# Classification of Quiver Hopf Algebras and Pointed Hopf Algebras of Type One

Shouchuan Zhang a,c, Hui-xiang Chen b, Yao-Zhong Zhang c,
$a$. Department of Mathematics, Hunan University
Changsha 410082, P.R. China
$b$. Department of Mathematics, Yangzhou University
Yangzhou 225002, P.R. China
$c$. Department of Mathematics, University of Queensland
Brisbane 4072, Australia.

###### Abstract

The quiver Hopf algebras are classified by means of ramification system with
irreducible representations. This leads to the classification of Nichols
algebras over group algebras and pointed Hopf algebras of type one.

2000 Mathematics Subject Classification: 16W30, 16G10

keywords: Quiver, Hopf algebra, Hopf bimodule, Nichols algebras

##  0 Introduction

Hopf algebras have important applications in mathematics and mathematical
physics. Indeed, quasitriangular Hopf algebras gives rise to braided tensor
categories through their categories of representations (see [ENO, EO]).
Semisimple Hopf algebras and non-semisimple Hopf algebras are related to
conformal field theories ( see [Ga]).

The classification of Hopf algebras is the main object in research of Hopf
algebras. So far, much important results have been obtained in the
classification of finite dimensional pointed Hopf algebras (see [AS98, AS02,
AS00, AS05, H1, H2, AZ07]). The classification of PM quiver Hopf algebras was
completed by means of ramification system with characters (RSC in short ) in
[ZZC]. The classification of RSC’s over symmetric group $\mathbb{S}_{n}$ with
$n\not=6$ was obtained in [ZWW]. Irreducible Hopf bimodules over a finite
group were described in [DPR]. They correspond to pairs
$(\mathcal{O}_{s},\rho)$, where $\mathcal{O}_{s}$ is a conjugacy class
containing $s$ in G and $\rho$ is an irreducible representation of the
centralizer $G^{s}$ in G. This result has been reobtained and reproved several
times; these data are used in a previous paper of Lusztig more or less in the
same direction as in [DPR]. The relation between bi-one arrow Nichols algebras
and $\mathfrak{B}({\mathcal{O}}_{s},\rho)$, which is the Nichols algebra
corresponding to pairs $(\mathcal{O}_{s},\rho)$, was given in [ZZWC].

In this paper, quiver Hopf algebras, Nichols algebras over group algebras and
pointed Hopf algebras of type one are classified by means of ramification
system with irreducible representations (RSR in short ). As examples we
classify ${\rm RSR}$’s over symmetric group $\mathbb{S}_{n}$ with $n\not=6.$

The quivers (see [ARS, CR02, CR97, ZZ, OZ] Ri84 ) and the tensor algebras of
Hopf bimodule (see [N, W]) have widely been applied in representation theory,
Hopf algebras and quantum groups. We use the quivers to describe Yetter-
Drinfeld $kG$-modules, $kG$-Hopf bimodules, Nichols algebras in braided tensor
category ${}^{kG}_{kG}{\mathcal{Y}D}$, pointed Hopf algebras of type one and
quiver Hopf algebras in this paper.

The main results in this paper are summarized in the following statement.

######  Theorem 1.

(Classification theorem) (i) Every Yetter-Drinfeld $kG$-module, (ii) every
$kG$-Hopf bimodule, (iii) every Nichols algebra in braided tensor category
${}^{kG}_{kG}{\mathcal{Y}D}$, (iv) every pointed Hopf algebra of type one with
coradical $kG$ and (v) every quiver Hopf algebra over $G$ are, respectively,
isomorphic to one of the following: $(kQ_{1}^{1},{\rm
ad}(G,r,\overrightarrow{\rho},u))$,
$(kQ_{1}^{c},G,r,\overrightarrow{\rho},u),$ ${\mathcal{B}}(kQ_{1}^{1},{\rm
ad}(G,r,\overrightarrow{\rho},u))$,
$kG[kQ_{1}^{c},G,r,\overrightarrow{\rho},u]$,
$kQ^{c}(G,r,\overrightarrow{\rho},u)$, $kQ^{s}(G,r,\overrightarrow{\rho},$
$u)$, $kG[kQ_{1}^{c},$ $G,r,$ $\overrightarrow{\rho},u],$
$kQ^{a}(G,r,\overrightarrow{\rho},u)$, $kQ^{sc}(G,r,\overrightarrow{\rho},u)$,
$(kG)^{*}[kQ_{1}^{a},G,r,\overrightarrow{\rho},u]$. Furthermore, they are
uniquely determined by ${\rm RSR}(G,r,\overrightarrow{\rho},u)$ up to
isomorphisms of ${\rm RSR}$’s.

Indeed, (i) follows from Proposition 2.4(i); (ii) follows from Proposition
1.3; (iii) follows from Proposition 2.4(ii); (iv) follows from Theorem 3; (v)
follows from definition of quiver Hopf algebras in [ZZC, the argument after
Theorem 2]. For the uniqueness, (i) and (iii) follow from Theorem 4; (ii),
(iv) and (v) follow from Theorem 2.

Note that the isomorphism in (i) is about Yetter-Drinfeld $kG$-modules; the
isomorphism in (ii) is about $kG$-Hopf bimodules; the isomorphism in (iii) is
about graded braided Hopf algebras in ${}^{kG}_{kG}{\mathcal{Y}D}$; the
isomorphism in (iv) and (v) are about graded Hopf algebras.

Furthermore, for any fixed map $u_{0}$ from ${\mathcal{K}}(G)$ to $G$ with
$u_{0}(C)\in C$ for any $C\in{\mathcal{K}}(G)$, if $u_{0}$ replaces $u$ in the
theorem above, then the theorem holds still but the isomorphisms in (i), (ii)
and (iii) need change into graded pull-push isomorphisms.

## Preliminaries

Throughout this paper we assume that $G$ is a finite group and $k$ is a field
with ${\rm char}k\nmid\ \mid\\!G\\!\mid$.

Let $\hat{{G}}$ denote the set of all isomorphic classes of irreducible
representations of group $G$ and $Z_{s}$ the centralizer of $s$ in $G$. For
$h\in G$ and an isomorphism $\phi$ from $G$ to $G^{\prime}$, define a map
$\phi_{h}$ from $G$ to $G^{\prime}$ by sending $x$ to $\phi(h^{-1}xh)$ for any
$x\in G$. ${\rm deg}(\rho)$ denotes the dimension of representation space of
representation $\rho$.

Let ${\mathbb{N}}$ and ${\mathbb{Z}}$ denote the sets of all non-negative
integers and all integers, respectively. For a set $X$, we denote by $|X|$ the
number of elements in $X$. If $X=\oplus_{i\in I}X_{(i)}$ as vector spaces,
then we denote by $\iota_{i}$ the natural injection from $X_{(i)}$ to $X$ and
by $\pi_{i}$ the corresponding projection from $X$ to $X_{(i)}$. We will use
$\mu$ to denote the multiplication of an algebra and use $\Delta$ to denote
the comultiplication of a coalgebra. For a (left or right) module and a (left
or right) comodule, denote by $\alpha^{-}$, $\alpha^{+}$, $\delta^{-}$ and
$\delta^{+}$ the left module, right module, left comodule and right comodule
structure maps, respectively. The Sweedler’s sigma notations for coalgebras
and comodules are $\Delta(x)=\sum x_{(1)}\otimes x_{(2)}$, $\delta^{-}(x)=\sum
x_{(-1)}\otimes x_{(0)}$, $\delta^{+}(x)=\sum x_{(0)}\otimes x_{(1)}$.

A quiver $Q=(Q_{0},Q_{1},s,t)$ is an oriented graph, where $Q_{0}$ and $Q_{1}$
are the sets of vertices and arrows, respectively; $s$ and $t$ are two maps
from $Q_{1}$ to $Q_{0}$. For any arrow $a\in Q_{1}$, $s(a)$ and $t(a)$ are
called its start vertex and end vertex, respectively, and $a$ is called an
arrow from $s(a)$ to $t(a)$. For any $n\geq 0$, an $n$-path or a path of
length $n$ in the quiver $Q$ is an ordered sequence of arrows
$p=a_{n}a_{n-1}\cdots a_{1}$ with $t(a_{i})=s(a_{i+1})$ for all $1\leq i\leq
n-1$. Note that a 0-path is exactly a vertex and a 1-path is exactly an arrow.
In this case, we define $s(p)=s(a_{1})$, the start vertex of $p$, and
$t(p)=t(a_{n})$, the end vertex of $p$. For a 0-path $x$, we have
$s(x)=t(x)=x$. Let $Q_{n}$ be the set of $n$-paths. Let ${}^{y}Q_{n}^{x}$
denote the set of all $n$-paths from $x$ to $y$, $x,y\in Q_{0}$. That is,
${}^{y}Q_{n}^{x}=\\{p\in Q_{n}\mid s(p)=x,t(p)=y\\}$.

A quiver $Q$ is finite if $Q_{0}$ and $Q_{1}$ are finite sets. A quiver $Q$ is
locally finite if ${}^{y}Q_{1}^{x}$ is a finite set for any $x,y\in Q_{0}$.

Let $G$ be a group. Let ${\mathcal{K}}(G)$ denote the set of conjugate classes
in $G$. A formal sum $r=\sum_{C\in{\mathcal{K}}(G)}r_{C}C$ of conjugate
classes of $G$ with cardinal number coefficients is called a ramification (or
ramification data ) of $G$, i.e. for any $C\in{\mathcal{K}}(G)$, $r_{C}$ is a
cardinal number. In particular, a formal sum
$r=\sum_{C\in{\mathcal{K}}(G)}r_{C}C$ of conjugate classes of $G$ with non-
negative integer coefficients is a ramification of $G$.

For any ramification $r$ and a $C\in{\mathcal{K}}(G)$, since $r_{C}$ is a
cardinal number, we can choice a set $I_{C}(r)$ such that its cardinal number
is $r_{C}$ without loss generality. Let
${\mathcal{K}}_{r}(G):=\\{C\in{\mathcal{K}}(G)\mid
r_{C}\not=0\\}=\\{C\in{\mathcal{K}}(G)\mid I_{C}(r)\not=\emptyset\\}$. If
there exists a ramification $r$ of $G$ such that the cardinal number of
${}^{y}Q_{1}^{x}$ is equal to $r_{C}$ for any $x,y\in G$ with $x^{-1}y\in
C\in{\mathcal{K}}(G)$, then $Q$ is called a Hopf quiver with respect to the
ramification data $r$. In this case, there is a bijection from $I_{C}(r)$ to
${}^{y}Q_{1}^{x}$, and hence we write ${\ }^{y}Q_{1}^{x}=\\{a_{y,x}^{(i)}\mid
i\in I_{C}(r)\\}$ for any $x,y\in G$ with $x^{-1}y\in C\in{\mathcal{K}}(G)$.
Denote by $(Q,G,r)$ the Hopf quiver of $G$ with respect to $r$.

If $\phi:A\rightarrow A^{\prime}$ is an algebra homomorphism and
$(M,\alpha^{-})$ is a left $A^{\prime}$-module, then $M$ becomes a left
$A$-module with the $A$-action given by $a\cdot x=\phi(a)\cdot x$ for any
$a\in A$, $x\in M$, called a pullback $A$-module through $\phi$, written as
${}_{\phi}M$. Dually, if $\phi:C\rightarrow C^{\prime}$ is a coalgebra
homomorphism and $(M,\delta^{-})$ is a left $C$-comodule, then $M$ is a left
$C^{\prime}$-comodule with the $C^{\prime}$-comodule structure given by
${\delta^{\prime}}^{-}:=(\phi\otimes{\rm id})\delta^{-}$, called a push-out
$C^{\prime}$-comodule through $\phi$, written as ${}^{\phi}M$.

If $B$ is a Hopf algebra and $M$ is a $B$-Hopf bimodule, then we say that
$(B,M)$ is a Hopf bimodules. For any two Hopf bimodules $(B,M)$ and
$(B^{\prime},M^{\prime})$, if $\phi$ is a Hopf algebra homomorphism from $B$
to $B^{\prime}$ and $\psi$ is simultaneously a $B$-bimodule homomorphism from
$M$ to ${}_{\phi}M^{\prime}{}_{\phi}$ and a $B^{\prime}$-bicomodule
homomorphism from ${}^{\phi}M^{\phi}$ to $M^{\prime}$, then $(\phi,\psi)$ is
called a pull-push Hopf bimodule homomorphism. If $\psi$ is a bijection, then
we say that $(\phi,\psi)$ is a a pull-push Hopf bimodule isomorphism, written
as $(B,M)\cong(B^{\prime},M^{\prime})$ as pull-push Hopf bimodules. In
particular, if $B=B^{\prime}$ we also write $M\cong M^{\prime}$ as pull-push
$B$-Hopf bimodules, in short. Similarly, we say that $(B,M)$ and $(B,X)$ are a
Yetter-Drinfeld module and a Yetter-Drinfeld Hopf algebra if $M$ is a Yetter-
Drinfeld $B$-module and $X$ is a braided Hopf algebra in Yetter-Drinfeld
category ${}^{B}_{B}{\mathcal{Y}D}$, respectively. For any two Yetter-Drinfeld
modules $(B,M)$ and $(B^{\prime},M^{\prime})$, if $\phi$ is a Hopf algebra
homomorphism from $B$ to $B^{\prime}$, and $\psi$ is simultaneously a left
$B$-module homomorphism from $M$ to ${}_{\phi}M^{\prime}$ and a left
$B^{\prime}$-comodule homomorphism from ${}^{\phi}M$ to $M^{\prime}$, then
$(\phi,\psi)$ is called a pull-push Yetter-Drinfeld module homomorphism. For
any two Yetter-Drinfeld Hopf algebra $(B,X)$ and $(B^{\prime},X^{\prime})$, if
$\phi$ is a Hopf algebra homomorphism from $B$ to $B^{\prime}$, $\psi$ is
simultaneously a left $B$-module homomorphism from $X$ to
${}_{\phi}X^{\prime}$ and a left $B^{\prime}$-comodule homomorphism from
${}^{\phi}X$ to $X^{\prime}$, meantime, $\psi$ also is algebra and coalgebra
homomorphism from $X$ to $X^{\prime}$, then $(\phi,\psi)$ is called a pull-
push Yetter-Drinfeld Hopf algebra homomorphism (see [ZZC, the remark after
Th.4]).

Let $A$ be an algebra and $M$ be an $A$-bimodule. Then the tensor algebra
$T_{A}(M)$ of $M$ over $A$ is a graded algebra with $T_{A}(M)_{(0)}=A$,
$T_{A}(M)_{(1)}=M$ and $T_{A}(M)_{(n)}=\otimes^{n}_{A}M$ for $n>1$. That is,
$T_{A}(M)=A\oplus(\bigoplus_{n>0}\otimes^{n}_{A}M)$ (see [N]). Let $D$ be
another algebra. If $h$ is an algebra map from $A$ to $D$ and $f$ is an
$A$-bimodule map from $M$ to $D=\ _{h}D_{h}$, then by the universal property
of $T_{A}(M)$ (see [N, Proposition 1.4.1]) there is a unique algebra map
$T_{A}(h,f):T_{A}(M)\rightarrow D$ such that $T_{A}(h,f)\iota_{0}=h$ and
$T_{A}(h,f)\iota_{1}=f$. One can easily see that
$T_{A}(h,f)=h+\sum_{n>0}\mu^{n-1}T_{n}(f)$. For the details, the reader is
directed to [N, Section 1.4] or [ZZC]. Dually, let $C$ be a coalgebra and let
$M$ be a $C$-bicomodule. Then the cotensor coalgebra $T_{C}^{c}(M)$ of $M$
over $C$ is a graded coalgebra with $T_{C}^{c}(M)_{(0)}=C$,
$T_{C}^{c}(M)_{(1)}=M$ and $T_{C}^{c}(M)_{(n)}=\Box^{n}_{C}M$ for $n>1$. That
is, $T_{C}^{c}(M)=C\oplus(\bigoplus_{n>0}\Box^{n}_{C}M)$ (see [N] or [ZZC] ).
If $B$ is a Hopf algebra and $M$ is a $B$-Hopf bimodule, then both $T_{B}(M)$
and $T_{B}^{c}(M)$ are graded Hopf algebras. Furthermore, subagebra generated
by $H$ and $M$ in $T_{B}^{c}(M)$, written as $H[M]$, is a Hopf subalgebra of
$T_{B}^{c}(M)$ and $H[M]$ is called a Hopf algebra of type one.

##  1 Classification of Quiver Hopf Algebras

######  Definition 1.1.

$(G,r,\overrightarrow{\rho},u)$ is called a ramification system with
irreducible representations (or RSR  in short ), if $r$ is a ramification of
$G$; $u$ is a map from ${\mathcal{K}}(G)$ to $G$ with $u(C)\in C$ for any
$C\in{\mathcal{K}}(G)$; $I_{C}(r,u)$ and $J_{C}(i)$ are sets with
$\mid\\!J_{C}(i)\\!\mid$ = ${\rm deg}(\rho_{C}^{(i)})$ and
$I_{C}(r)=\\{(i,j)\mid i\in I_{C}(r,u),j\in J_{C}(i)\\}$ for any
$C\in{\mathcal{K}}_{r}(G)$, $i\in I_{C}(r,u)$;
$\overrightarrow{\rho}=\\{\rho_{C}^{(i)}\\}_{i\in
I_{C}(r,u),C\in{\mathcal{K}}_{r}(G)}\
\in\prod_{C\in{\mathcal{K}}_{r}(G)}(\widehat{{Z_{u(C)}}})^{\mid
I_{C}(r,u)\mid}$ with $\rho_{C}^{(i)}\in\widehat{{Z_{u(C)}}}$ for any $i\in
I_{C}(r,u),C\in{\mathcal{K}}_{r}(G)$.

${\rm RSR}(G,r,\overrightarrow{\rho},u)$ and ${\rm
RSR}(G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})$ are
said to be isomorphic if the following conditions are satisfied:

$\bullet$ There exists a group isomorphism $\phi:G\rightarrow G^{\prime}$.

$\bullet$ For any $C\in{\mathcal{K}}(G)$, there exists an element $h_{C}\in G$
such that $\phi(h_{C}^{-1}u(C)h_{C})=u^{\prime}(\phi(C))$.

$\bullet$ For any $C\in{\mathcal{K}}_{r}(G)$, there exists a bijective map
$\phi_{C}:I_{C}(r,u)\rightarrow I_{\phi(C)}(r^{\prime},u^{\prime})$ such that
$\rho_{C}^{(i)}\cong\rho^{\prime}{}_{\phi(C)}^{(\phi_{C}(i))}\phi_{h_{c}}$ as
representations of $kZ_{u(C)}$ for all $i\in I_{C}(r,u)$, where
$\phi_{h_{C}}(h)=\phi(h^{-1}_{C}hh_{C})$ for any $h\in G$.

Remark. Assume that $G=G^{\prime}$, $r=r^{\prime}$, $u(C)=u^{\prime}(C)$ and
$I_{C}(r,u)=I_{C}(r^{\prime},u^{\prime})$ for any $C\in{\mathcal{K}}_{r}(G)$.
If there is a permutation $\phi_{C}$ on $I_{C}(r,u)$ for any
$C\in{\mathcal{K}}_{r}(G)$ such that
$\rho^{\prime}{{}_{C}^{(\phi_{C}(i))}}\cong\rho_{C}^{(i)}$ for all $i\in
I_{C}(r,u)$, then obviously ${\rm RSR}(G,r,\overrightarrow{\rho},u)\cong{\rm
RSR}(G,r,\overrightarrow{\rho^{\prime}},u)$.

######  Example 1.2.

Assume that $k$ is a complex field and $G=\mathbb{S}_{3}$, then there are 3
elements in ${\mathcal{K}}(G)$, which are $\\{(1)\\}$, $\\{(12),(13),(23)\\}$,
$\\{(123),(132)\\}$, and there are 3 non-isomorphic irreducible
representations, which are two 1 dimensional irreducible representations
$\epsilon$ and $\rm{sgn}$; one 2 dimensional irreducible representation
$\rho$. Obviously $Z_{u(\\{1\\})}=G$. The set $\\{{\rm
RSR}(G,r,\overrightarrow{\rho},u)$
$\mid\overrightarrow{\rho}=\rho,(sgn,sgn),(\epsilon,sgn),(sgn,\epsilon),(\epsilon,\epsilon)\\}$
is all ${\rm RSR}$’s with $r=r_{C}r$ and $C=\\{(1)\\}$. The set $\\{{\rm
RSR}(G,r,\overrightarrow{\rho},u)$
$\mid\overrightarrow{\rho}=\rho,(sgn,sgn),(\epsilon,sgn),(\epsilon,\epsilon)\\}$
is all representatives of isomorphic classes of ${\rm RSR}$’s with $r=r_{C}r$
and $C=\\{(1)\\}$. Furthermore, when $\overrightarrow{\rho}=\rho$, we can set
$I_{C}(r,u)=\\{1\\}$ and $J_{C}(1)=\\{1\\}$. In this case
$\rho_{C}^{(1)}=\rho$. When $\overrightarrow{\rho}=(\epsilon,sgn)$, we can set
$I_{C}(r,u)=\\{1,2\\}$ and $J_{C}(1)=J_{C}(2)=\\{1\\}$. In this case
$\rho_{C}^{(1)}=\epsilon$, $\rho_{C}^{(2)}=sgn$.

Let

$\displaystyle G$ $\displaystyle=$
$\displaystyle\bigcup_{\theta\in\Theta_{C}}Z_{u(C)}g_{\theta},$ (1.1)

where $\Theta_{C}$ is an index set, be a coset decomposition of $Z_{u(C)}$ in
$G$. It is easy to check that $|\Theta_{C}|=|C|$. We always assume that the
representative element of the coset $Z_{u(C)}$ is the identity $1$ of $G$. For
any $h\in G$ and $\theta\in\Theta_{C}$, there exist unique $h^{\prime}\in
Z_{u(C)}$ and $\theta^{\prime}\in\Theta_{C}$ such that
$g_{\theta}h=h^{\prime}g_{\theta^{\prime}}$. Let
$\zeta_{\theta}(h)=h^{\prime}$. Then we have

$\displaystyle g_{\theta}h$ $\displaystyle=$
$\displaystyle\zeta_{\theta}(h)g_{\theta^{\prime}}.$ (1.2)

######  Proposition 1.3.

If $N$ is a $kG$-Hopf bimodule, then there exist a Hopf quiver $(Q,G,r)$, an
${\rm RSR}(G,r,\overrightarrow{\rho},u)$ and a $kG$-Hopf bimodule
$(kQ_{1}^{c},\alpha^{-},\alpha^{+})$ with

$\displaystyle\alpha^{-}(h\otimes a^{(i,j)}_{y,x}):=h\cdot
a^{(i,j)}_{y,x}=a^{(i,j)}_{hy,hx},\ \ \alpha^{+}(a^{(i,j)}_{y,x}\otimes
h):=a^{(i,j)}_{y,x}\cdot h=\sum_{s\in
J_{C}(i)}k_{C,h}^{(i,j,s)}a^{(i,s)}_{yh,xh}$ (1.3)

for some $k_{C,h}^{(i,j,s)}\in k$ such that
$N\cong(kQ_{1}^{c},\alpha^{-},\alpha^{+})$ as $kG$-Hopf bimodules, where
$x,y,h\in G$ with $x^{-1}y=g^{-1}_{\theta}u(C)g_{\theta}$, $\zeta_{\theta}$ is
given by [ZZC, (0.3)], $C\in{\mathcal{K}}_{r}(G)$, $i\in I_{C}(r,u)$, $j\in
J_{C}(i)$, $x_{C}^{(i,j)}\cdot\zeta_{\theta}(h)=\sum_{s\in
J_{C}(i)}k_{C,h}^{(i,j,s)}x_{C}^{(i,s)}$.

Proof. Since $N$ is a $kG$-Hopf bimodule, there exists an object
$\prod_{C\in{\mathcal{K}}(G)}M(C)$ in
$\prod_{C\in{\mathcal{K}}(G)}{\mathcal{M}}_{kZ_{u(C)}}$ such that $M(C)$ is a
$kZ_{u(C)}$-module for any $C\in{\mathcal{K}}(G)$ and
$N\cong\bigoplus_{y=xg_{\theta}^{-1}u(C)g_{\theta},\ x,y\in G}\ x\otimes
M(C)\otimes_{kZ_{u(C)}}g_{\theta}$ as $kG$-Hopf bimodules by [CR97] or [ZZC,
Th. 1]. Let $r=\sum_{C\in{\mathcal{K}}(G)}r_{C}C$ with $r_{C}={\rm dim}M(C)$
for any $C\in{\mathcal{K}}(G)$. Notice that ${\rm dim}M(C)$ denotes the
cardinal number of a basis of $M(C)$ when $M(C)$ is infinite dimensional.
Since $M(C)$ is a $kZ_{u(C)}$-module and $kZ_{u(C)}$ is semisimple, there
exists a family of irreducible representations
$\\{(X_{C}^{(i)},\rho_{C}^{(i)})\mid i\in I_{C}(r,u)\\}$ such that
$M(C)=\bigoplus_{i\in I_{C}(r,u)}(X_{C}^{(i)},\rho_{C}^{(i)})$. Let
$\\{{x_{C}^{(i,j)}\mid j\in J_{C}(i)}\\}$ be a k-basis of $X_{C}^{(i)}$ for
any $i\in I_{C}(r,u)$. Then for any $h\in G$ there are some
$k_{C,h}^{(i,j,s)}\in k$ such that
$x_{C}^{(i,j)}\cdot\zeta_{\theta}(h)=\sum_{s\in
J_{C}(i)}k_{C,h}^{(i,j,s)}x_{C}^{(i,s)}$ for all $i\in I_{C}(r,u)$ and $j\in
J_{C}(i)$ since $x_{C}^{(i,j)}\cdot\zeta_{\theta}(h)\in X_{C}^{(i)}$.

It remains to show that $(kQ_{1}^{c},\alpha^{-},\alpha^{+})$ is isomorphic to
$\bigoplus_{y=xg_{\theta}^{-1}u(C)g_{\theta},\ x,y\in G}\ x\otimes
M(C)\otimes_{kZ_{u(C)}}g_{\theta}$ as $kG$-Hopf bimodules. Observe that there
is a canonical $kG$-bicomodule isomorphism
$\varphi:kQ_{1}\rightarrow\bigoplus_{y=xg_{\theta}^{-1}u(C)g_{\theta},\ x,y\in
G}\ x\otimes M(C)\otimes_{kZ_{u(C)}}g_{\theta}$ given by

$\displaystyle\varphi(a^{(i,j)}_{y,x})=x\otimes
x_{C}^{(i,j)}\otimes_{kZ_{u(C)}}g_{\theta}$ (1.4)

where $x,y\in G$ with $x^{-1}y=g^{-1}_{\theta}u(C)g_{\theta}$,
$C\in{\mathcal{K}}_{r}(G)$ , $i\in I_{C}(r,u)$ and $j\in J_{C}(r)$. See

$\displaystyle\varphi(\alpha^{-}(h\otimes a^{(i,j)}_{y,x}))$ $\displaystyle=$
$\displaystyle\varphi(a^{(i,j)}_{hy,hx})=hx\otimes
x^{(i,j)}_{C}\otimes_{kZ_{u(C)}}g_{\theta}\ \ $ $\displaystyle=$
$\displaystyle h\cdot(x\otimes x^{(i,j)}_{C}\otimes_{kZ_{u(C)}}g_{\theta})\ \
\ (\hbox{see }\cite[cite]{[\@@bibref{}{ZZC04}{}{}, (1.2)]})$ $\displaystyle=$
$\displaystyle h\cdot\varphi(a^{(i,j)}_{y,x})\ \ .$

Thus $\varphi$ is a left $kG$-module isomorphism. See

$\displaystyle\alpha^{+}(\varphi(a^{(i,j)}_{y,x})\otimes h)$ $\displaystyle=$
$\displaystyle xh\otimes x_{C}^{(i,j)}\cdot\zeta_{\theta}(h)\otimes
g_{\theta^{\prime}}$ $\displaystyle=$ $\displaystyle xh\otimes(\sum_{s\in
J_{C}(i)}{k_{C,h}^{(i,j,s)}}x_{C}^{(i,s)})\otimes g_{\theta^{\prime}})$
$\displaystyle=$ $\displaystyle\varphi(\sum_{s\in
J_{C}(i)}{k_{C,h}^{(i,j,s)}}a_{yh,xh}^{(i,s)})$ $\displaystyle=$
$\displaystyle\varphi(\alpha^{+}(a_{y,x}^{(i,j)}\otimes h))\ \ (\hbox{by
}(\ref{e1.10.11})).$

Consequently, $\varphi$ is a $kG$-Hopf bimodule isomorphism. $\Box$

Let $(kQ_{1}^{c},G,r,\overrightarrow{\rho},u)$ denote the $kG$-Hopf bimodule
$(kQ_{1}^{c},\alpha^{-},\alpha^{+})$ given in Proposition 1.3. Furthermore, if
$(kQ_{1}^{c},kQ_{1}^{a})$ is an arrow dual pairing, i.e. $kQ_{1}^{c}$ is
isomorphic to the dual of $kQ_{1}^{a}$ as $kG$-Hopf bimodules or $kQ_{1}^{a}$
is isomorphic to the dual of $kQ_{1}^{c}$ as $(kG)^{*}$-Hopf bimodules under
the isomorphisms in [ZZC, Lemma 1.7] (cf [ZZC, the argument before Def. 1.8]),
then we denote the $(kG)^{*}$-Hopf bimodule $kQ_{1}^{a}$ by
$(kQ_{1}^{a},G,r,\overrightarrow{\rho},u)$. We obtain six quiver Hopf algebras
$kQ^{c}(G,r,\overrightarrow{\rho},u)$, $kQ^{s}(G,r,\overrightarrow{\rho},u)$,
$kG[kQ_{1}^{c},G,r,\overrightarrow{\rho},u]$,
$kQ^{a}(G,r,\overrightarrow{\rho},u)$, $kQ^{sc}(G,r,\overrightarrow{\rho},u),$
$(kG)^{*}[kQ_{1}^{a},G,r,\overrightarrow{\rho},$ $u]$, called the quiver Hopf
algebras determined by ${\rm RSR}(G,$ $r,$ $\overrightarrow{\rho},u)$.

From Proposition 1.3, it seems that the right $kG$-action on
$(kQ_{1}^{c},G,r,\overrightarrow{\rho},u)$ depends on the choice of the set
$\\{g_{\theta}\mid\theta\in\Theta_{C}\\}$ of coset representatives of
$Z_{u(C)}$ in $G$ (see, (1.1) or [ZZC, (0.1)]). The following lemma shows that
$(kQ_{1}^{c},G,r,\overrightarrow{\rho},u)$ is, in fact, independent of the
choice of the coset representative set
$\\{g_{\theta}\mid\theta\in\Theta_{C}\\}$, up to $kG$\- Hopf bimodule
isomorphisms. For a while, we write
$(kQ_{1}^{c},G,r,\overrightarrow{\rho},u)=(kQ_{1}^{c},G,r,\overrightarrow{\rho},u,\\{g_{\theta}\\})$
given before. Now let $\\{h_{\theta}\in G\mid\theta\in\Theta_{C}\\}$ be
another coset representative set of $Z_{u(C)}$ in $G$ for any
$C\in{\mathcal{K}}(G)$. That is,

$\displaystyle G=\bigcup_{\theta\in\Theta_{C}}Z_{u(C)}h_{\theta}.$ (1.5)

######  Lemma 1.4.

With the above notations,
$(kQ_{1}^{c},G,r,\overrightarrow{\rho},u,\\{g_{\theta}\\})$ and
$(kQ_{1}^{c},G,r,\overrightarrow{\rho},u,$ $\\{h_{\theta}\\})$ are isomorphic
$kG$-Hopf bimodules.

Proof. We may assume $Z_{u(C)}h_{\theta}=Z_{u(C)}g_{\theta}$ for any
$C\in{\mathcal{K}}(G)$ and $\theta\in\Theta_{C}$. Then
$g_{\theta}h^{-1}_{\theta}\in Z_{u(C)}$. Now let $x,y,h\in G$ with
$x^{-1}y=g^{-1}_{\theta}u(C)g_{\theta}$. Then $x^{-1}y=$
$h^{-1}_{\theta}(g_{\theta}h^{-1}_{\theta})^{-1}u(C)(g_{\theta}h^{-1}_{\theta})h_{\theta}$
$=h^{-1}_{\theta}u(C)h_{\theta}$ and

$\displaystyle
h_{\theta}h=(h_{\theta}g^{-1}_{\theta})g_{\theta}h=(h_{\theta}g^{-1}_{\theta})\zeta_{\theta}(h)g_{\theta^{\prime}}=(h_{\theta}g^{-1}_{\theta})\zeta_{\theta}(h)(g_{\theta^{\prime}}h^{-1}_{\theta^{\prime}})h_{\theta^{\prime}}\
,$ (1.6)

where $g_{\theta}h=\zeta_{\theta}(h)g_{\theta^{\prime}}$.

For any $C\in{\mathcal{K}}(G)$, let $M(C)$ be a right $kZ_{u(C)}$-module. Let
$N:=\bigoplus_{y=xg_{\theta}^{-1}u(C)g_{\theta},\ x,y\in G}\ x\otimes
M(C)\otimes_{kZ_{u(C)}}g_{\theta}$ and
$M:=\bigoplus_{y=xh_{\theta}^{-1}u(C)h_{\theta},\ x,y\in G}\ x\otimes
M(C)\otimes_{kZ_{u(C)}}h_{\theta}$ be two $kG$-Hopf bimodules. It is
sufficient to show $N\cong M$ as $kG$-Hopf bimodules by the proof of
Proposition 1.3.

Considering $x\otimes w\otimes_{kZ_{u(C)}}g_{\theta}=x\otimes w\cdot
g_{\theta}h_{\theta}^{-1}\otimes_{kZ_{u(C)}}h_{\theta}$, we have that
$f:N\rightarrow M$ given by

$f(x\otimes w\otimes_{kZ_{u(C)}}g_{\theta})=x\otimes w\cdot
g_{\theta}h_{\theta}^{-1}\otimes_{kZ_{u(C)}}h_{\theta}$

for any $w\in M(C)$, any $x,y\in G$ with
$x^{-1}y=g^{-1}_{\theta}u(C)g_{\theta}$, $C\in{\mathcal{K}}_{r}(G)$ and $i\in
I_{C}(r,u)$, is a $k$-linear isomorphism. It is clear that $f$ is a
$kG$-bicomodule isomorphism and a left $kG$-module isomorphism from $N$ to
$M$. See

$\displaystyle(f(x\otimes x_{C}^{(i,j)}\otimes_{kZ_{u(C)}}g_{\theta}))\cdot h$
$\displaystyle=$
$\displaystyle(x\otimes(x_{C}^{(i,j)})\rho_{C}^{(i)}(g_{\theta}h_{\theta}^{-1})\otimes_{kZ_{u(C)}}h_{\theta})\cdot
h$ $\displaystyle=$ $\displaystyle
xh\otimes(x_{C}^{(i,j)})\rho_{C}^{(i)}(\zeta_{\theta}(h))\rho_{C}^{(i)}(g_{\theta^{\prime}}h_{\theta^{\prime}}^{-1})\otimes_{kZ_{u(C)}}h_{\theta^{\prime}}\
\ (\hbox{by (\ref{e1.131})})$ $\displaystyle=$ $\displaystyle f((x\otimes
x_{C}^{(i,j)}\otimes_{kZ_{u(C)}}g_{\theta})\cdot h),$

for any $x,y,h\in G$, $i\in I_{C}(r,u)$, $j\in J_{C}(i)$,
$C\in{\mathcal{K}}_{r}(G)$ with $x^{-1}y=g^{-1}_{\theta}u(C)g_{\theta}$. Thus
$f$ is a right $kG$-module homomorphism. $\Box$

Now we state one of our main result, which classifies quiver Hopf algebras.

######  Theorem 2.

Let $(G,r,\overrightarrow{\rho},u)$ and
$(G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})$ be two
RSR’s. Then the following statements are equivalent:

(i) ${\rm RSR}(G,r,\overrightarrow{\rho},u)$ $\cong$ ${\rm
RSR}(G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})$.

(ii) There exists a Hopf algebra isomorphism $\phi:kG\rightarrow kG^{\prime}$
such that $(kQ_{1}^{c},G,r,\overrightarrow{\rho},u)\cong\
_{\phi}^{\phi^{-1}}((kQ_{1}^{\prime}{}^{c},G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})){}_{\phi}^{\phi^{-1}}$
as $kG$-Hopf bimodules.

(iii) $kQ^{c}(G,r,\overrightarrow{\rho},u)\cong
k{Q^{\prime}}^{c}(G,r,\overrightarrow{\rho},u)$.

(iv) $kQ^{s}(G,r,\overrightarrow{\rho},u)\cong
k{Q^{\prime}}^{s}(G^{\prime},r^{\prime},\overrightarrow{\rho}^{\prime},u^{\prime})$.

(v) $kG[kQ_{1}^{c},G,r,\overrightarrow{\rho},u]$ $\cong
kG^{\prime}[kQ_{1}^{\prime}{}^{c},G^{\prime},r^{\prime},\overrightarrow{\rho}^{\prime},u^{\prime}]$.
Furthermore, if $Q$ is finite, then the above are equivalent to the following:

(vi) $kQ^{a}(G,r,\overrightarrow{\rho},u)$ $\cong
k{Q^{\prime}}^{a}(G^{\prime},r^{\prime},\overrightarrow{\rho}^{\prime},u^{\prime})$.

(vii) $kQ^{sc}(G,r,\overrightarrow{\rho},u)\cong
k{Q^{\prime}}^{sc}(G^{\prime},r^{\prime},\overrightarrow{\rho}^{\prime},u^{\prime})$.

(viii)
$(kG)^{*}[kQ_{1}^{a},G,r,\overrightarrow{\rho},u]\cong(kG^{\prime})^{*}[kQ_{1}^{\prime}{}^{a},G^{\prime},r^{\prime},\overrightarrow{\rho}^{\prime},u^{\prime}]$.
Notice that the isomorphisms above are ones of graded Hopf algebras but (i)
(ii).

Proof. By [ZZC, Lemma 1.5, Lemma 1.6], we only have to prove (i)
$\Leftrightarrow$ (ii).

(i) $\Rightarrow$ (ii). Assume that ${\rm RSR}(G,r,\overrightarrow{\rho},u)$
$\cong$ ${\rm
RSR}(G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})$. Let
$(X_{C}^{(i)},\rho_{C}^{(i)})$ and
$(X^{\prime}{}_{C^{\prime}}^{(i^{\prime})},\rho^{\prime}{}_{C^{\prime}}^{(i^{\prime})})$
be irreducible representations over $Z_{u(C)}$ and
$Z_{u^{\prime}(C^{\prime})}$, respectively. Then there exist a group
isomorphism $\phi:G\rightarrow G^{\prime}$, an element $h_{C}\in G$ such that
$\phi(h^{-1}_{C}u(C)h_{C})=u^{\prime}(\phi(C))$ for any $C\in{\mathcal{K}}(G)$
and a bijective map $\phi_{C}:I_{C}(r,u)\rightarrow
I_{\phi(C)}(r^{\prime},u^{\prime})$ such that
$(X^{(i)}_{C},\rho_{C}^{(i)})\stackrel{{\scriptstyle\xi_{C}^{(i)}}}{{\cong}}(X^{\prime}{}_{\phi(C)}^{(\phi_{C}(i))},\rho^{\prime}{}_{\phi(C)}^{(\phi_{C}(i))}\phi_{h_{c}})$
as right $kZ_{u(C)}$-modules for all $i\in I_{C}(r,u)$.

Now let $G=\bigcup_{\theta\in\Theta_{C}}Z_{u(C)}g_{\theta}$ as in (1.1) or
[ZZC, (0.1)] for any $C\in{\mathcal{K}}(G)$. Let $x,y,h\in G$ with
$x^{-1}y=g^{-1}_{\theta}u(C)g_{\theta}$, $C\in{\mathcal{K}}_{r}(G)$ and
$\theta\in\Theta_{C}$. Assume that
$g_{\theta}h^{-1}_{C}=\zeta_{\theta}(h^{-1}_{C})g_{\eta}$,
$g_{\theta}h=\zeta_{\theta}(h)g_{\theta^{\prime}}$,
$g_{\eta}(h_{C}hh^{-1}_{C})=\zeta_{\eta}(h_{C}hh^{-1}_{C})g_{\eta^{\prime}}$
and
$g_{\theta^{\prime}}h^{-1}_{C}=\zeta_{\theta^{\prime}}(h^{-1}_{C})g_{\theta^{\prime\prime}}$
with $\zeta_{\theta}(h^{-1}_{C}),$ $\zeta_{\theta}(h),$
$\zeta_{\eta}(h_{C}hh^{-1}_{C}),$ $\zeta_{\theta^{\prime}}(h^{-1}_{C})\in
Z_{u(C)}$ and
$\eta,\theta^{\prime},\eta^{\prime},\theta^{\prime\prime}\in\Theta_{C}$. Then
we have

$\displaystyle g_{\theta}hh^{-1}_{C}$ $\displaystyle=$
$\displaystyle\zeta_{\theta}(h)g_{\theta^{\prime}}h^{-1}_{C}\
=\zeta_{\theta}(h)\zeta_{\theta^{\prime}}(h^{-1}_{C})g_{\theta^{\prime\prime}}$
(1.7)

and

$\displaystyle g_{\theta}hh^{-1}_{C}$ $\displaystyle=$
$\displaystyle(g_{\theta}h^{-1}_{C})(h_{C}hh^{-1}_{C})=\zeta_{\theta}(h^{-1}_{C})g_{\eta}(h_{C}hh^{-1}_{C})\
=\zeta_{\theta}(h^{-1}_{C})\zeta_{\eta}(h_{C}hh^{-1}_{C})g_{\eta^{\prime}}.$
(1.8)

It follows that

$\displaystyle\theta^{\prime\prime}=\eta^{\prime}\ \ \ \hbox{and }\ \ \
\zeta_{\theta}(h)\zeta_{\theta^{\prime}}(h^{-1}_{C})=\zeta_{\theta}(h^{-1}_{C})\zeta_{\eta}(h_{C}hh^{-1}_{C}).$
(1.9)

Moreover, we have
$(xh)^{-1}(yh)=h^{-1}g^{-1}_{\theta}u(C)g_{\theta}h=g^{-1}_{\theta^{\prime}}u(C)g_{\theta^{\prime}}$
and
$g_{\theta}=g_{\theta}h^{-1}_{C}h_{C}=\zeta_{\theta}(h^{-1}_{C})g_{\eta}h_{C}$.
Thus

$\begin{array}[]{rcl}\phi(x)^{-1}\phi(y)&=&\phi(x^{-1}y)=\phi(g^{-1}_{\theta}u(C)g_{\theta})=\phi(h^{-1}_{C}g^{-1}_{\eta}u(C)g_{\eta}h_{C})\\\
&=&\phi(h^{-1}_{C}g^{-1}_{\eta}h_{C})\phi(h^{-1}_{C}u(C)h_{C})\phi(h^{-1}_{C}g_{\eta}h_{C})\\\
&=&\phi(h^{-1}_{C}g_{\eta}h_{C})^{-1}u^{\prime}(\phi(C))\phi(h^{-1}_{C}g_{\eta}h_{C}).\end{array}$

We also have

$\displaystyle\phi(h^{-1}_{C}g_{\eta}h_{C})\phi(h)$ $\displaystyle=$
$\displaystyle\phi(h^{-1}_{C}g_{\eta}h_{C}hh^{-1}_{C}h_{C})$ (1.10)
$\displaystyle=$
$\displaystyle\phi(h^{-1}_{C}\zeta_{\eta}(h_{C}hh^{-1}_{C})g_{\eta^{\prime}}h_{C})$
$\displaystyle=$
$\displaystyle\phi(h^{-1}_{C}\zeta_{\eta}(h_{C}hh^{-1}_{C})h_{C})\phi(h^{-1}_{C}g_{\eta^{\prime}}h_{C}).$

Since

$\begin{array}[]{rcl}\phi_{h_{C}}(g_{\eta^{\prime}})\phi(xh)^{-1}\phi(yh)&=&\phi(h_{C}^{-1}(\zeta_{\eta}(h_{C}hh_{C}^{-1}))^{-1}(\zeta_{\theta}(h_{C}^{-1}))^{-1}u(C)g_{\theta}h)\
\ \ (\hbox{by }\ref{e1.101})\\\
&=&u^{\prime}(\phi(C))\phi_{h_{C}}(g_{\eta^{\prime}})\ \ \ (\hbox{by
}\ref{e1.101}),\\\ \end{array}$ $\displaystyle\phi(xh)^{-1}\phi(yh)$
$\displaystyle=$
$\displaystyle\phi_{h_{C}}(g_{\eta^{\prime}})^{-1}u^{\prime}(\phi(C))\phi_{h_{C}}(g_{\eta^{\prime}}).$
(1.11)

It is clear

$\displaystyle
G^{\prime}=\bigcup_{\theta\in\Theta_{C}}Z_{u^{\prime}(\phi(C))}(\phi(h^{-1}_{C}g_{\theta}h_{C}))$
(1.12)

is a coset decomposition of $Z_{u^{\prime}(\phi(C))}$ in $G^{\prime}$ for any
$\phi(C)\in{\mathcal{K}}(G^{\prime})$.

Let

$\displaystyle N:$
$\displaystyle=\bigoplus_{y=xg_{\theta}^{-1}u(C)g_{\theta},\ x,y\in G}\
x\otimes N(C)\otimes_{kZ_{u(C)}}g_{\theta}\ \ \ \ \ \hbox{ and }$
$\displaystyle M:$
$\displaystyle=\bigoplus_{\phi(y)=\phi(x)\phi_{h_{C}}(g_{\eta}^{-1})u^{\prime}(\phi(C)\phi_{h_{C}}(g_{\eta}),\
x,y\in G}\ \phi(x)\otimes
M(\phi(C))\otimes_{kZ_{u^{\prime}(\phi(C))}}\phi_{h_{C}}(g_{\eta})$

with $N(C):=\bigoplus_{i\in I_{C}(r,u)}X_{C}^{(i)}$ and
$M(\phi(C)):=\bigoplus_{i\in
I_{C}(r,u)}X^{\prime}{}_{\phi(C)}^{(\phi_{C}(i))}$. It is suffices to show
$N\cong\ ^{\phi^{-1}}_{\phi}M^{\phi^{-1}}_{\phi}$ as $kG$-Hopf bimodules by
the proof of Proposition 1.3.

Considering
$x\otimes(w)\rho^{(i)}_{C}((\zeta_{\theta}(h_{C}^{-1}))^{-1})\otimes_{kZ_{u(C)}}g_{\theta}=x\otimes
w\otimes_{kZ_{u(C)}}g_{\eta}h_{c}$, we have that $\psi:N\rightarrow M$ given
by

$\psi(x\otimes(w)\rho^{(i)}_{C}((\zeta_{\theta}(h_{C}^{-1}))^{-1})\otimes_{kZ_{u(C)}}g_{\theta})=\phi(x)\otimes\xi_{C}^{(i)}(w)\otimes_{kZ_{u^{\prime}(\phi(C))}}\phi_{h_{C}}(g_{\eta})$

for any $x,y\in G$ with $x^{-1}y=g^{-1}_{\theta}u(C)g_{\theta}$, and $i\in
I_{C}(r,u)$, $w\in X_{C}^{(i)}$, where $C\in{\mathcal{K}}_{r}(G)$ and
$g_{\theta}h^{-1}_{C}=\zeta_{\theta}(h^{-1}_{C})g_{\eta}$ with
$\zeta_{\theta}(h^{-1}_{C})\in Z_{u(C)}$ and $\theta,\eta\in\Theta_{C}$, is a
$k$-linear isomorphism. It is clear that $\psi$ is a homomorphism not only of
$kG$-bicomodules from $N$ to $\ {}^{\phi^{-1}}M^{\phi^{-1}}$ but also of left
$kG$-modules from $N$ to $\ {}_{\phi}M$.

For any $h\in G$ and $w\in X_{C}^{(i)}$, see

$\displaystyle\psi((x\otimes(w)\rho^{(i)}_{C}((\zeta_{\theta}(h_{C}^{-1}))^{-1})\otimes_{kZ_{u(C)}}g_{\theta})\cdot
h)$
$\displaystyle=\psi((xh\otimes(w)\rho^{(i)}_{C}((\zeta_{\theta}(h_{C}^{-1}))^{-1}\zeta_{\theta}(h))\otimes_{kZ_{u(C)}}g_{\theta^{\prime}})$
$\displaystyle=\phi(xh)\otimes\xi_{C}^{(i)}((w)\rho^{(i)}_{C}((\zeta_{\theta}(h_{C}^{-1}))^{-1}\zeta_{\theta}(h)\zeta_{\theta^{\prime}}(h_{C}^{-1})))\otimes_{kZ_{u^{\prime}(\phi(C))}}\phi_{h_{C}}(g_{\eta^{\prime}})\
\ \ (\hbox{ by (\ref{e1.102})})$
$\displaystyle=\phi(x)\phi(h)\otimes(\xi_{C}^{(i)}(w))\rho^{(\phi_{C}(i))}_{\phi(C)}\phi_{h_{C}}((\zeta_{\theta}(h_{C}^{-1}))^{-1}\zeta_{\theta}(h)\zeta_{\theta^{\prime}}(h_{C}^{-1})))\otimes_{kZ_{u^{\prime}(\phi(C))}}\phi_{h_{C}}(g_{\eta^{\prime}})$
$\displaystyle\ \ (\hbox{ by Def. \ref{1.1}})$

and

$\displaystyle\psi(x\otimes(w)\rho^{(i)}_{C}((\zeta_{\theta}(h_{C}^{-1}))^{-1})\otimes_{kZ_{u(C)}}g_{\theta})\cdot\phi(h)$
$\displaystyle=(\phi(x)\otimes\xi_{C}^{(i)}(w)\otimes_{kZ_{u^{\prime}(\phi(C))}}\phi_{h_{C}}(g_{\eta}))\cdot\phi(h)$
$\displaystyle=\phi(x)\phi(h)\otimes(\xi_{C}^{(i)}(w))\rho^{(\phi_{C}(i))}_{\phi(C)}\phi_{h_{C}}(\zeta_{\eta}(h_{C}hh_{C}^{-1}))\otimes_{kZ_{u^{\prime}(\phi(C))}}\phi_{h_{C}}(g_{\eta^{\prime}})\
\ \ (\hbox{ by (\ref{e1.1311})})$
$\displaystyle=\phi(x)\phi(h)\otimes(\xi_{C}^{(i)}(w))\rho^{(\phi_{C}(i))}_{\phi(C)}\phi_{h_{C}}((\zeta_{\theta}(h_{C}^{-1}))^{-1}\zeta_{\theta}(h)\zeta_{\theta^{\prime}}(h_{C}^{-1})))\otimes_{kZ_{u^{\prime}(\phi(C))}}\phi_{h_{C}}(g_{\eta^{\prime}})$
$\displaystyle\ \ \ (\hbox{ by (\ref{coeff})}),$

which show that $\psi$ is a right $kG$-module homomorphism.

(ii) $\Rightarrow$ (i). Assume that there exist a Hopf algebra isomorphism
$\phi:kG\rightarrow kG^{\prime}$ and a $kG$-Hopf bimodule isomorphism
$\psi:(kQ_{1}^{c},G,r,\overrightarrow{\rho},u)\rightarrow\
^{\phi^{-1}}_{\phi}(kQ_{1}^{\prime}{}^{c},G^{\prime},r^{\prime},\overrightarrow{\rho}^{\prime},u^{\prime})^{\phi^{-1}}_{\phi}$.
Then $\phi:G\rightarrow G^{\prime}$ is a group isomorphism. Let
$C\in{\mathcal{K}}(G)$. Then $\phi(u(C))$,
$u^{\prime}(\phi(C))\in\phi(C)\in{\mathcal{K}}(G^{\prime})$, and hence
$u^{\prime}(\phi(C))=\phi(h_{C})^{-1}\phi(u(C))\phi(h_{C})=\phi(h^{-1}_{C}u(C)h_{C})$
for some $h_{C}\in G$. Since $\psi$ is a $kG^{\prime}$-bicomodule isomorphism
from ${}^{\phi}(kQ_{1}^{c},G,r,\overrightarrow{\rho},u)^{\phi}$ to
$(kQ_{1}^{\prime}{}^{c},G^{\prime},r^{\prime},\overrightarrow{\rho}^{\prime},u^{\prime})$
and $\phi(h^{-1}_{C}u(C)h_{C})=u^{\prime}(\phi(C))$, by restriction one gets a
$k$-linear isomorphism

$\psi_{C}:\ ^{h^{-1}_{C}u(C)h_{C}}(kQ_{1})^{1}\rightarrow\
^{u^{\prime}(\phi(C))}\\!(kQ^{\prime}_{1})^{1},\ x\mapsto\psi(x).$

We also have a $k$-linear isomorphism

$f_{C}:\ ^{u(C)}\\!(kQ_{1})^{1}\rightarrow\
^{h^{-1}_{C}u(C)h_{C}}(kQ_{1})^{1},\ x\mapsto h^{-1}_{C}\cdot x\cdot h_{C}.$

Since $\phi(h^{-1}_{C}u(C)h_{C})=u^{\prime}(\phi(C))$ and
$h^{-1}_{C}Z_{u(C)}h_{C}=Z_{h^{-1}_{C}u(C)h_{C}}$, one gets
$\phi(h^{-1}_{C}Z_{u(C)}h_{C})=Z_{u^{\prime}(\phi(C))}$. Hence $\phi_{h_{C}}$
is an algebra isomorphism from $kZ_{u(C)}$ to $kZ_{u^{\prime}(\phi(C))}$ by
sending $h$ to $\phi(h^{-1}_{C}hh_{C})$. Using the hypothesis that $\psi$ is a
$kG$-bimodules homomorphism from $(kQ_{1}^{c},G,r,\overrightarrow{\rho},u)$ to
${}_{\phi}(kQ_{1}^{\prime}{}^{c},$
$G^{\prime},r^{\prime},\overrightarrow{\rho}^{\prime},u^{\prime})_{\phi}$, one
can easily check that the composition $\psi_{C}f_{C}$ is a right
$kZ_{u(C)}$-module isomorphism from $(^{u(C)}\\!(kQ_{1})^{1},\lhd)$ to
$((^{u^{\prime}(\phi(C))}\\!(kQ^{\prime}_{1})^{1})_{\phi_{h_{C}}},\lhd)$.
Indeed, For any $z\in^{u(C)}\\!(kQ_{1})^{1}$, see

$\displaystyle\psi_{C}f_{C}(z)\lhd\phi_{h_{C}}(h)$
$\displaystyle=\psi_{C}(z\lhd h_{C})\lhd\phi_{h_{C}}(h)$
$\displaystyle=\psi_{C}(z\lhd h_{C}\phi_{h_{C}}(h))\ \ \ (\hbox{since
}\psi\hbox{ is a bimodule homomorphism})$ $\displaystyle=\psi_{C}f_{C}(z\lhd
h).$

Obviously, both ${}^{u(C)}\\!(kQ_{1})^{1}$ and
$(^{u^{\prime}(\phi(C))}\\!(kQ^{\prime}_{1})^{1})_{\phi_{h_{C}}}$ are
semisimple right $kZ_{u(C)}$-modules. Assume
${}^{u(C)}\\!(kQ_{1})^{1}\cong\oplus_{i\in
I_{C}(r,u)}(X_{C}^{(i)},\rho_{C}^{(i)})$ as right $kZ_{u(C)}$-modules and
${}^{u^{\prime}(\phi(C))}\\!(kQ^{\prime}_{1})^{1}\cong\oplus_{j\in
I_{\phi(C)}(r^{\prime},u)}(X^{\prime}{}_{\phi(C)}^{(j)},\rho^{\prime}{}_{\phi(C)}^{(j)})$
as right $kZ_{u^{\prime}(\phi(C))}$-modules, where
$(X_{C}^{(i)},\rho_{C}^{(i)})$ is an irreducible right $kZ_{u(C)}$-module for
any $i\in I_{C}(r,u)$ and
$(X^{\prime}{}_{\phi(C)}^{(j)},\rho^{\prime}{}_{\phi(C)}^{(j)})$ is an
irreducible right $kZ_{u^{\prime}(\phi(C))}$-module for any $j\in
I_{\phi(C)}(r^{\prime},u^{\prime})$. Therefore, there exists a bijective map
$\phi_{C}:I_{C}(r,u)\rightarrow I_{\phi(C)}(r^{\prime},u^{\prime})$ such that
$(X_{C}^{(i)},\rho_{C}^{(i)})\cong(X^{\prime}{}_{\phi(C)}^{(\phi_{C}(i))},\rho_{\phi(C)}^{(\phi_{C}(i))}\phi_{h_{C}})$
as right $kZ_{u(C)}-modules$ for all $i\in I_{C}(r,u)$. It follows that ${\rm
RSR}(G,r,\overrightarrow{\rho},u)$ $\cong$ ${\rm
RSR}(G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},$ $u^{\prime}).$
$\Box$

Up to now we have classified the quiver Hopf algebras by means of RSR’s. In
other words, ramification systems with irreducible representations uniquely
determine their corresponding quiver Hopf algebras up to graded Hopf algebra
isomorphisms.

######  Proposition 1.5.

Let ${\rm RSR}(G,r,\overrightarrow{\rho},u)$ and ${\rm
RSR}(G,r,\overrightarrow{\rho^{\prime}},u^{\prime})$ be two RSR’s. If
$u^{\prime}(C)=h_{C}^{-1}u(C)h_{C}$ and
$\rho_{C}{}^{(i)}=\rho^{\prime}{}_{C}^{(i)}{\rm ad}^{+}_{h_{C}}$ with
$I_{C}(r,u)=I_{C}(r,u^{\prime})$ for any $C\in{\mathcal{K}}_{r}(G)$, $i\in
I_{C}(r,u)$, where $h_{C}\in G$ and ${\rm
ad}^{+}_{h_{C}}(g)=h_{C}^{-1}gh_{C}$, then ${\rm
RSR}(G,r,\overrightarrow{\rho},u)$ $\cong$ ${\rm
RSR}(G,r,\overrightarrow{\rho^{\prime}},u^{\prime})$.

Proof. Let $\phi=id_{G}$ and $\phi_{C}=id_{I_{C}(r,u)}$ for any
$C\in{\mathcal{K}}_{r}(G).$ It is clear that ${\rm
RSR}(G,r,\overrightarrow{\rho},u)$ $\cong$ ${\rm
RSR}(G,r,\overrightarrow{\rho^{\prime}},u^{\prime})$. $\Box$

Remark: This proposition means that the choice of map $u$ doesn’t affect the
classification of ${\rm RSR}^{\prime}$s. That is, if we fix a map $u_{0}$ from
${\mathcal{K}}(G)$ to $G$ with $u_{0}(C)\in C$ for any $C\in{\mathcal{K}}(G)$,
then for any ${\rm RSR}(G,r,\overrightarrow{\rho},u)$, there exists ${\rm
RSR}(G,r,\overrightarrow{\rho^{\prime}},u_{0})$ such that ${\rm
RSR}(G,r,\overrightarrow{\rho},u)$ $\cong$ ${\rm
RSR}(G,r,\overrightarrow{\rho^{\prime}},u_{0})$.

##  2 The classification of pointed Hopf algebras of type one

A graded Hopf algebra $A=\oplus_{n=0}^{\infty}A_{(n)}$ is called to be of
Nichols type, if diagram of $A$ is a Nichols algebra over $A_{(0)}$ (the
definition of diagram was in [AS98] and [ZZC, Subsection 3.1]). Furthermore,
if the coradical of $A$ is a group algebra, then $A$ is called a pointed Hopf
algebra of Nichols type.

For an ${\rm RSR}(G,r,\overrightarrow{\rho},u)$ and a $kG$-Hopf bimodule
$(kQ_{1}^{c},G,r,\overrightarrow{\rho},u)$ with the module operations
$\alpha^{-}$ and $\alpha^{+}$, define a new left $kG$-action on $kQ_{1}$ by

$g\rhd x:=g\cdot x\cdot g^{-1},\ g\in G,x\in kQ_{1},$

where $g\cdot x=\alpha^{-}(g\otimes x)$ and $x\cdot g=\alpha^{+}(x\otimes g)$
for any $g\in G$ and $x\in kQ_{1}$. With this left $kG$-action and the
original left (arrow) $kG$-coaction $\delta^{-}$, $kQ_{1}$ is a Yetter-
Drinfeld $kG$-module. Let $Q_{1}^{1}:=\\{a\in Q_{1}\mid s(a)=1\\}$, the set of
all arrows with starting vertex $1$. It is clear that $kQ_{1}^{1}$ is a
Yetter-Drinfeld $kG$-submodule of $kQ_{1}$, denoted by
$(kQ_{1}^{1},ad(G,r,\overrightarrow{\rho},u))$.

######  Lemma 2.1.

(i) If $H$ is a Hopf algebra with bijective antipode and
$(B,\alpha_{B}^{-},\delta_{B}^{-})$ is a graded braided Hopf algebra in
${}^{H}_{H}{\mathcal{Y}D}$ with $B_{(0)}=k1_{B}$, then ${\rm
diag}(B\\#H)=B\\#1_{H}\cong B$ as graded braided Hopf algebras in
${}^{H}_{H}{\mathcal{Y}D}$.

(ii) $A$ is a pointed Hopf algebra of Nichols type if and only if $A$ is
isomorphic to biproduct ${\mathcal{B}}(V)\\#kG$ with Nichols algebra
${\mathcal{B}}(V)$ over group agebra $kG$, $A_{(0)}=kG$ and $V=A_{(1)}$.

Proof. (i) Obviously, ${\rm diag}(B\\#H)=B\otimes 1_{H}$. Define a map $\psi$
from ${\rm diag}(B\\#H)$ to $B$ by sending $x\otimes 1_{H}$ to $x$ for any
$x\in B$. It is easy to check that $\psi$ is a graded braided Hopf algebra
isomorphism in ${}^{H}_{H}{\mathcal{Y}D}$.

(ii) If $A$ is a pointed Hopf algebra of Nichols type, then ${\rm
diag}(A)={\mathcal{B}}(V)$ is a Nichols algebra in
${}^{kG}_{kG}{\mathcal{Y}D}$ and the coradical of $A$ is group algebra $kG$.
Therefore $A\cong{\mathcal{B}}(V)\\#kG$ as graded Hopf algebras. By [AS98,
Lemma 2.5], $A_{(0)}=kG$ and $A_{(1)}=V.$

Conversely, clearly, ${\rm diag}({\mathcal{B}}(V)\\#kG)=$
$({\mathcal{B}}(V)\\#kG)^{{\rm co}kG}=$ ${\mathcal{B}}(V)\\#1\cong$
${\mathcal{B}}(V)$ by (i) and the coradical of ${\mathcal{B}}(V)\\#kG$ is
$({\mathcal{B}}(V)\\#kG)_{(0)}=kG$. $\Box$

######  Lemma 2.2.

If $H=kG$ is a group algebra and $M$ is an $H$-Hopf bimodule, then pointed
Hopf algebra $H[M]$ of type one is a Hopf algebra of Nichols type. In
particular, one-type-co-path Hopf algebra
$kG[kQ_{1}^{c},G,r,\overrightarrow{\rho},u]$ is a pointed Hopf algebra of
Nichols type and ${\rm diag}(kG[kQ_{1}^{c},G,r,\overrightarrow{\rho},u])$
$={\mathcal{B}}(kQ_{1}^{1},{\rm ad}(G,r,\overrightarrow{\rho},u))$.

Proof. By Proposition 1.3 there exists an ${\rm
RSR}(G,r\overrightarrow{\rho},u)$ such that
$M\cong(kQ_{1}^{c},G,r\overrightarrow{\rho},u)$ as $kG$-Hopf bimodules. Thus
$kG[M]\cong kG[kQ_{1}^{c}]$ as graded Hopf algebras by [ZZC, Lemma 1.6].
Therefore, it is enough to show that $kG[kQ_{1}^{c}]$ is of Nichols type. Let
$A:=kG[kQ_{1}^{c}]$ and $R:={\rm diag}(kG[kQ_{1}^{c}])$. Obviously,
$R_{(0)}=k.$ Now we show $R_{(1)}=kQ_{1}^{1}$. Obviously, $kQ_{1}^{1}\subseteq
R_{(1)}$. Let $\alpha=\sum_{p=1}^{n}k_{p}b^{(p)}\in R_{(1)}$, where $b^{(p)}$
is an arrow from $x^{(p)}$ to $y^{(p)}$ with $0\not=k_{p}\in k$, and
$b^{(1)},b^{(2)},\cdots,b^{(n)}$ are different each other. Therefore
$\sum_{p=1}^{n}k_{p}b^{(p)}\otimes 1=\sum_{p=1}^{n}k_{p}b^{(p)}\otimes x^{p}$,
which implies $x^{(p)}=1$ for $1\leq p\leq n.$ Thus $\alpha\in kQ_{1}^{1}.$ We
next show that $R$ is generated by $R_{(1)}$ as algebras. Let $\mu$ denote the
multiplication and let $B$ denote the algebra generated by $kQ_{1}^{1}$ in
$kG[kQ_{1}^{c}]$. Obviously, $B\subseteq R.$ It follows from argument in [ZZC,
Subsection 3.1] that $\alpha^{+}:=\mu(id\otimes\iota_{0})$ is an algebraic
isomorphism from $R\\#kG$ to $kG[kQ_{1}^{c}]$. For any $x,y\in G$ and any
arrow $a_{y,x}$ from $x$ to $y$, see

$\displaystyle a_{y,x}$ $\displaystyle=$ $\displaystyle x\cdot
a_{x^{-1}y,1}=\mu(x\otimes
a_{x^{-1}y,1})=\mu(\alpha^{+}(1\\#x)\otimes\alpha^{+}(a_{x^{-1}y,1}\\#1))$
$\displaystyle=$
$\displaystyle\alpha^{+}(\mu((1\\#x)\otimes(a_{x^{-1}y,1}\\#1)))=\alpha^{+}(x\rhd
a_{x^{-1}y,1}\\#x)\in\alpha^{+}(B\\#kG).$

Therefore $\alpha^{+}(B\\#kG)=\alpha^{+}(R\\#kG)$ and so $B=R.$

It is enough to show $P(R)=kQ_{1}^{1}$, where $P(R)$ denotes the set of all
primitive elements in $R$. For any $a\in Q_{1}^{1}$ with $\delta^{-}(a)=y$ and
$\delta^{+}(a)=1$, see $\Delta_{R}(a)=$ $(\omega\otimes id)\Delta_{A}(a)=$
$1\otimes a+a\otimes 1$ (see [ZZC, Section 3] or [Ra]), i.e.
$kQ_{1}^{1}\subseteq P(R)$, where
$\omega=\mu_{A}(id\otimes\iota_{0}\pi_{0}S)\Delta_{A}$.

Conversely, we shall show $P(R)\subseteq kQ_{1}^{1}$ by the following two
steps. Obviously, $kG\cap P(R)=0$ and $P(R)$ is a graded subspace of $R$.

(i) Assume that $\alpha=a_{x_{n}x_{n-1}}a_{x_{n-1}x_{n-2}}\cdots
a_{x_{1}x_{0}}$ is a path from vertex $x_{0}$, via arrows $a_{x_{1}x_{0}}$
$,\cdots,$ $a_{x_{n-1}x_{n-2}}$, $a_{x_{n}x_{n-1}}$, to vertex $x_{n}$. Then
$\omega(\alpha)=\alpha\cdot x_{0}^{-1}$.

(ii) Let $v=\sum_{p=1}^{m}k_{p}\alpha_{p}\in P(R)$, where
$\alpha_{p}=b_{x_{n}x_{n-1}}^{(p)}b_{x_{n-1}x_{n-2}}^{(p)}\cdots
b_{x_{1}x_{0}}^{(p)}$ is a path with $n>1$, $k_{p}\in k$ for $p=1,2,\cdots,m$,
and $b_{x_{j}x_{j-1}}^{(p)}$ is an arrow from vertex $x_{j-1}$ to vertex
$x_{j}$ for $j=1,2,\cdots,n$. We shall show that $k_{p}=0$ for
$p=1,2,\cdots,m.$ Indeed,

$\displaystyle\Delta_{R}(v)$ $\displaystyle=$
$\displaystyle\sum_{p=1}^{m}\sum_{j=0}^{n}k_{p}(b_{x_{n}x_{n-1}}^{(p)}b_{x_{n-1}x_{n-}}^{(p)}\cdots
b_{x_{j+1}x_{j}}^{(p)})\cdot(x^{(p)}_{j})^{-1}\otimes
b_{x_{j}x_{j-1}}^{(p)}b_{x_{j-1}x_{j-2}}^{(p)}\cdots b_{x_{1}x_{0}}^{(p)}$
$\displaystyle=$ $\displaystyle\sum_{p=1}^{m}k_{p}(\alpha_{p}\otimes
1+1\otimes\alpha_{p}).$

This implies

$\displaystyle\sum_{p=1}^{m}k_{p}(b_{x_{n}x_{n-1}}^{(p)}b_{x_{n-1}x_{n-2}}^{(p)}\cdots
b_{x_{j+1}x_{j}}^{(p)})\cdot(x^{(p)}_{j})^{-1}\otimes
b_{x_{j}x_{j-1}}^{(p)}b_{x_{j-1}x_{j-2}}^{(p)}\cdots b_{x_{1}x_{0}}^{(p)}$
$\displaystyle=$ $\displaystyle 0,$ (2.1)

for $j=1,2,\cdots,n-1$, because of their length. For any $j$ with $1\leq j\leq
n-1$, assume that $\\{p\mid
b_{x_{j}x_{j-1}}^{(p)}b_{x_{j-1}x_{j-2}}^{(p)}\cdots b_{x_{1}x_{0}}^{(p)}$
$=b_{x_{j}x_{j-1}}^{(1)}b_{x_{j-1}x_{j-2}}^{(1)}\cdots
b_{x_{1}x_{0}}^{(1)}\\}$ $=\\{1,2,\cdots,m_{1}\\}$ without loss generality.
Therefore, by (2.1),

$\sum_{p=1}^{m_{1}}k_{p}(b_{x_{n}x_{n-1}}^{(p)}b_{x_{n-1}x_{n-2}}^{(p)}\cdots
b_{x_{j+1}x_{j}}^{(p)})\cdot(x^{(p)}_{j})^{-1}\otimes
b_{x_{j}x_{j-1}}^{(p)}b_{x_{j-1}x_{j-2}}^{(p)}\cdots b_{x_{1}x_{0}}^{(p)}=0,$

since $b_{x_{j}x_{j-1}}^{(q)}b_{x_{j-1}x_{j-2}}^{(q)}\cdots
b_{x_{1}x_{0}}^{(q)}\not=b_{x_{j}x_{j-1}}^{(1)}b_{x_{j-1}x_{j-2}}^{(1)}\cdots
b_{x_{1}x_{0}}^{(1)}$ for $q=m_{1}+1,\cdots,m.$ This implies
$\sum_{p=1}^{m_{1}}k_{p}b_{x_{n}x_{n-1}}^{(p)}b_{x_{n-1}x_{n-2}}^{(p)}\cdots
b_{x_{j+1}x_{j}}^{(p)}=0$ and $k_{p}=0$ for $p=1,2,\cdots,m_{1}$. Similarly,
we can show $k_{p}=0$ for $p=m_{1}+1,\cdots,m.$ $\Box$

By [BD, Theorem 4.3.2], the category ${}^{H}_{H}{\mathcal{Y}D}$ of Yetter-
Drinfeld modules is equivalent to the category
${}^{H}_{H}\\!{\mathcal{M}}^{H}_{H}$ of $H$-Hopf bimodule, where $H$ is a Hopf
algebra with bijective antipode. Let $T$ and $U$ be the two corresponding
functors. For any $N\in^{H}_{H}\\!{\mathcal{Y}D}$, according to [BD,
Prop.4.2.1], $T(N):=N\rtimes H=N\otimes H$ as vector spaces, and the actions
and coactions are given as follows: the left (co)actions are diagonal and
right (co)actions are induced by $H$. Explicitly, $g\cdot(x\otimes h):=g\cdot
x\otimes gh$, $(x\otimes h)\cdot g=x\otimes hg$, $\delta^{-}_{N\rtimes
H}(x\otimes h):=\sum_{x}x_{(-1)}h\otimes x_{(0)}\otimes h;$
$\delta^{+}_{N\rtimes H}(x\otimes h):=x\otimes h\otimes h$, where
$\delta_{N}^{-}(x)=\sum_{x}x_{(-1)}\otimes x_{(0)}$, $x\in N,h,g\in H.$ For
any $M\in^{H}_{H}\\!{\mathcal{M}}^{H}_{H}$, according to [BD, Equ. (7) and
(21)], $U(M)$ is the coinvariant of $M$ as a vector space, i.e.,
$U(M):=M^{{\rm co}H}:=\\{x\in M\mid\delta_{N}^{+}(x)=x\otimes 1\\}$. The left
action is left adjoint action and the left coaction is the restricted coaction
of the original coaction of $M$. That is,

$\alpha_{U(M)}^{-}(h\otimes x)=h\triangleright_{\rm
ad}x:=\alpha_{M}^{+}(\alpha_{M}^{-}(h\otimes x)\otimes h^{-1})=(h\cdot x)\cdot
h^{-1}$

and $\delta_{U(M)}^{-}(x)=\delta_{M}^{-}(x)$ for any $h\in H,x\in U(M).$ In
fact, $TU(M)=U(M)\rtimes H$ and $UT(N)=N\otimes 1_{H}$. Let $\lambda_{N}$ be a
map from $N\otimes 1_{H}$ to $N$ by sending $x\otimes 1_{H}$ to $x$ for any
$x\in N$, and let $\nu_{M}$ be a map from $U(M)\rtimes H$ to $M$ by sending
$x\otimes h$ to $\alpha_{M}^{+}(x\otimes h)=x\cdot h$ for any $x\in U(M)$ and
$h\in H$. $\lambda$ and $\nu$ are the natural isomorphisms from functor $UT$
to $id$ and from functor $TU$ to $id$, respectively. Note that the inverse of
$\nu_{M}$ is $(\alpha^{+}_{M}\otimes id)(id\otimes S\otimes
id)(\delta^{+}_{M}\otimes id)\delta_{M}^{+}.$

Remark: We have $U(kQ_{1}^{c},G,r,\overrightarrow{\rho},u)=(kQ_{1}^{1},{\rm
ad}(G,r,\overrightarrow{\rho},u))$ by the proof of Lemma 2.2.

######  Lemma 2.3.

Assume that $\phi$ is a Hopf algebra isomorphism from $H$ to $H^{\prime}$. Let
$N\in{}^{H^{\prime}}_{H^{\prime}}{\mathcal{Y}D}$ and
$M\in{}^{H^{\prime}}_{H^{\prime}}\\!{\mathcal{M}}^{H^{\prime}}_{H^{\prime}}$.
Then

$T({}{{}^{\phi}}^{-1}_{\phi}\\!N)\cong{}{{}^{\phi}}^{-1}_{\phi}T(N){{}^{\phi}}^{-1}_{\phi}\hbox{\
\ in \ }{}^{H}_{H}\\!{\mathcal{M}}^{H}_{H}\hbox{ \ \ and \ \
}U({}{{}^{\phi}}^{-1}_{\phi}M{{}^{\phi}}^{-1}_{\phi})\cong{{}^{\phi}}^{-1}_{\phi}U(M)\hbox{\
\ in }\ {}^{H}_{H}{\mathcal{Y}D}.$

Proof. The first isomorphism is given by sending $x\otimes h$ to
$x\otimes\phi(h)$ for any $x\in N,h\in H$; the second one is identity. $\Box$

######  Proposition 2.4.

(i) If $N$ is a Yetter-Drinfeld $kG$-module, then there exists an ${\rm
RSR}(G,r,$ $\overrightarrow{\rho},u)$ such that
$N\cong(kQ_{1}^{1},ad(G,r,\overrightarrow{\rho},u))$ as Yetter-Drinfeld
$kG$-modules.

(ii) If ${\mathcal{B}}(N)$ is a Nichols algebra in
${}^{kG}_{kG}{\mathcal{Y}D}$, then there exists an ${\rm
RSR}(G,r,\overrightarrow{\rho},u)$ such that
${\mathcal{B}}(N)\cong{\mathcal{B}}(kQ_{1}^{1},ad(G,r,\overrightarrow{\rho},u))$
as graded braided Hopf algebras in ${}^{kG}_{kG}{\mathcal{Y}D}$.

Proof. (i) Since $T(N)$ is a $kG$-Hopf bimodule, it follows from Proposition
1.3 that there exists an ${\rm RSR}(G,r,\overrightarrow{\rho},u)$ such that
$T(N)\cong(kQ_{1}^{c},G,r,\overrightarrow{\rho},u)$ as $kG$-Hopf bimodules.
Thus, $N\cong UT(N)$ $\cong
U(kQ_{1}^{c},G,r,\overrightarrow{\rho},u)=(kQ_{1}^{1},ad(G,r,\overrightarrow{\rho},u))$
as Yetter-Drinfeld $kG$-modules by [BD, Equ. (7) and (21)].

(ii) It follows from (i) and [AS02, Cor.2.3 ]. $\Box$

######  Lemma 2.5.

Assume that $\phi$ is a Hopf algebra isomorphism from $H$ to $H^{\prime}$. Let
$R$ and $R^{\prime}$ be graded braided Hopf algebras in
${}^{H}_{H}{\mathcal{Y}D}$ and ${}^{H^{\prime}}_{H^{\prime}}{\mathcal{Y}D}$
with $R_{(0)}=k1_{R}$ and $R_{(0)}^{\prime}=k1_{R^{\prime}}$, respectively. If
$R$ and ${}^{\phi^{-1}}_{\phi}R^{\prime}$ are isomorphic as graded braided
Hopf algebras in ${}^{H}_{H}{\mathcal{Y}D}$, then biproducts $R\\#H\cong
R^{\prime}\\#H^{\prime}$ as graded Hopf algebras.

Proof. Let $\psi$ be a graded braided Hopf algebra isomorphism from $R$ to
${}^{\phi^{-1}}_{\phi}R^{\prime}$ in ${}^{H}_{H}{\mathcal{Y}D}$. Define a map
$\nu$ from $R\\#H$ to $R^{\prime}\\#H^{\prime}$ by sending $x\otimes h$ to
$\psi(x)\otimes\phi(h)$ for any $x\in R,h\in H$. It is easy to check that
$\nu$ is an isomorphism of graded Hopf algebras. $\Box$

######  Theorem 3.

If $A$ is a pointed Hopf algebra of Nichols type with coradical $kG$, a group
algebra, then there exists a unique ${\rm RSR}(G,r,\overrightarrow{\rho},u)$,
up to isomorphism, such that $A\cong
kG[kQ_{1}^{c},G,r,\overrightarrow{\rho},u]$ as graded Hopf algebras.

Proof. By Lemma 2.1, $A\cong{\mathcal{B}}(V)\\#kG$ as graded Hopf algebras. By
Proposition 2.4 (ii), there exists a ${\rm RSR}(G,r,\overrightarrow{\rho},u)$
such that ${\mathcal{B}}(V)\cong{\mathcal{B}}(kQ_{1}^{1},{\rm
ad}(G,r,\overrightarrow{\rho},u))$ as graded braided Hopf algebras in
${}^{kG}_{kG}\\!{\mathcal{Y}D}$. Thus

$\displaystyle kG[kQ_{1}^{c},G,r,\overrightarrow{\rho},u]$
$\displaystyle\cong$ $\displaystyle{\mathcal{B}}(kQ_{1}^{1},{\rm
ad}(G,r,\overrightarrow{\rho},u))\\#kG\ \ (\hbox{by Lemma \ref{2.2}})$
$\displaystyle\cong$ $\displaystyle{\mathcal{B}}(V)\\#kG\ \ (\hbox{by Lemma
\ref{2.5}})$ $\displaystyle\cong$ $\displaystyle A.$

The uniqueness follows from [ZZC, Theorem 3]. $\Box$

Considering Theorem 3 and Lemma 2.2, we have that $A$ is a pointed Hopf
algebra of Nichols type if and only if $A$ is a pointed Hopf algebra of type
one.

##  3 Classification of Nichols Algebras

######  Theorem 4.

Let $(G,r,\overrightarrow{\rho},u)$ and
$(G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})$ be two
RSR’s. Then the following statements are equivalent:

(i) ${\rm RSR}(G,r,\overrightarrow{\rho},u)$ $\cong$ ${\rm
RSR}(G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})$.

(ii) There exists a Hopf algebra isomorphism $\phi:kG\rightarrow kG^{\prime}$
such that $(kQ_{1}^{1},ad(G,r,\overrightarrow{\rho},u))$ $\cong\ $
${}_{\phi}^{\phi^{-1}}(kQ_{1}^{\prime}{}^{1},ad(G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime}))$
as Yetter-Drinfeld $kG$-modules.

(iii) There is a Hopf algebra isomorphism $\phi:kG\rightarrow kG^{\prime}$
such that ${\mathcal{B}}(kQ_{1}^{1},{\rm ad}(G,r,\overrightarrow{\rho},u))$
$\cong\ $ ${}_{\phi}^{\phi^{-1}}{\mathcal{B}}(kQ_{1}^{\prime}{}^{1},{\rm
ad}(G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime}))$ as
graded braided Hopf algebra in ${}^{kG}_{kG}{\mathcal{Y}D}$.

Proof. ${\rm(i)}\Rightarrow{\rm(ii)}$. See

${}_{\phi}^{\phi^{-1}}(kQ_{1}^{\prime}{}^{1},ad(G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime}))$
$\displaystyle=$
${}_{\phi}^{\phi^{-1}}U(kQ_{1}^{\prime}{}^{c},G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})\
\ \ (\hbox{ by the remark before Lemma \ref{2.3}})$ $\displaystyle\cong$
$\displaystyle
U(_{\phi}^{\phi^{-1}}(kQ_{1}^{\prime}{}^{c},G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})_{\phi}^{\phi^{-1}})\
\ \ (\hbox{ by Lemma \ref{2.3}})$ $\displaystyle\cong$ $\displaystyle
U((kQ_{1}^{c},G,r,\overrightarrow{\rho},u))\ \ \ (\hbox{ by Theorem \ref{1}})$
$\displaystyle=$ $\displaystyle(kQ_{1}^{1},ad(G,r,\overrightarrow{\rho},u)).$

${\rm(ii)}\Rightarrow{\rm(iii)}$. By [ZZC, Lemma 2.7],
${\mathcal{B}}(kQ_{1}^{1},{\rm ad}(G,r,\overrightarrow{\rho},u))$ $\cong$
${\mathcal{B}}(^{\phi^{-1}}_{\phi}({kQ_{1}^{\prime}}^{1},{\rm
ad}(G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})))$
$\cong$ $\ {}^{\phi^{-1}}_{\phi}\\!{\mathcal{B}}({kQ_{1}^{\prime}}^{1},{\rm
ad}(G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime}))$ as
graded braided Hopf algebras in ${}^{kG}_{kG}{\mathcal{Y}D}$.

${\rm(iii)}\Rightarrow{\rm(i)}$.
${\mathcal{B}}(kQ_{1}^{1})\\#kG\cong{\mathcal{B}}(kQ_{1}^{\prime}{}^{1})\\#kG^{\prime}$
as graded Hopf algebras by Lemma 2.5. Thus
$kG[kQ_{1}^{c}]\cong{\mathcal{B}}(kQ_{1}^{1})\\#kG\cong{\mathcal{B}}(kQ_{1}^{\prime}{}^{1})\\#kG^{\prime}\cong
kG^{\prime}[kQ^{\prime}_{1}{}^{c}]$ as graded Hopf algebras by Lemma 2.2. Now
(i) follows from Theorem 2. $\Box$

Up to now we have classified all Nichols algebras by means of RSR’s. In other
words, ramification systems with irreducible representations uniquely
determine their corresponding Nichols algebras up to pull-push graded braided
Hopf algebra isomorphisms.

##  4 The classification of RSR over symmetric groups

Let ${\rm ad}_{h}^{-}$ and ${\rm ad}_{h}^{+}$ denote the left and right
adjoint actions, respectively. That is, ${\rm ad}_{h}^{-}(x):=hxh^{-1}$ for
any $x\in G.$ Let $\gamma_{C}=\mid\\!Z_{s}\\!\mid$ when $s\in
C\in{\mathcal{K}}(G)$. Aut$G$ and Inn$G$ denote the automorphism group and
inner automorphism group of $G.$

######  Definition 4.1.

Let ${\rm RSR}(G,r,\overrightarrow{\rho},u)$ and ${\rm
RSR}(G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})$ be two
${\rm RSR}$’s. If there exists a bijective map $\phi_{C}$ from $I_{C}(r,u)$ to
$I_{C}(r^{\prime},u^{\prime})$ such that
$\rho^{(i)}_{C}\cong\rho^{\prime}{}^{(\phi_{C}(i))}_{C}$ for any $i\in
I_{C}(r,u)$ and $C\in{\mathcal{K}}_{r}(G)$ with $G=G^{\prime}$, $r=r^{\prime}$
and $u=u^{\prime}$, then ${\rm RSR}(G,r,\overrightarrow{\rho},u)$ and ${\rm
RSR}(G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})$ are
said to be of the same type. Furthermore, if let
$\widehat{Z_{u(C)}}=\\{\xi_{u(C)}^{(i)}\mid i=1,2,\cdots,\gamma_{C}\\}$ and
$n_{C}^{(i)}:=$ $\mid\\{j\mid\rho_{C}^{(j)}\cong\xi_{u(C)}^{(i)}\\}\\!\mid$
for any $C\in\mathcal{K}_{r}(G)$ and $1\leq i\leq\gamma_{u(C)}\ $, then
$\\{(n_{C}^{(1)},n_{C}^{(2)},\cdots,n_{C}^{(\gamma_{C})})\\}_{C\in\mathcal{K}_{r}(G)}$
is called the type of ${\rm RSR}(G,r,\overrightarrow{\rho},$ $u)$.

######  Lemma 4.2.

If $\mathrm{Aut}G=\mathrm{Inn}G$, for example, $G=\mathbb{S}_{n}$ with
$n\not=6$, then ${\rm RSR}(G,r,\overrightarrow{\rho},u)$ and ${\rm
RSR}(G,r,\overrightarrow{\rho^{\prime}},u)$ are isomorphic if and only if they
have the same type.

Proof.  By [Zh82, Theorem 1.12.7] or [ZWW, Proposition 1.1 (ii)],
$\mathrm{Aut}G=\mathrm{Inn}G$ when $G=\mathbb{S}_{n}$ with $n\not=6$.

Let ${\rm
RSR}(G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})$ denote
${\rm RSR}(G,r,\overrightarrow{\rho^{\prime}},u)$ with $G=G^{\prime}$,
$r=r^{\prime}$ and $u=u^{\prime}$ for convenience. If ${\rm
RSR}(G,r,\overrightarrow{\rho},u)$ and ${\rm
RSR}(G,r,\overrightarrow{\rho^{\prime}},u)$ have the same type, then there
exists a bijective map $\phi_{C}$ from $I_{C}(r,u)$ to
$I_{\phi(C)}(r^{\prime},u^{\prime})$ such that
$\rho^{\prime}{}^{(\phi_{C}(i))}_{C}\cong\rho^{(i)}_{C}$ for any $i\in
I_{C}(r,u)$, $C\in{\mathcal{K}}_{r}(G)$. Therefore they are isomorphic.

Conversely, if ${\rm RSR}(G,r,\overrightarrow{\rho},u)$ and ${\rm
RSR}(G,r,\overrightarrow{\rho^{\prime}},u)$ are isomorphic, then there exist a
$\phi\in{\rm Aut}(G)$, $h_{C}\in G$ and a bijective map
$\phi_{C}:I_{C}(r,u)\rightarrow I_{\phi(C)}(r^{\prime},u^{\prime})$ such that
$u^{\prime}(\phi(C))=\phi{\rm ad}^{+}_{h_{C}}(u(C))$ and
$\rho^{\prime}{}_{\phi(C)}^{(\phi_{C}(i))}\phi{\rm
ad}^{+}_{h_{C}}\cong\rho_{C}^{(i)}$ for any $C\in{\mathcal{K}}_{r}(G)$ and
$i\in I_{C}(r,u)$. Since $\phi{\rm ad}_{h_{C}}^{+}\in{\rm Aut}G$ $={\rm
Inn}G$, there exists a $g_{C}\in G$ such that $\phi{\rm ad}_{h_{C}}^{+}={\rm
ad}_{g_{C}}^{+}$. Therefore $u(C)={\rm ad}^{+}_{g_{C}}(u(C))$ and
$\rho^{\prime}{}_{C}^{(\phi_{C}(i))}{\rm ad}^{+}_{g_{C}}\cong\rho_{C}^{(i)}$.
That is, $g_{C}\in Z_{u(C)}$ and $\chi^{\prime}{}_{C}^{(\phi_{C}(i))}{\rm
ad}^{+}_{g_{C}}(h)$ $=\chi^{\prime}{}_{C}^{(\phi_{C}(i))}(g_{C}^{-1}hg_{C})$
$=\chi^{\prime}{}_{C}^{(\phi_{C}(i))}(h)=\chi_{C}^{(i)}(h)$ for any $h\in
Z_{u(C)}$, where $\chi^{\prime}{}_{C}^{(\phi_{C}(i))}$ and $\chi_{C}^{(i)}$
denote the characters of $\rho^{\prime}{}_{C}^{(\phi_{C}(i))}$ and
$\rho_{C}^{(i)}$, respectively. Consequently,
$\chi^{\prime}{}_{C}^{(\phi_{C}(i))}=\chi_{C}^{(i)}$ and
$\rho^{\prime}{}_{C}^{(\phi_{C}(i))}\cong\rho_{C}^{(i)}$. This implies that
${\rm RSR}(G,r,\overrightarrow{\rho},u)$ and ${\rm
RSR}(G,r,\overrightarrow{\rho^{\prime}},u)$ have the same type. $\Box$

For a given ramification $r$ of $G$, let $\Omega(G,r)$ be the set of all RSR’s
of G with the ramification $r$, namely,
$\Omega(G,r):=\\{(G,r,\overrightarrow{\rho},u)\mid(G,r,\overrightarrow{\rho},u)\mbox{\
is\ an }{\rm RSR}\\}$. Let ${\mathcal{N}}(G,r)$ be the number of isomorphic
classes in $\Omega(G,r)$.

######  Theorem 5.

Given a group $G$ and a ramification $r$ of $G$. Assume
$\mathrm{Aut}G=\mathrm{Inn}G$, for example, $G=\mathbb{S}_{n}$ with $n\not=6$.
Let $u_{0}$ be a fixed map from ${\mathcal{K}}(G)\rightarrow G$ with
$u_{0}(C)\in C$ for any $C\in{\mathcal{K}}(G)$. Let $\bar{\Omega}(G,r,u_{0})$
denote the set consisting of all elements with distinct type in
$\\{(G,r,\overrightarrow{\rho},u_{0})\mid(G,r,\overrightarrow{\rho},u_{0})$ is
an RSR $\\}$. Then $\bar{\Omega}(G,r,u_{0})$ becomes the representative system
of isomorphic classes in $\Omega(G,r).$

Proof. For any ${\rm RSR}(G,r,\overrightarrow{\rho},u)$, there exists ${\rm
RSR}(G,r,\overrightarrow{\rho^{\prime}},u_{0})$ such that ${\rm
RSR}(G,r,\overrightarrow{\rho},u)$ $\cong{\rm
RSR}(G,r,\overrightarrow{\rho^{\prime}},u_{0})$ by Proposition 1.5. Using
Lemma 4.2, we complete the proof. $\Box$

######  Corollary 4.3.

$\displaystyle{\mathcal{N}}(G,r)$ $\displaystyle=$
$\displaystyle\prod\limits_{C\in\mathcal{K}_{r}(G)}\tau_{C},$

where $\tau_{C}=$ the number of elements in set
$\\{(n_{1},n_{2},\cdots,n_{\gamma_{C}})\in{\mathbb{N}}^{\gamma_{C}}\mid
n_{1}{\rm deg}\xi_{1}+n_{2}{\rm deg}\xi_{2}+\cdots+n_{\gamma_{C}}{\rm
deg}\xi_{\gamma_{C}}=r_{C}\\}$ for any $C\in{\mathcal{K}}_{r}(G)$.

Remark: For a given finite Hopf quiver $(Q,G,r)$ over symmetric group
$G={\mathbb{S}}_{n}$ with $n\not=6$, every path algebra
$T_{(kG)^{*}}(kQ_{1}^{a})$ over quiver $(Q,G,r)$ exactly admits
${\mathcal{N}}(G,r)$ non-isomorphic graded Hopf algebra structures; every path
coalgebra $T_{kG}^{c}(kQ_{1}^{c})$ over quiver $(Q,G,r)$ exactly admits
${\mathcal{N}}(G,r)$ non-isomorphic graded Hopf algebra structures.

##  5 Appendix

We now consider the dual case of Theorem 4. If $Q$ is finite, then
$(kQ_{1}^{a},G,r,\overrightarrow{\rho},u)$ is a $(kG)^{*}$-Hopf bimodule with
comodule operations $\delta^{-}$ and $\delta^{+}$. Define a new left
$(kG)^{*}$-coaction on $kQ_{1}^{a}$ given by

$\delta^{-}_{\rm coad}(x):=\sum_{x}x_{(-1)}S(x_{(0)(1)})\otimes x_{(0)(0)},\ \
\ \hbox{ for any }x\in kQ_{1}^{a},$

i.e. adjoint coaction. With this left $(kG)^{*}$-coaction and the original
left (arrow) $(kG)^{*}$-action $\alpha^{-}$, $kQ_{1}^{a}$ is a Yetter-Drinfeld
$(kG)^{*}$-module. Let $kQ^{1a}_{1}$ denote the subspace spanned by
$Q_{1}^{1}$ in $kQ_{1}^{a}$. It is clear that
$\xi_{kQ_{1}^{c}}(kQ_{1}^{1})=(kQ_{1}^{1a})^{*}$, where $\xi_{kQ_{1}^{c}}$ was
defined in [ZZC, Lemma 1.7]. Thus $kQ_{1}^{1a}$ is a Yetter-Drinfeld
$(kG)^{*}$-submodule of $kQ_{1}^{a}$, denoted by $(kQ_{1}^{1a},$ ${\rm
coad}(G,r,\overrightarrow{\rho},u))$, which is isomorphic to the dual of
$(kQ_{1}^{1},ad(G,r,$ $\overrightarrow{\rho},u))$ as Yetter-Drinfeld
$(kG)^{*}$-modules.

Therefore we have the dual case of Theorem 4.

######  Proposition 5.1.

Let $(G,r,\overrightarrow{\rho},u)$ and
$(G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})$ be two
RSR’s. Then the following statements are equivalent:

(i) ${\rm RSR}(G,r,\overrightarrow{\rho},u)$ $\cong$ ${\rm
RSR}(G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime})$.

(ii) There exists a Hopf algebra isomorphism
$\phi:(kG)^{*}\rightarrow(kG^{\prime})^{*}$ such that $(kQ_{1}^{1a},$ ${\rm
coad}(G,r,\overrightarrow{\rho},u))\cong\
_{\phi}^{\phi^{-1}}(kQ_{1}^{\prime}{}^{1a},{\rm
coad}(G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime}))$ as
Yetter-Drinfeld $(kG)^{*}$-modules.

(iii) There is a Hopf algebra isomorphism
$\phi:(kG)^{*}\rightarrow(kG^{\prime})^{*}$ such that
${\mathcal{B}}(kQ_{1}^{1a},$ ${\rm coad}(G,r,\overrightarrow{\rho},u)$ $\cong$
${}_{\phi}^{\phi^{-1}}{\mathcal{B}}(kQ_{1}^{\prime}{}^{1a},{\rm
coad}(G^{\prime},r^{\prime},\overrightarrow{\rho^{\prime}},u^{\prime}))$ as
graded braided Hopf algebras in ${}^{(kG)^{*}}_{(kG)^{*}}{\mathcal{Y}D}$.

Proof. Obviously, (ii) and Theorem 4 (ii) are equivalent; (iii) and Theorem 4
(iii) are equivalent. $\Box$

Acknowledgement: We would like to thank Prof. N. Andruskiewitsch for his help.
The first author and the third author were financially supported by the
Australian Research Council, and the second author was supported by NSF of
China (10771183) and Sino-German project (GZ310). S.C.Z thanks the Department
of Mathematics, University of Queensland for hospitality.

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