2 \in
\Gamma$ and characters $\chi_1,\chi_2 \in \widehat{\Gamma}$ of
$\Gamma$, an odd integer $p > 1$ (which is not assumed to be prime
in this section) and a primitive $p$-th root of unity $q$ in $\k$
such that
\begin{equation}\eqlabel{4.1}
q = \chi_1(g_1) = \chi_2(g_2) ,\quad
\chi_1(g_2)\chi_2(g_1) = q^{-1}.
\end{equation}
Thus the braiding $b_{ij} = \chi_j(g_i), 1 \leq i, j \leq 2$, is of
type $A_2.$

If $C$ is a coalgebra, we denote  by $G(C)$ the set of group-like
elements of $C$. If $g$, $h \in G(C)$, then we let $P_{g,h}(C) =
 \{x\in C \mid \Delta(x) =  g\otimes x + x\otimes h \}$ be the
space of all 
$(g,h)$-primitive elements. If $A$ is a Hopf algebra and $\Gamma =
G(A)$ is abelian, then $\Gamma$ acts on each $P_{g,h}(A)$ by
conjugation, and if $\chi$ is a character of $\Gamma$, then we
define $P_{g,h}(A)^{\chi} = \{a\in P_{g,h}(A) \mid uau^{-1} =
\chi(u) a,$ for all $u \in \Gamma \}.$

\begin{lemma}\lelabel{4.1}
Let $A$ be a Hopf algebra with coradical
$\k[\Gamma]$ and $a_1,a_2 \in A$ such that $$\Delta(a_i) =
g_i\otimes a_i + a_i\otimes 1, \quad ha_ih^{-1} = \chi_i(h)a, \quad
{\rm for ~ all} \quad h \in \Gamma, \quad 1 \leq i \leq 2.$$  Define
$c = a_1a_2 - \chi_2(g_1)a_2a_1.$
Then\\
\hspace*{5mm}(1) $\Delta(c) = g_1g_2 \otimes c + c\otimes 1 + (1 -
q^{-1})a_1g_2 \otimes a_2$.\\
\hspace*{5mm}(2) $ca_1 - \chi_1(g_2)a_1c\in
P_{g_1^2g_2,1}(A)^{\chi_1^2\chi_2}.$
\\
\hspace*{5mm}(3) $a_2c - \chi_1(g_2)ca_2\in
P_{g_1g_2^2,1}(A)^{\chi_1\chi_2^2}$.
\\
\hspace*{5mm}(4) $a_i^p \in P_{g_i^p,1}(A)^{\chi_i^p}, 1 \leq i \leq 2.$
\\
\hspace*{5mm}(5) Assume the Serre relations
$$ca_1 - \chi_1(g_2)a_1c = 0,~~{\rm and}~~
a_2c - \chi_1(g_2)ca_2 = 0.$$
\hspace*{11mm}Then
$$ \Delta(c^p) = g_1^pg_2^p
\otimes c^p + c^p \otimes 1 + (q - 1)^p\chi_1(g_2)^{\frac{p(p -
1)}{2}}a_1^pg_2^p \otimes a_2^p.$$
\end{lemma}

\begin{proof} (1) follows easily from the assumptions. (2) and (3) follow
by direct
computation using also \eqref{4.1}.

(4) follows from the $q$-binomial formula: let $u_1 = a_i \otimes 1, u_2 =
g_i \otimes
a_i$. Then $$ u_2u_1 = g_ia_i \otimes a_i = qa_ig_i \otimes a_i =
qu_1u_2.$$ Hence by
the $q$-binomial formula, $$\Delta(a_i)^p = (u_1 + u_2)^p = u_1^p + u_2^p =
g_i^p
\otimes a_i^p + a_i^p \otimes 1,$$ since $q$ is a $p$-th root of unity.

(5) Let $u_1 = c \otimes 1, u_2 = (1 - q^{-1})a_1g_2 \otimes a_2$ and $u_3
= g_1g_2 \otimes c$. Then
\begin{eqnarray*}
u_2u_1 & =& (1 -q^{-1})a_1g_2c \otimes a_2\\
& =& (1 - q^{-1})\chi_1(g_2)\chi_2(g_2)a_1cg_2 \otimes a_2, \\
& =& (1 - q^{-1})\chi_1(g_2)q\chi_1(g_2)^{-1}ca_1g_2 \otimes a_2,\\
&& \quad\quad {\rm since~}a_1c = \chi_1(g_2)^{-1}ca_1,\\
& =& qu_1u_2.
\end{eqnarray*}
In the same way we see $u_3u_2 = qu_2u_3$ using the relation $a_2c =
\chi_1(g_2)ca_2.$

Finally $$ u_3u_1 = g_1g_2c \otimes c =
(\chi_1\chi_2)(g_1g_2)cg_1g_2\otimes c =
qu_1u_3,$$ since $(\chi_1\chi_2)(g_1g_2) =
\chi_1(g_1)\chi_1(g_2)\chi_2(g_1)\chi_2(g_2)
= q$ by \eqref{4.1}.

Hence by the $q$-binomial formula,
$$\Delta(c^p) = (u_1 + u_2 + u_3)^p = (u_1 + u_2)^p + u_3^p = u_1^p + u_2^p
+ u_3^p.$$
This proves (5) since
$$(a_1g_2)^p = \chi_1(g_2)^{\frac{p(p - 1)}{2}}a_1^pg_2^p.$$
\end{proof}

To compute all liftings of type $A_2$
we first list the following properties of our elements $g_1,g_2,
\chi_1, \chi_2$ satisfying \eqref{4.1}.

\begin{lemma}\lelabel{4.2}
(1) $g_1^2g_2 \ne g_i$ and $g_1g_2^2 \ne g_i$, $1 \leq i \leq 2$.\\
\hspace*{5mm}(2) If $p > 3$, then $\chi_1^2\chi_2 \ne \varepsilon,
\chi_1\chi_2^2
\ne \varepsilon.$\\
\hspace*{5mm}(3) If $\Gamma$ is cyclic of order 3 and $\chi_1^2\chi_2 =
\varepsilon$ or $\chi_1\chi_2^2 = \varepsilon$, then $g_1^2g_2 = 1 =
g_1g_2^2.$
\end{lemma}

\begin{proof} (1) Assume $g_1^2g_2 = g_2$. Then $g_1^2 = 1$, and we
get the contradiction $\chi_1(g_1)^2 = 1$. If $g_1^2g_2 = g_1$, then
$g_1g_2 = 1$, and we see from \eqref{4.1} that $\chi_2(g_1) = 1$, and
$\chi_1(g_2) = 1.$ By \eqref{4.1} this implies $q = 1$ which is
impossible. In the same way we see that $g_1g_2^2 \ne g_i.$


(2) Assume $\chi_1^2\chi_2 = \varepsilon.$ Then $\chi_1\chi_2 =
\chi_1^{-1}$, and $\chi_1^{-1}(g_1)\chi_1(g_2) = 1$ and
$\chi_1^{-1}(g_2)\chi_2(g_1) = 1$ by \eqref{4.1}. Hence $q = \chi_1(g_2) =
\chi_2(g_1)$ which implies by \eqref{4.1} the contradiction $q^3 = 1$. In
the same way we prove $\chi_1\chi_2^2 \ne \varepsilon.$

(3) We consider the case $\chi_1^2\chi_2 = \varepsilon.$ Since
$\widehat\Gamma \cong \Gamma $ is of order 3, we get $\chi_1 =
\chi_2$. Then $\chi_1(g_1^2g_2) = \chi_1(g_1)\chi_2(g_2)\chi_1(g_2)
= 1$ by \eqref{4.1}. Therefore $g_1^2g_2 = 1.$ Hence $g_1 = g_2$, and
$g_1g_2^2 = 1$.
\end{proof}

We also need

\begin{lemma}\lelabel{4.3} Let $\Gamma$ be any group, $g,h \in \Gamma,
\alpha \in \k$ and $x \in \k[\Gamma].$ Then the following are
equivalent:\\
\hspace*{5mm}(1) $\Delta(x) = gh \otimes x + x \otimes 1 + \alpha(1 - g)h
\otimes (1 -
h).$\\
\hspace*{5mm}(2) $x = \alpha(1 - h) + \lambda(1 - gh)$ for some $\lambda
\in \k$.
\end{lemma}

\begin{proof} Both statements mean that $x - \alpha(1 - h)$ is
$(gh,1)$-primitive in the group algebra $\k[\Gamma]$.
\end{proof}

\begin{lemma}\lelabel{4.4} Let $x,y,z$ be elements in an algebra, $\alpha,
\beta$ scalars in $\k$ and $n$ a natural number. Assume $yx = \alpha
xy + z$.

(1) Asssume $zx = \beta xz$. Then $yx^n = \alpha^n x^ny +
 (\sum_{i=0}^{n-1} \alpha^i\beta^{n-1-i})x^{n-1}z,$ and if $\alpha
\ne \beta$ and $\alpha^n = \beta^n$, then $(\alpha xy + z)x^{n-1} =
\alpha^n x^ny.$

(2) Assume $yz = \beta zy$.  Then $y^nx = \alpha^n xy^n +
 (\sum_{i=0}^{n-1} \alpha^i\beta^{n-1-i})zy^{n-1},$ and if $\alpha
\ne \beta$ and $\alpha^n = \beta^n$, then $y^{n-1}(\alpha xy + z) =
\alpha^n xy^n.$
\end{lemma}

\begin{proof} The first formula in (1) is easily proved by induction
and the second formula follows from the first since $
\sum_{i=0}^{n-1} \alpha^i \beta^{n-1-i} = 0.$ (2) is dual to (1).
\end{proof}

\begin{definition}\delabel{4.5}
Define
\begin{eqnarray*}
U^+ := \k\langle x_1,z,x_2 ~\mid~
x_2x_1 & =& \chi_2(g_1)^{-1} x_1x_2 - \chi_2(g_1)^{-1}z,\\
zx_1 &=&
\chi_1(g_2)x_1z,\\ x_2z & =& \chi_1(g_2)zx_2\rangle
\end{eqnarray*}
as a Hopf
algebra in the category of left Yetter-Drinfel'd modules over
$\k[\Gamma]$ with primitive elements $x_1,x_2$ and action and
coaction of $\Gamma$ given by
$$ h\cdot x_i = \chi_i(h)x_i,\quad
{\rm for~ all} \quad h \in \Gamma, \quad   \delta(x_i) = g_i \otimes
x_i,\quad 1 \leq i \leq 2.$$
Let $U := U^+ \# \k[\Gamma]$ be the
Radford biproduct. Identifying $y\#u = yu$ , $y \in U^+, u \in
\k[\Gamma]$ we have in $U$ $$\Delta(x_i) = g_i \otimes x_i + x_i
\otimes 1, \quad hx_ih^{-1} = \chi_i(h)x_i, \quad {\rm for ~ all}
\quad h \in \Gamma, 1 \leq i \leq 2.$$
\end{definition}

In quantum group theory our $U$ is usually called $U^{\geq 0}$. 
Note that $U^+$ is a well-defined braided Hopf algebra since by
\leref{4.1} the Serre relations are primitive in the free algebra
generated by the primitive elements $x_1, x_2$. It follows easily
from the diamond lemma that $U^+$ has the PBW-basis $x_1^iz^jx_2^l$,
$i,j,l \geq 0.$

Then $V := \k x_1 + \k x_2 \subset U^+$ is a Yetter-Drinfel'd module
over $\k[\Gamma]$ of type $A_2$.


\begin{theorem}\thlabel{4.6}
Let  $\mu_1, \mu_2 \in \{0,1\}$ and $\lambda \in \k$ be scalars such
that
\begin{eqnarray}
\mu_i = 0,& if& g_i^p = 1~ or ~\chi_i^p \ne
\varepsilon, \quad 1 \leq i \leq 2.\eqlabel{4.6.1}\\
\lambda = 0,& if &g_1^pg_2^p = 1~ or
~\chi_1^p\chi_2^p \ne \varepsilon.\eqlabel{4.6.2}
\end{eqnarray}
Define $u_i:= \mu_i(1 - g_i^p)$, $1 \leq i \leq 2$, and $u:= \alpha
(1 - g_2^p) + \lambda (1 - g_1^pg_2^p)$ in $\k[\Gamma]$, where
$\alpha:= \mu_1\mu_2(q - 1)^p$. Then
$$A:= U/(x_1^p - u_1, z^p - u, x_2^p - u_2)$$ is a quotient Hopf
algebra of $U$ of dimension $\ord (\Gamma)p^3$, $\BBBB(V) \cong
U^+/(x_1^p, z^p, x_2^p)$, and $$\gr(A) \cong \BBBB(V) \# \k[\Gamma]
\cong U/(x_1^p, z^p, x_2^p).$$
\end{theorem}

\begin{proof} 1) We first show that $A$ is a quotient Hopf algebra of
$U$. By \leref{4.1} (4), $x_i^p - u_i$ is $(g_i^p,1)$-primitive, since
$u_i$ is $(g_i^p,1)$-primitive by construction. Let $\sigma := (q -
1)^p\chi_1(g_2)^{\frac{p(p-1)}{2}}$. Note that $\mu_1 \mu_2 \sigma = \alpha$ since $p$ is odd and $\chi_1^p = \varepsilon$ if $\mu_1 \neq 0$. By  \leref{4.1} (5) and 
\leref{4.3} (1), $$\Delta(z^p) = g_1^pg_2^p \otimes z^p + \sigma x_1^pg_2^p
\otimes x_2^p + z^p \otimes 1$$ and $$\Delta (u) = g_1^pg_2^p
\otimes u + \sigma u_1g_2^p \otimes u_2 + u \otimes 1.$$ Hence
$$\Delta(z^p - u) = g_1^pg_2^p \otimes (z^p - u) + \sigma x_1^pg_2^p
\otimes (x_2^p - u_2) + \sigma(x_1^p - u_1)g_2^p \otimes u_2 + (z^p
- u) \otimes 1.$$ Thus $(x_1^p - u_1,z^p - u,x_2^p - u_{2})$ is a
biideal of $U$, and $A$ is a quotient Hopf algebra of $U$ since $U$
is pointed.

2) Now we show that $\dim A = \ord(\Gamma)p^3$. Let $a_1,c,a_2$ be
the canonical images of $x_1,z,x_2$ in $A$. Then the elements
$ha_1^ic^ja_2^l$, $h \in \Gamma$, $i,j,l \geq 0$ form a $\k$-basis
of $A$. This can be seen using the diamond lemma. As in the proof of
\cite[Proposition 5.2]{AS2} we use the ordering $h_1 < \ldots <h_s <
a_1 < c < a_2$ where $h_1, \ldots, h_s$ are generators of the group
$\Gamma$. If $M$ is a monomial in $h_1, \ldots,a_2$ we let $\Phi(M)$
be its $\{a_1,c,a_2\}$-part. If $M,N$ are monomials we define $M <
N$ if $\Phi(M) < \Phi(N)$ in the standard degree-lexicographic order
or $\Phi(M) = \Phi(N)$ and $M$ is smaller than $N$ in the standard
order. The relations of $A$ are the relations of $\Gamma$, the
relations of $U$, namely for all $h \in \Gamma$, $1 \leq i \leq 2,$
\begin{eqnarray*}
a_2a_1 &=& \chi_2(g_1)^{-1}a_1a_2 - \chi_2(g_1)^{-1}c,\\
ca_1 &=& \chi_1(g_2)a_1c,\\
a_2c &=& \chi_1(g_2)ca_2,\\
a_ih &=&\chi_i(h)^{-1}ha_i,\\
ch &=& (\chi_1\chi_2)(h)^{-1}hc,
\end{eqnarray*} and
the additional relations $a_1^p = u_1,\quad c^p = u,\quad a_2^p =
u_2.$

We only need to consider overlaps involving the additional
relations. In the overlapping pair $(a_2a_1^p, (a_2a_1)a_1^{p-1})$
the first element is reduced to $a_2u_1$. The second element is
reduced to $(\chi_2(g_1)^{-1}a_1a_2 - \chi_2(g_1)^{-1}c)a_1^{p-1}$.
By \leref{4.4} (1) this expression is equal to
$\chi_2(g_1)^{-p}a_1^pa_2 = \chi_2(g_1)^{-p}u_1a_2.$ To see that we
have reached the same normal form we need $a_2u_1 =
\chi_2(g_1)^{-p}u_1a_2$. This equality holds in case $u_1 = 0$. If
$u_1 \ne 0$, we have $\chi_1^p = \varepsilon$ by \eqref{4.6.2}, hence by
\eqref{4.6.1} also $\chi_2(g_1)^p = 1$, and the desired equality
again follows. In the same way we treat
$(a_2a_1^p,a_2^{p-1}(a_2a_1))$ using \leref{4.4} (2).

The remaining overlaps
$$(a_ih,a_i^{p-1}(a_ih)), \quad(c^ph,c^{p-1}ch),
\quad(ca_1^p,(ca_1)a_1^{p-1}), \quad  (c^pa_1,c^{p-1}(ca_1)),$$
$$(a_2^pc,a_2^{p-1}(a_2c)),\quad (a_2c^p,(a_2c)c^{p-1}), \quad
(a_i^pa_i^r,a_i^ra_i^p) \quad{\rm and} \quad(c^pc^r,c^rc^p)$$
are easily reduced to the same normal form since respectively
$$u_i = \chi_i(h)^{-1}u_i, ~~ u = (\chi_1\chi_2)(h)^{-p}u, ~~
 cu_1 = \chi_1(g_2)^pu_1c, ~~ ua_1 = \chi_1(g_2)^pa_1u,$$
$$ u_2c = \chi_1(g_2)^pcu_2, \quad
 a_2u = \chi_1(g_2)ua_2, \quad u_1a_i = a_iu_1, \quad {\rm and}
\quad uc = cu.$$

3) The braided Hopf algebra $\BBBB(V)$ is a quotient of the free
algebra $T(V)$ generated by the primitive elements $x_1,x_2$ with
action and coaction of $\Gamma$ as in $V$. Define $z := x_1x_2 -
\chi_2(g_1)x_2x_1$ in the free algebra. By \leref{4.1} applied to the
biproduct $T(V) \# \k[\Gamma]$ it follows that the elements $zx_1 -
\chi_1(g_2)x_1z, \quad x_2z = \chi_1(g_2)zx_2 \quad {\rm and} \quad
x_1^p, x_2^p$ are primitive in $T(V)$. Since $P(\BBBB(V)) = V$ by
definition of $\BBBB(V)$, these elements are mapped onto $0$ in
$\BBBB(V)$. Then it follows from \leref{4.1} (5) that also $z^p$ is
primitive in $\BBBB(V)$ hence $0$. Thus $\BBBB(V)$ is a braided
Hopf algebra quotient of $U^+/(x_1^p,z^p,x_2^p)$. We know that
$\BBBB(V)$ has dimension $p^3$ \cite{AS3}. Hence $\BBBB(V) \cong
U^+/(x_1^p,z^p,x_2^p)$ since the last algebra also has dimension
$p^3$ for example by the proof in 2).

Finally consider the Hopf algebra map $U \to \gr(A)$ which maps
$x_i$ onto the residue class $\bar{a_i}$ of $a_i$ in $\gr(A)(1) =
A_1/A_0$, $1 \leq i \leq 2$, $z$ onto $\bar{c}$ in $A_2/A_1$ and $h$
onto $h$ for the elements in the group $\Gamma$. This map is
surjective since $A$ is generated by $a_1,a_2$ and $\Gamma$, and
$x_1^p,x_2^p,z^p$ are in the kernel since $\bar{a_1}^p, \bar{a_2}^p,
\bar{c}^p$ are all $0$ since by definition $a_1^p, a_2^p, c^p$ are
in the group algebra $A_0 = \k[\Gamma]$. Hence $$U/(x_1^p,x_2^p,z^p)
\cong \BBBB(V) \# \k[\Gamma] \cong \gr(A).$$
\end{proof}

As an example of dimension $p^5$ consider $\Gamma = \langle g \rangle$
cyclic of
order $p^2$. Let $q$ be a primitive $p$-th root of unity and $b,d$
integers such that $bd \equiv 1, b + d \equiv -1 \mod p$. Such
integers exist if $p = 3$ or $p \equiv 1 \mod 3$ and $-3$ is a square
$\mod p$ by the quadratic reciprocity law. Define $g_1:= g,
g_2:= g^b$, and $\chi_1(g):= q, \chi_2(g):= q^d$. Then we can choose
in \thref{4.6} $\mu_1 = 1, \mu_2 = 1$ and $\lambda$ arbitrary in
$\k$.

\begin{theorem}\thlabel{4.7} Let $A$ be a pointed Hopf algebra with
$$\gr(A) \cong \BBBB(V) \# \k[\Gamma]$$

where $V$ is of type $A_2$ as above.\\
(1) If $p > 3$ or if $\Gamma$ is cyclic of order 3, then $A$ is one
of the Hopf algebras described in \thref{4.6}.\\
(2) If $\Gamma$ has exponent $p$ where $p >3$ is a prime, or $\Gamma$ is cyclic of order $3$, then $A
\cong \BBBB(V) \# \k[\Gamma]$.
\end{theorem}

\begin{proof} We show (1); (2) follows immediately.

Let $A_0 \subset A_1 \subset \ldots$ be the coradical
filtration of $A$. Then we know from \cite[Lemma 5.4]{AS2} that

\begin{equation}\eqlabel{4.4}
A_1 = \k[\Gamma] \oplus \bigoplus_{g,h,\chi}
P_{g,h}(A)^{\chi}
\end{equation}

where $g,h$ run through all elements of $\Gamma$ and $\chi$ runs
through all non-trivial characters of $\Gamma$. Since
 $\gr(A) \cong \BBBB(V) \# \k[\Gamma]$ we
conclude for all elements $g \in \Gamma$ and non-trivial characters
$\chi$ of $\Gamma$
\begin{equation}\eqlabel{4.5}
P_{g,1}(A)^{\chi} \ne 0
\quad {\rm if ~and ~only ~if} \quad g =g_i, \chi = \chi_i, \quad i \in
\{1,2\}.
\end{equation}

Choose $a_i \in P_{g_i,1}(A)^{\chi_i}$ with residue class $x_i$ in
$A_1/A_0$,  $1 \leq i \leq 2$, and define $c:= a_1a_2 -
\chi_2(g_1)a_2a_1$. By \leref{4.1}, $ca_1 - \chi_1(g_2)a_1c \in
P_{g_1^2g_2,1}(A)^{\chi_1^2\chi_2}$ and $a_2c - \chi_1(g_2)ca_2 \in
P_{g_1g_2^2,1}(A)^{\chi_1\chi_2^2}.$

If $\chi_1^2\chi_2$ is non-trivial, then
$P_{g_1^2g_2,1}(A)^{\chi_1^2\chi_2} = 0$ by \eqref{4.5} since $g_1^2g_2
\ne g_i$ by \leref{4.2} (1). In particular,
$P_{g_1^2g_2,1}(A)^{\chi_1^2\chi_2} = 0$ if $p > 3$ by \leref{4.2}
(2). If $\Gamma$ is cyclic of order $3$, and $\chi_1^2\chi_2 =
\varepsilon$, then $g_1^2g_2 = 1$ by Lemma 3.2 (3). Hence
$P_{g_1^2g_2,1}(A) = P_{1,1}(A) = P(A) = 0$ since $A$ is
finite-dimensional and the characteristic of $\k$ is 0. Similarly, $P_{g_1g_2^2,1}(A)^{\chi_1 \chi_2^2} = 0$. Thus we have
shown the Serre relations $$ca_1 = \chi_1(g_2)a_1c, \quad a_2c =
\chi_1(g_2)ca_2.$$ By \leref{4.1} (4), $a_i^p \in
P_{g_i^p,1}(A)^{\chi_i^p}, 1 \leq i \leq 2.$ Since $\chi_i^p(g_i^p) = 1$, $1 \leq i \leq 2$, $g_i^p$ and $a_i^p$ commute. Thus by Lemma 3.1 in \cite{AS2},
 $$a_i^p = \mu_i(1 - g_i^p) \quad {\rm  for~ some} \quad \mu_i
\in \k, 1 \leq i \leq 2.$$ If $\mu_i \neq 0$, $i \in \{1,2 \}$, then we can choose $\mu_i = 1$ by replacing $a_i$ by a scalar multiple of $a_i$. If $\chi_i^p \neq \varepsilon$, and $g_i^p \neq 1$, then $\mu_i$ must be zero.
 From \leref{4.1} (5) we
then obtain $$\Delta(c^p) = g_1^pg_2^p \otimes c^p + c^p \otimes 1 +
\alpha (1 - g_1^p)g_2^p \otimes (1 - g_2^p),$$ where $\alpha:=
\mu_1\mu_2(q - 1)^p \chi_1(g_2)^{\frac{p(p - 1)}{2}}.$ Then $\alpha = \mu_1 \mu_2 (q-1)^p$ since $p$ is odd and $\chi_1^p = \varepsilon$ if $\mu_1 \neq 0$. Hence
$\Delta(c^p) \in A_0 \otimes A + A \otimes A_0$, and $c^p \in A_1$. Since
$\Gamma$ acts on $c^p$ via the character $\chi_1^p\chi_2^p$, we see
from \eqref{4.4} that $c^p \in \k[\Gamma]$ if $\chi_1^p\chi_2^p =
\varepsilon$.  Hence if $\chi_1^p \chi_2^p = \varepsilon$, then by \leref{4.3}
$$c^p =
\alpha(1 - g_2^p) + \lambda(1 - g_1^pg_2^p)$$ for some scalar
$\lambda$.
Assume that $\chi_1^p\chi_2^p$ is non-trivial. Then
$\chi_1^p$ or $\chi_2^p$ is non-trivial, and $a_1^p = 0$ or $a_2^p =
0$ by the above argument. Hence $\mu_1\mu_2 = 0$. Therefore $\alpha
= 0$, and $c^p \in
P_{g_1^pg_2^p,1}(A)^{\chi_1^p\chi_2^p}.$
Thus again by Lemma 3.1 in \cite{AS2}, $c^p$ is a scalar multiple of $1 - g_1^p g_2^p$.
\end{proof}

\section{Pointed Hopf algebras of dimension $p^4$}\selabel{5}
Let $p$ be an odd prime and $\Gamma = \langle g \rangle$ the cyclic group
of order $p$.
We describe Nichols algebras of type $A_2$ over $\Gamma$. Let $q$ be a
primitive $p$-th
root of unity in $\k$, $b,d$ integers and
$$g_1 = g,\quad g_2 = g^b, \quad \chi_1(g) = q, \quad \chi_2(g) = q^d.$$
Type $A_2$ means $b + d \equiv -1 \equiv - bd \mod p$. Therefore
$b^{2} + b + 1\equiv 0 \mod p$. Hence $p = 3$ and $b \equiv 1 \mod p$, or $b
\not \equiv 1\mod p$ is a cubic root of 1. In the last case, $b$ exists if and
only if $p \equiv 1 \mod 3$. The corresponding Radford biproduct $U^+
\#\k[\Gamma]$ is
the Hopf algebra
\begin{eqnarray*}
U(q,b) := \k \langle g,x_1,z,x_2 &\mid &
gx_1g^{-1} = qx_1, ~gx_2g^{-1} = q^{-1-b}x_2,\\
&&x_1x_2-q^{-1-b}x_2x_1=z,~zx_1=q^bx_1z,~x_2z=q^bzx_2,\\
&& g^p=1,~x_1^p=0,~z^p=0,~x_2^p=0\rangle
\end{eqnarray*}
with $g$ group-like, $x_1$ $(g,1)$-primitive and $x_2$ $(g^b,1)$-primitive.

\begin{theorem}\thlabel{5.1} Let $p$ be an odd prime. Let $A$ be a pointed
non-semisimple Hopf algebra of dimension $p^4$ with coradical $\k[\Gamma]$.
Then $A$ is
isomorphic to one of the Hopf algebras in the list of Section 3 if $\Gamma$
has order
$p^3$ or $p^2$, or to one of the Hopf algebras $U(q,b)$ if $\Gamma$ has
order $p$,
where $p = 3$ or $p\equiv 1\mod 3$.
\end{theorem}

\begin{proof} This follows from \thref{2.1}, \seref{3} and \thref{4.7}.
\end{proof}

In \seref{3}, the Hopf algebras are classified up to isomorphism.
We know from \cite[Lemma 6.4]{AS3} that two Hopf algebras $U(q,b)$ and
$U(\widetilde{q},
\widetilde{b})$ are isomorphic if and only if $\widetilde{q} = q$, and
$\widetilde{b}
\equiv b \mod p$ or $\widetilde{b} \equiv {-1-b} \mod p$. Hence there are
exactly
$2(p-1)$ non-isomorphic Hopf algebras of the form $U(q,b)$ if $p \equiv 1
\mod 3$, and
$2$ of dimension $81$ if $p=3$.

\section{Appendix. Quasi-isomorphism types.}
 We now prove
that there are only a finite number of {\it quasi-isomorphism types}
of pointed Hopf algebras of order $p^{4}$.\vspace*{3mm}\\
{\it Twisting the multiplication by a cocycle.}
Let $A$ be a Hopf algebra and let $\sigma: A\otimes A\to k$ be a
normalized cocycle; that is, $\sigma$ is a convolution invertible
linear map such that $\sigma(x, 1) = \varep(x) = \sigma(1, x)$ and
$$ \sigma(x_{(1)}, y_{(1)})\sigma(x_{(2)}y_{(2)}, z)
=\sigma(y_{(1)}, z_{(1)}) \sigma(x, y_{(2)}z_{(2)}) $$ for all
$x,y,z\in A$. Let $._{\sigma}:  A\otimes A\to A$ be the linear map
given by
\begin{equation}\eqlabel{A.1}
x._{\sigma} y := \sigma(x_{(1)}, y_{(1)}) x_{(2)}
y_{(2)} \sigma^{-1}(x_{(3)}, y_{(3)})
\end{equation}
 Let $A_{\sigma}$
be the vector space underlying $A$ with the comultiplication of $A$
and the multiplication $._{\sigma}$; it is again a Hopf algebra.

\begin{definition} We shall say that two Hopf algebras $A$ and
$B$ are {\it quasi-isomorphic} if there exists an isomorphism $\phi:\
A_{\sigma}\to B$ for some  normalized cocycle $\sigma$.
\end{definition}

\begin{remarks}\rm (i). Dually, it is possible to twist the
comultiplication via $F\in A\otimes A$; we obtain a new Hopf algebra
$A^{F}$ and a corresponding notion of {\it co-quasi-isomorphic.}

(ii). If $C$ is a coalgebra, $\widetilde\sigma: C\otimes C \to k$ is
a normalized cocycle and $\psi: A\to C$ is a coalgebra map, then
evidently  $\sigma: A\otimes A \to k$, $\sigma =
\widetilde\sigma(\psi\otimes \psi)$  is a normalized cocycle. If $A$
is pointed with coradical $\k[\Gamma]$ then there exists always
coalgebra maps $A\to \k[\Gamma]$; hence, we get  normalized cocycles
on $A$ from normalized group cocycles on $\Gamma$.

(iii). It is known that two finite dimensional Hopf algebras are
quasi-isomorphic if and only if their categories of comodules are
equivalent as braided categories.
\end{remarks}

{\it Lifting of quantum linear spaces.}\\
Let $\Gamma$ be a finite abelian group. We fix a decomposition
$\Gamma = \langle  y_{1}\rangle\oplus \dots\oplus  \langle
y_{\nu}\rangle$ and we denote by $M_{\ell}$ the order of $y_{\ell}$,
$1\leq \ell \leq \nu$.\vspace*{3mm}\\
Let $g_{1}, \dots, g_{\theta} \in \Gamma$, $\chi_{1}, \dots,
\chi_{\theta} \in \widehat{\Gamma}$ be a datum for quantum linear
space; {\it i. e.},  the following conditions hold
\begin{equation}\eqlabel{A.2}\begin{array}{l}
q_{i}:= \chi_{i} (g_{i}) \neq 1,{\rm for~ all~} i~.{\rm
We~ denote~ by~} N_{i}~{\rm  the~ order~ of~} q_{i}.\\
\chi_{j} (g_{i}) \chi_{i} (g_{j}) = 1,
\quad {\rm for ~all~}  i \ne j
\end{array}\end{equation}
We abbreviate ${\cal R}   :=  {\cal R}  (g_1, \dots, g_{\theta}; \chi_1,
\dots, \chi_{\theta})$ for the quantum linear space as in
\cite[Section 3]{AS2}. The integer $\theta$ is called the {\it rank} of
${\cal R}$.
\vspace*{5mm}\\
A compatible datum ${\cal D}$ for $\Gamma$ and ${\cal R}$ consists of
\vspace*{3mm} \\
1) a scalar $\mu_{i} \in \{0, 1\}$ for each $i$, $1\leq i
\leq \theta$;  it is arbitrary if $g_{i}^{N_{i}} \neq 1$ and
$\chi_{i}^{N_{i}} = 1$, but 0 otherwise;\\
2) a scalar $\lambda_{ij} \in k$ for each $i, j$, $1\leq i
< j \leq \theta$;  it is is arbitrary if
$g_{i}g_{j} \neq 1$ and
$\chi_{i}\chi_{j} = 1$, but 0 otherwise.

\begin{remark}\rm   If $\lambda_{ij}\neq 0$ and  $\lambda_{ih}\neq 0$,
then $\chi_{i}\chi_{j} = 1$ and $\chi_{i}\chi_{h} = 1$; hence
$\chi_{j} = \chi_{h}.$  If in addition the order $N_i$ of
$q_{i}:=\chi_{i} (g_{i})$ is odd, then  $j = h$ \cite{AS2}.
\end{remark}

\begin{definition} Let $\Gamma$ be a finite abelian group,
${\cal R}  $ a quantum linear space and ${\cal D}  $ a compatible datum.
Keep the notation above. Let ${\cal A}(\Gamma,{\cal R},{\cal D})$ be the
algebra
presented by generators $h_{\ell}$, $1\leq \ell \leq \nu$, and $a_{i}$,
$1\leq i
\leq \theta$ with defining relations
\begin{eqnarray}
&&h_{\ell}^{M_{\ell}} = 1,~~1\leq \ell \leq \nu;\eqlabel{A.6}\\
&&h_{\ell}h_{t} = h_{t}h_{\ell},~~1\leq t <\ell \leq \nu;\eqlabel{A.7}\\
&&a_{i}h_{\ell} = \chi_{i}^{-1}(y_{\ell})h_{\ell} a_{i},~~1\leq
\ell \leq \nu,~~1\leq i \leq \theta;\eqlabel{A.8}\\
&&a_{i}^{N_{i}} = \mu_{i}\left(1 - g_{i}^{N_{i}}\right),~~1\leq i
\leq \theta;\eqlabel{A.9}\\
&&a_{j}a_{i} = \chi_{i}(g_{j}) a_{i}a_{j} + \lambda_{ij} \left(1 -
g_{i} g_{j} \right),~~1\leq i < j \leq \theta.\eqlabel{A.10}
\end{eqnarray}
\end{definition}

There exists a unique Hopf algebra structure on
${\cal A}(\Gamma,{\cal R},{\cal D})$ such that
\begin{equation}\eqlabel{A.11}
\Delta(h_{\ell}) = h_{\ell}\otimes
h_{\ell},
\qquad \Delta(a_{i}) = a_{i} \otimes 1 + g_{i}\otimes a_{i}, \quad
1\leq \ell \leq \nu,\quad 1\leq i \leq \theta.
\end{equation}
The set of monomials
$h_{1}^{r_{1}}\dots h_{\nu}^{r_{\nu}} a_{1}^{s_{1}}\dots
a_{\theta}^{r_{\theta}}$,  $0\le r_{\ell} < M_{\ell}$,   $0\le s_{i}
< N_{i}$,  $1\leq \ell \leq \nu$,  $1\leq i \leq \theta$, is a basis
of ${\cal A}(\Gamma,{\cal R},{\cal D})$. See \cite{AS2}.\vspace*{5mm}\\
Let $\widetilde\sigma:\ \Gamma\times \Gamma \to \k^{\times}$ be a
normalized group cocycle and let $\psi: {\cal A}(\Gamma,{\cal R},{\cal
D})\to \k[\Gamma]$ be the
coalgebra map given by $$\psi(h_{1}^{r_{1}}\dots h_{\nu}^{r_{\nu}}
a_{1}^{s_{1}}\dots a_{\theta}^{r_{\theta}}) = h_{1}^{r_{1}}\dots
h_{\nu}^{r_{\nu}} \delta_{s_{1}, 0}\dots \delta_{r_{\theta}, 0};$$
then   $\sigma: {\cal A}(\Gamma,{\cal R},{\cal D})\otimes {\cal
A}(\Gamma,{\cal R},{\cal D}) \to k$, $\sigma =
\widetilde\sigma(\psi\otimes \psi)$  is a normalized cocycle. Here
we extended linearly $\widetilde\sigma$ to $\k[\Gamma]\times
\k[\Gamma] \to \k$. It is harmless to write $\sigma$ instead of
$\widetilde\sigma$ in what follows.\vspace*{5mm}\\
Let  $\Psi_{i,n - 1} = \prod_{1\le h
\le n}  \sigma(g_{i}, g_{i}^{h}).$ Let $\omega_{i}^{-1}$ be a fixed
$N_{i}$-th root of
$\psi_{i, N_{i}}$.

\begin{lemma}\lelabel{A.1} If $\sigma$ is symmetric, then
${\cal A}(\Gamma,{\cal
R},{\cal D})_{\sigma}$ is isomorphic to
${\cal A}(\Gamma,{\cal R},{\cal D}_{\sigma})$, where
${\cal D}_{\sigma} = (\mu_{i},
\omega_{i}\omega_{j} \sigma(g_{i}, g_{j})\lambda_{ij})$.
\end{lemma}

\begin{proof}
Let $\widehat{h_{\ell}}$, $\widehat{a_{i}}$ be the elements
$h_{\ell}$, $a_{i}$ when regarded in
${\cal A}(\Gamma,{\cal R},{\cal D})_{\sigma}$. We claim that
\begin{eqnarray}
&&\widehat{h_{\ell}}._{\sigma} \widehat{h_{t}} =
h_{\ell}h_{t};\eqlabel{A.12}\\
&&\widehat{a_{i}}._{\sigma}  \widehat{h_{t}} =
\sigma(g_{i}, h_{t}) a_{i}h_{t},  \quad \widehat{h_{t}}._{\sigma}
\widehat{a_{i}} = \sigma(h_{t}, g_{i}) a_{i}h_{t};\eqlabel{A.13}\\
&&\widehat{a_{i}} ._{\sigma}\widehat{a_{j}} = \sigma(g_{i}, g_{j})
a_{i}a_{j};\eqlabel{A.14}\\
&&\widehat{a_{i}} ^{n} =
\Psi_{i,n}a_{i}^{n}.\eqlabel{A.15}
\end{eqnarray}
Indeed, \eqref{A.12}, \eqref{A.13} and \eqref{A.14}
follow easily from the definition \eqref{A.1} of the twisted
multiplication using that $\Delta^{2}(h) = h\otimes h\otimes h$,
$\Delta^{2}(a_{i}) = a_{i} \otimes 1\otimes 1 + g_{i}\otimes
a_{i}\otimes 1 + g_{i}\otimes g_{i}\otimes a_{i}$. In turn, \eqref{A.15}
follows from \eqref{A.1}  by induction; the first step is \eqref{A.14} and the
inductive step uses the expression $$\Delta^{2}(a_{i}^{n}) =
\sum_{0\le h \le n} \sum_{0\le s \le h} {{n}\choose{h}}_{q_{i}}
{{h}\choose{s}}_{q_{i}} a_{i}^{s}g_{i}^{n - s}\otimes  a_{i}^{h - s}
g_{i}^{n - h}\otimes a_{i}^{n - h} g_{i}^{n - h}, $$ which follows
applying twice the quantum binomial formula.

Let now $\widetilde h_{t}$, $\widetilde a_{i}$ be the generators of
${\cal A}(\Gamma,{\cal R},{\cal D})s$ satisfying the relations
$\widetilde{\eqref{A.6}}$, $\cdots$,
$\widetilde{\eqref{A.10}}$ analogous to \eqref{A.6}, \dots, \eqref{A.10}
but with
respect to the data ${\cal D} _{\sigma}$. We claim that there exists a
Hopf algebra isomorphism ${\cal F}:\ {\cal A}(\Gamma,{\cal R},{\cal D})s
\to {\cal
A}(\Gamma,{\cal R},{\cal D})_{\sigma}$ such that
$${\cal F}(\widetilde h_{t}) = \widehat{h_{t}}, \quad{\cal F}(\widetilde
a_{i}) = \omega_{i}\widehat{a_{i}}, \quad  1\le t \le \nu, \quad
1\le i \le \theta.$$  We need only to verify that relations
$\widetilde{\eqref{A.6}}$, $\cdots$,
$\widetilde{\eqref{A.10}}$ hold. Relations
$\widetilde{\eqref{A.6}}, \widetilde{\eqref{A.7}}$, follow immediately from
\eqref{A.12}; $\widetilde{\eqref{A.8}}$ from \eqref{A.13}  and
$\widetilde{\eqref{A.10}}$
from \eqref{A.14}  using the symmetry of $\sigma$; $\widetilde{\eqref{A.9}}$
from \eqref{A.15} by the choice of $\omega_{i}$.
\end{proof}

\begin{lemma}\lelabel{A.2} Let ${\cal D}   = (\mu_{i}, \lambda_{ij})$ be a
compatible datum of rank $\theta = 2$. Let ${\cal E}$ be the
compatible datum $(\mu_{i}, \eta_{ij})$ where $\eta_{ij} = 1$ if
$\lambda_{ij} \neq 0$ and 0 otherwise.  Then ${\cal A}(\Gamma,{\cal
R},{\cal D})$ is
quasi-isomorphic to ${\cal A}(\Gamma, {\cal R}  , {{\cal E}})$ .
\end{lemma}

\begin{proof} By \leref{A.1}, it is enough to find a symmetric
cocycle $\sigma:\ \Gamma\times \Gamma \to \k^{\times}$  such
that  $\omega_{1}\omega_{2} \sigma(g_{1},
g_{2})\lambda_{12} = 1$ whenever $\lambda_{12}\neq 0$. Assume
that $\sigma$ is the coboundary of
a map $f:\ \Gamma \to \k^{\times}$ satisfying $f(1) = 1$. Then
$\sigma(g_{1}, g_{2}) = f(g_{1}) f(g_{2}) f(g_{1}g_{2})^{-1}$ and
\begin{eqnarray*}
\Psi_{1,n} &=&  \sigma(g_{1}, g_{1}) \sigma(g_{1},
g_{1}^{2})\dots \sigma(g_{1}, g_{1}^{n - 1}) \\
&=& f(g_{1}) f(g_{1})
f(g_{1}^{2})^{-1}f(g_{1}) f(g_{1}^{2}) f(g_{1}^{3})^{-1} \dots
f(g_{1}) f(g_{1}^{n-1}) f(g_{1}^{n})^{-1} \\
&=& f(g_{1})^{n}
f(g_{1}^{n})^{-1}.
\end{eqnarray*}
 We can choose $\omega_{1} =
f(g_{1})^{-1}\root N_{1}\of{f(g_{1}^{N_{1}})}$ for some $N_{1}$-th
root of $f(g_{1}^{N_{1}})$ and similarly for $\omega_{2}$. So that
we are reduced to find a function $f$ such that $f(1) = 1$ and
$$\lambda_{12}^{-1} =
\root N_{1}\of{f(g_{1}^{N_{1}})} \root N_{2}\of{f(g_{2}^{N_{2}})}
f(g_{1}g_{2})^{-1}.  $$
Since $\lambda_{12} \neq 0$, $g_{1}g_{2} \neq 1$ and such $f$
exists.
\end{proof}

\begin{proposition}\prlabel{A.3} There are only a finite number of
quasi-isomorphism types of pointed Hopf algebras of order
$p^{4}$.
\end{proposition}

\begin{proof} This follows at once from \leref{A.2} and the
classification results in the previous Sections of the paper.
\end{proof}

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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
FAMAF. UNC. (5000),
C. Universitaria, C\'ordoba,
Argentina,\\
e-mail andrus@mate.uncor.edu\\

Mathematisches Seminar der Universit\"at  M\"unchen,
Theresienstrasse 39,
D-8033 M\"unchen,
Germany,\\
e-mail hanssch\@rz.mathematik.uni-muenchen.de
\end{document}









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