# Generalized McKay Quivers, Root System and Kac-Moody Algebras

Bo Hou  College of Applied Science, Beijing University of Technology, Beijing
100124, People’s Republic China<EMAIL_ADDRESS>and  Shilin Yang
College of Applied Science, Beijing University of Technology, Beijing 100124,
People’s Republic China<EMAIL_ADDRESS>
###### Abstract.

Let $Q$ be a finite quiver and $G\subseteq\mbox{Aut}(\mathbbm{k}Q)$ a finite
abelian group. Assume that $\widehat{Q}$ and $\Gamma$ is the generalized Mckay
quiver and the valued graph corresponding to $(Q,G)$ respectively. In this
paper we discuss the relationship between indecomposable
$\widehat{Q}$-representations and the root system of Kac-Moody algebra
$\mathfrak{g}(\Gamma)$. Moreover, we may lift $G$ to
$\overline{G}\subseteq\mbox{Aut}(\mathfrak{g}(\widehat{Q}))$ such that
$\mathfrak{g}(\Gamma)$ embeds into the fixed point algebra
$\mathfrak{g}(\widehat{Q})^{\overline{G}}$ and
$\mathfrak{g}(\widehat{Q})^{\overline{G}}$ as $\mathfrak{g}(\Gamma)$-module is
integrable.

###### Key words and phrases:

Generalized McKay quiver, Representation of quiver, Root system, Kac-Moody
algebra

###### 2000 Mathematics Subject Classification:

Primary 16G10, 16G20, 17B67

The second author was supported by the Science and Technology Program of
Beijing Education Committee (Grant No. KM200710005013) and Foundation of
Selected Excellent Science and Technology Activity for Returned Scholars of
Beijing.

##  1\. Introduction

Thirty years ago, McKay introduced a class of quivers, now called the McKay
quivers, for some finite subgroups of the general linear group [16]. Let
$\mathbb{C}$ denote the complex number field. McKay observed that the McKay
quiver for $G\subseteq\mathrm{SL}(2,\mathbb{C})$ is the double quiver of the
extended Dynkin quiver $\widetilde{A}_{n}$, $\widetilde{D}_{n}$,
$\widetilde{E}_{6}$, $\widetilde{E}_{7}$, $\widetilde{E}_{8}$ respectively.
Furthermore, the corresponding Dynkin diagram is the same as the one occurring
in the minimal resolution of singularities for the quotient surface
$\mathbb{C}/G$ (see [4]). McKay quiver has played an important role in many
mathematical fields such as quantum group, algebraic geometry, mathematics
physics and representation theory (see, for examples [2, 5, 9, 8, 15, 17]).

Let $V$ be a finite vector space over a field $\mathbbm{k}$ of characteristic
0 and $G\subseteq\mathrm{GL}_{\mathbbm{k}}(V)$ a finite group. Assume that
$\mathrm{T}_{\mathbbm{k}}(V)$ is the tensor algeba of $V$ over $\mathbbm{k}$.
It is well-known that the skew group algebra $\mathrm{T}_{\mathbbm{k}}(V)\ast
G$ is Morita equivalent to the path algebra $\mathbbm{k}\widehat{Q}$, where
$\widehat{Q}$ is the McKay quiver of $G$ (see [9]). In other words, the McKay
quiver realizes the Gabriel quiver of $\mathrm{T}_{\mathbbm{k}}(V)\ast G$. It
is natural to ask how to determine the Gabriel quiver of skew group algebra
$\Lambda\ast G$ for any algebra $\Lambda$. Recently, for any path algebra
$\mathbbm{k}Q$ over an algebraically closed field $\mathbbm{k}$ and a finite
group $G$ such that char$\mathbbm{k}\nmid|G|$, if the action of $G$ on
$\mathbbm{k}Q$ permutes the set of primitive idempotents and stabilizing the
vector space spanned by the arrows, Demonet in [6] has constructed a quiver
$\widehat{Q}$ such that the path algebra $\mathbbm{k}\widehat{Q}$ is Morita
equivalent to the skew group algebra $\mathbbm{k}Q\ast G$. The quiver
$\widehat{Q}$ can be viewed as a generalization of McKay quiver, which is
called the generalized McKay quiver of $(Q,G)$ in this paper.

Given a finite quiver $Q$ with an admissible automorphism ${\bf a}$. Hubery in
[11, 12] described the correspondence between dimension vectors of the
isomorphically invariant $Q$-indecomposables and the positive root system of
${\mathfrak{g}}(\Gamma)$, where $\Gamma$ is the valued graph of $(Q,{\bf a})$.
Motivated by Hubery’s work, the aim of this paper is to establish the
correspondence between the indecomposable $\widehat{Q}$-representations and
the positive roots of the symmetrizable Kac-Moody algebra
$\mathfrak{g}(\Gamma)$ of the valued graph $\Gamma$ associated to $(Q,G)$,
where $Q$ is a finite quiver and $G$ is a finite abelian automorphism group of
$\mathbbm{k}Q$. Moreover, we can lift $G$ to an automorphism group
$\overline{G}$ of Kac-Moody algebra $\mathfrak{g}:=\mathfrak{g}(\widehat{Q})$
of $\widehat{Q}$, such that $\mathfrak{g}(\Gamma)$ can be embedded into the
fixed point subalgebra $\mathfrak{g}^{\overline{G}}$. In this case, we also
show that $\mathfrak{g}^{\overline{G}}$ as a $\mathfrak{g}(\Gamma)$-module is
integrable. Compared with Hubery’s work, a more general description is given
by approach of the generalized McKay quiver.

For a finite quiver $Q=(I,E)$ and a finite abelian group
$G\subseteq\mbox{Aut}(\mathbbm{k}Q)$ (the algebra automorphism group of
$\mathbbm{k}Q$). We always assume that the action of $G$ on $Q$ is admissible,
i.e., no arrow connects to vertices in the same orbit. Then we can get a
valued graph $\Gamma$ without loops and a generalized McKay quiver
$\widehat{Q}$ corresponding to $(Q,G)$. By [18], we can define an action of
$G$ on $\mathbbm{k}\widehat{Q}$ due to the Morita equivalence between the skew
group algebra $\mathbbm{k}Q\ast G$ and $\mathbbm{k}\widehat{Q}$. Therefore
this action induces an action on $\widehat{Q}$-representations. Let $G_{X}$ be
a complete set of left coset representatives of $H_{X}=\\{g\in
G\mid{{}^{g}X}\cong X\\}$ in $G$ for any $\widehat{Q}$-representation $X$, let
$\mathbb{Z}I$, $\mathbb{Z}\widehat{I}$ and $\mathbb{Z}\mathcal{I}$ be the root
lattice of $Q$, $\widehat{Q}$ and $\Gamma$, respectively. Applying the
equivalence between representation category of $\widehat{Q}$ and module
category of the skew group algebra $\mathbbm{k}Q\ast G$ and the fact that each
$\mathbbm{k}Q\ast G$ module as a $Q$-representation is $G$-invariant, we
define a map

$h:\quad\mathbb{Z}\widehat{I}\longrightarrow(\mathbb{Z}I)^{G}\longrightarrow\mathbb{Z}\mathcal{I}$

where $(\mathbb{Z}I)^{G}$ is the fixed point set of $\mathbb{Z}I$ under the
action of $G$. The map $h$ builds a bridge between the dimension vectors of
indecomposable $\widehat{Q}$-representations and the root system of Kac-Moody
algebra $\mathfrak{g}(\Gamma)$. The first main result of this paper is
described as follows.

######  Theorem 1.1.

Let $Q$ be a quiver without loops and with an admissible action of a finite
abelian subgroup $G\subseteq{\rm Aut}(\mathbbm{k}Q)$, where $\mathbbm{k}$ is
an algebraically closed field with ${\rm char}\mathbbm{k}\nmid|G|$. Assume
that $\Gamma$ and $\widehat{Q}$ is the valued graph and generalized McKay
quiver associated to $(Q,G)$. Then

(1) the images under $h$ of the dimension vectors of all the indecomposable
$\widehat{Q}$-representations give the positive root system of the
symmetrisable Kac-Moody algebra $\mathfrak{g}(\Gamma)$;

(2) for each positive real root $\alpha$ of $\mathfrak{g}(\Gamma)$, let $X$ be
a $\widehat{Q}$-representation such that $h({\bf dim}X)=\alpha$. Then there
are $|G_{X}|$ indecomposable $\widehat{Q}$-representations (up to isomorphism)
such that their dimension vectors under $h$ are $\alpha$.

The proof of this theorem is based on understanding the relationship among
indecomposable $\widehat{Q}$-representations, indecomposable $\mathbbm{k}Q\ast
G$-modules and indecomposable $G$-invariant $Q$-representations. In the proof,
we also need the dual between $(Q,G)$ and $(\widehat{Q},G)$. This duality is
first discussed in [18] for a finite quiver with an automorphism. Here we give
a general and strict proof by the generalized McKay quiver.

Next we consider the relationship between Kac-Moody algebra
$\mathfrak{g}(\Gamma)$ and the fixed point subalgebra
$\mathfrak{g}^{\overline{G}}$. The action of $G$ on $\widehat{Q}$ naturally
induces an action on the derived algebra $\mathfrak{g}^{\prime}$ of
$\mathfrak{g}$. Let $\Omega=\\{g_{1},g_{2},\cdots,g_{n}\\}$ be a set of
generators of $G$. Following from [14], we lift $G$ to
$\overline{G}\subseteq\mbox{Aut}(\mathfrak{g})$ corresponding to a family of
linear maps
$\\{\psi_{i}:=\psi_{g_{i}}:\mathfrak{h}/\mathfrak{h}^{\prime}\rightarrow\mathfrak{c}\mid
g_{i}\in\Omega\\}$, where $\mathfrak{c}$ is the center of $\mathfrak{g}$,
$\mathfrak{h}$ and $\mathfrak{h}^{\prime}$ is the Cartan subalgebra of
$\mathfrak{g}$ and $\mathfrak{g}^{\prime}$ respectively. Denote by $C$ the
symmetrisable generalized Cartan matrix of the valued graph $\Gamma$. Then, we
can give a realization
$(\mathcal{H}^{\overline{G}},\\{\epsilon_{i}\\},\\{h_{i}\\})$ of $C$ by the
fixed point set $\mathfrak{h}^{\overline{G}}$ of $\mathfrak{h}$, and we obtain
that

######  Theorem 1.2.

For the lifting $\overline{G}$ of $G$ corresponding to
$\\{\psi_{i}:\mathfrak{h}/\mathfrak{h}^{\prime}\rightarrow\mathfrak{c}\mid
g_{i}\in\Omega\\}$ such that
$\psi_{i}\big{(}(\mathcal{H}+\mathfrak{h}^{\prime})/\mathfrak{h}^{\prime}\big{)}=0$,
there is a monomorphism

$\mathfrak{g}(\Gamma)\rightarrow\mathfrak{g}^{\overline{G}}.$

Moreover this monomorphism endows $\mathfrak{g}^{\overline{G}}$ with an
integrable $\mathfrak{g}(\Gamma)$-module structure under the adjoint action of
$\mathfrak{g}(\Gamma)$. In particular, if $Q$ is a finite union of Dynkin
quivers, then $\mathfrak{g}(\Gamma)\cong\mathfrak{g}^{\overline{G}}$ as Lie
algebras.

In the end of this paper, two examples are given to elucidate our results.

Throughout this paper, let $\mathbbm{k}$ denote an algebraic closed field and
$\mathbb{Z}$ denote the set of integers. We denote by $G$ the finite group
such that char$\mathbbm{k}\nmid|G|$, denote by ${\bf mod}$-$\Lambda$ the
category of (right) $\Lambda$-modules for any $\mathbbm{k}$-algebra $\Lambda$.

##  2\. Preliminaries

2.1. Recall that a quiver $Q=(I,E)$ is an oriented graph with $I$ the set of
vertices and $E$ the set of arrows. A quiver $Q$ is said to be finite if $I$
and $E$ are all finite set. An arrow in $Q$ is called a loop if its staring
vertex coincides with its terminating vertex. In this paper we only consider a
finite quiver without loops. Therefore we have a path algebra $\mathbbm{k}Q$
for a quiver $Q$ (see [1, 3]).

A representation $X=(X_{i},X_{\alpha})$ of the quiver $Q=(I,E)$ consists of a
family of $\mathbbm{k}$-vector spaces $X_{i}$ for $i\in I$, together with a
family of $\mathbbm{k}$-linear maps $X_{\alpha}:X_{i}\rightarrow X_{j}$ for
$\alpha:i\rightarrow j$ in $E$. Given two representations $X$ and $Y$ of $Q$,
a morphism $\varphi:X\rightarrow Y$ is given by a family of
$\mathbbm{k}$-linear maps $\varphi_{i}:X_{i}\rightarrow Y_{i}~{}(i\in I)$ such
that $\varphi_{j}\circ X_{\alpha}=Y_{\alpha}\circ\varphi_{i}$ for each arrow
$\alpha:i\rightarrow j$. It is well-known that the category of representations
of $Q$ is naturally equivalent to the category of $\mathbbm{k}Q$-modules (see
[1, 3]). Thus we always identify a $\mathbbm{k}Q$-module $X$ with a
$Q$-representation $(X_{i},X_{\alpha})$ in this paper.

2.2. Assume that $\Lambda$ is a $\mathbbm{k}$-algebra and $G$ acts on
$\Lambda$, the skew group algebra of $\Lambda$ under the action of $G$ is by
definition the $\mathbbm{k}$-algebra whose underlying $\mathbbm{k}$-vector
space is $\Lambda\otimes_{\mathbbm{k}}\mathbbm{k}[G]$ and whose multiplication
is defined by

$(\lambda\otimes g)(\lambda^{\prime}\otimes g^{\prime})=\lambda
g(\lambda^{\prime})\otimes gg^{\prime}$

for all $\lambda,\lambda^{\prime}\in\Lambda$ and $g,g^{\prime}\in G$ (see
[18]). For convenience, we denote this algebra by $\Lambda\ast G$, denote the
element $\lambda\otimes g$ in $\Lambda\ast G$ by $\lambda g$. Note that
$\Lambda$ and $\mathbbm{k}[G]$ can be viewed as subalgebras of $\Lambda\ast
G$.

Let $\Lambda=\mathbbm{k}Q$ be the path algebra for a quiver $Q=(I,E)$. Assume
that $G$ acts on $\mathbbm{k}Q$ permuting the set of primitive idempotents
$\\{e_{i}\mid i\in I\\}$ and stabilizing the vector space spanned by the
arrows. Let $\mathcal{I}$ denote a set of representatives of class of $I$
under the action of $G$. For any $i\in I$, let $G_{i}$ denote the subgroup of
$G$ stabilizing $e_{i}$, For each $i\in I$, there exist some $g\in G$ such
that $g^{-1}(i)\in\mathcal{I}$. We fix such a $g$ and denote it by
$\kappa_{i}$. Let $\mathcal{O}_{i}$ be the orbit of $i$ under the action of
$G$. For $(i,j)\in\mathcal{I}^{2}$, $G$ acts on
$\mathcal{O}_{i}\times\mathcal{O}_{j}$ by the diagonal action. A set of
representatives of the classes of this action will be denoted by
${\mathcal{F}}_{ij}$.

For $i,j\in I$, we denote by $E_{ij}\subseteq\mathbbm{k}Q$ the vector space
spanned by the arrows from $i$ to $j$ and regard it as a left and right
$\mathbbm{k}[G_{ij}]:=\mathbbm{k}[G_{i}\cap G_{j}]$-module by restricting the
action of $G$. In [6] Demonet defined the quiver
$\widehat{Q}=(\widehat{I},\widehat{E})$ as follows

$\widehat{I}=\bigcup_{i\in\mathcal{I}}\\{i\\}\times\mbox{irr}G_{i},$

where $\mbox{irr}G_{i}$ is a set of representatives of isomorphism classes of
irreducible representations of $G_{i}$. The set of arrows of $\widehat{Q}$
from $(i,\rho)$ to $(j,\sigma)$ is a basis of

$\bigoplus_{(i^{\prime},j^{\prime})\in{\mathcal{F}}_{ij}}\mbox{Hom}_{\mathbbm{k}[G_{i^{\prime}j^{\prime}}]}\left((\rho\cdot\kappa_{i^{\prime}})|_{G_{i^{\prime}j^{\prime}}},~{}(\sigma\cdot\kappa_{j^{\prime}})|_{G_{i^{\prime}j^{\prime}}}\otimes_{\mathbbm{k}}E_{i^{\prime}j^{\prime}}\right),$

where the representation $\rho\cdot\kappa_{i^{\prime}}$ of $G_{i^{\prime}}$ is
the same as $\rho$ as a $\mathbbm{k}$-vector space, and
$(\rho\cdot\kappa_{i^{\prime}})g=\rho\kappa_{i^{\prime}}g\kappa_{i^{\prime}}^{-1}$
for each $g\in
G_{i^{\prime}}=\kappa_{i^{\prime}}^{-1}G_{i}\kappa_{i^{\prime}}$. Demonet
yielded the following theorem.

######  Theorem 2.1.

(see [6]) The category mod-$\mathbbm{k}\widehat{Q}$ is equivalent to the
category mod-$\mathbbm{k}Q\ast G$.

In particular, if the quiver $Q$ is a singular vertex with $m$ loops, we can
view $G$ as a subgroup of ${\bf GL}_{m}(\mathbbm{k})$. Then the quiver
$\widehat{Q}$ is just the McKay quiver of $G$. Thus, we view $\widehat{Q}$ as
a generalization of McKay quiver in general. Furthormore, for any factor
algebra $\mathbbm{k}Q/J$, the shew group algebra $(\mathbbm{k}Q/J)\ast G$ is
Morita equivalent to a factor algebra of $\mathbbm{k}\widehat{Q}$. This
implies that the generalized McKay quiver can realize the Garbiel quiver of
$\Lambda\ast G$ for any basic algebra $\Lambda$.

2.3. For a quiver $Q=(I,E)$, there is a corresponding symmetric generalized
Cartan matrix $A=(a_{ij})$ indexed by $I$ with entries

$a_{ij}=\left\\{\begin{array}[]{ll}2,&i=j;\\\ -|\\{\mbox{edges between
vertices }i\mbox{ and }j\\}|,&i\neq j.\end{array}\right.$

It is obvious that $A$ is independent of the orientation of $Q$.

Denote by $\mathfrak{g}(Q)$ for the associated symmetric Kac-Moody algebra
corresponding to $A$ with the simple root set $\Pi=\\{\varepsilon_{i}\mid i\in
I\\}$ and root system $\Delta_{Q}$. The root lattice $\mathbb{Z}I$ of $Q$ is
the free abelian group on $\Pi$, with the partially order such that
$\alpha=\sum_{i\in I}\alpha_{i}\varepsilon_{i}\geq 0$ if and only if
$\alpha_{i}\geq 0$ for all $i\in I$. We endow $\mathbb{Z}I$ with a symmetric
bilinear form $(-,-)_{Q}$ via $(\varepsilon_{i},\varepsilon_{j})_{Q}=a_{ij}$.
Then, for each vertex $i\in I$, we have a reflection
$r_{i}:\alpha\mapsto\alpha-(\alpha,\varepsilon_{i})_{Q}\varepsilon_{i}$. These
reflections generate the Weyl group $\mathcal{W}(Q)$ of $Q$. The real roots of
$Q$ are given by the images under $\mathcal{W}(Q)$ of the simple roots
$\varepsilon_{i}$ and the imaginary roots are given by $\pm$ the images under
$\mathcal{W}(Q)$ of the fundamental set

$F_{Q}:=\\{\alpha>0\mid(\alpha,\varepsilon_{i})_{Q}\leq 0\mbox{ for all
}i\mbox{ and the support of }\alpha\mbox{ is connected}\\}.$

Suppose that the action of $G$ on path algebra $\mathbbm{k}Q$ permutes the set
of primitive idempotents. The action of $G$ is said to be admissible if no
arrow connects to vertices in the same $G$-orbit. For any quiver $Q$ with an
admissible action of $G$, we can construct a symmetric matrix $B=(b_{ij})$
indexed by $\mathcal{I}$, where

$b_{ij}=\left\\{\begin{array}[]{ll}2|\mathcal{O}_{i}|,&i=j;\\\
-|\\{\mbox{edges between vertices in }\mathcal{O}_{i}\mbox{ and
}\mathcal{O}_{j}\\}|,&i\neq j.\end{array}\right.$

Let $d_{i}:=b_{ii}/2=|\mathcal{O}_{i}|$ and $D=\mbox{diag}(d_{i})$. Then
$C=(c_{ij})=D^{-1}B$ is a symmetrisable generalized Cartan matrix indexed by
$\mathcal{I}$. It is well-known that there is a unique valued graph $\Gamma$
corresponding to the matrix $C$ by [7]. The valued graph $\Gamma$ has the
vertex set $\mathcal{I}$ and an edge $i$—-$j$ equipped with the ordered pair
$(|c_{ji}|,|c_{ij}|)$ whenever $c_{ij}\neq 0$. Since the action of $G$ is
admissible, $\Gamma$ has no loops. For each connected component
$\Gamma^{\prime}$ of the graph $\Gamma$, we always take the representative set
$\mathcal{I}$ such that the underlying graph of the full subquiver generated
by the vertices in $\Gamma^{\prime}$ is connected.

Denote by $\mathfrak{g}(\Gamma)$ for the associated symmetric Kac-Moody
algebra corresponding to $C$. The simple root set and root system of $\Gamma$
are denoted by $\Pi_{\Gamma}=\\{\overline{\varepsilon}_{i}\mid
i\in\mathcal{I}\\}$ and $\Delta_{\Gamma}$. Let $\mathbb{Z}\mathcal{I}$ denote
the root lattice of $\Gamma$. There is a symmetric bilinear form
$(-,-)_{\Gamma}$ determined by $B$ on $\mathbb{Z}\mathcal{I}$ such that
$(\overline{\varepsilon}_{i},\overline{\varepsilon}_{j})_{\Gamma}=b_{ij}$, and
a reflection $\gamma_{i}$ on $\mathbb{Z}\mathcal{I}$ defined by

$\gamma_{i}:\alpha\mapsto\alpha-\frac{1}{d_{i}}(\alpha,\overline{\varepsilon}_{i})_{\Gamma}\overline{\varepsilon}_{i}$

for each $i\in{\mathcal{I}}$. These reflections generate the Weyl group
$\mathcal{W}(\Gamma)$ of $\Gamma$. Similarly, we have the real roots and the
imaginary roots associated to $\Gamma$ (see [13]).

##  3\.  Proof of Theorem 1.1

From now on, unless otherwise stated we fix a finite group
$G\subseteq\mbox{Aut}(\mathbbm{k}Q)$ and assume that the action of $G$ is
admissible. Let $\widehat{Q}$ and $\Gamma$ be the generalized Mckay quiver and
the valued graph corresponding to $(Q,G)$. In this section, we show that the
correspondence between indecomposable representations of $\widehat{Q}$ and the
positive root system of $\Gamma$.

3.1. The group $G$ acts naturally on the root lattice $\mathbb{Z}I$, i.e.,
$g(\varepsilon_{i})=\varepsilon_{g(i)}$ for any $g\in G$. It is easy to check
that this action preserves the partial order $\geq$ and the bilinear form
$(-,-)_{Q}$ is $G$-invariant. Let

$(\mathbb{Z}I)^{G}:=\\{\alpha\in\mathbb{Z}I\mid g(\alpha)=\alpha\mbox{ for any
}g\in G\\}.$

There is a canonical bijection

$f:\quad(\mathbb{Z}I)^{G}\longrightarrow\mathbb{Z}\mathcal{I}$

given by

$f\Big{(}\sum_{i\in
I}\alpha_{i}\varepsilon_{i}\Big{)}=\sum_{i\in\mathcal{I}}\alpha_{i}\overline{\varepsilon}_{i}.$

The admissibility of the action of $G$ implies that the reflections $r_{i}$
and $r_{j}$ commute whenever $i$ and $j$ lie in the same $G$-orbit. Therefore
the element

$S_{i}:=\prod_{i^{\prime}\in\mathcal{O}_{i}}r_{i^{\prime}}\in\mathcal{W}(Q)$

is well-defined for any $i\in\mathcal{I}$. Note that $g\circ
r_{i}=r_{g(i)}\circ g$ for any $g\in G$, we have $S_{i}\in
C_{G}(\mathcal{W}(Q))$, the set of elements in the Weyl group commuting with
the action of $G$. By induction on the length of the element in
$C_{G}(\mathcal{W}(Q))$, it is easy to check that $C_{G}(\mathcal{W}(Q))$ is
generated by $S_{i}$, $i\in\mathcal{I}$.

Similar to [12, Lemma 3], we have

######  Lemma 3.1.

For any $\alpha,\beta\in(\mathbb{Z}I)^{G}$, we have

(1) $(\alpha,\beta)_{Q}=(f(\alpha),f(\beta))_{\Gamma}$;

(2) $f(S_{i}(\alpha))=\gamma_{i}(f(\alpha))\in\mathbb{Z}\mathcal{I}$ for
$i\in\mathcal{I}$.

(3) The map $\gamma_{i}\mapsto S_{i}$ induces a group isomorphism
$\mathcal{W}(\Gamma)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}C_{G}(\mathcal{W}(Q))$.

###### Proof.

(1) Set
$\varepsilon^{i}:=\sum_{i^{\prime}\in\mathcal{O}_{i}}\varepsilon_{i^{\prime}}$.
Then $\\{\varepsilon^{i}\mid i\in\mathcal{I}\\}$ is a basis of
$(\mathbb{Z}I)^{G}$. Since

$(\varepsilon^{i},\varepsilon^{j})_{Q}=\sum_{i^{\prime}\in\mathcal{O}_{i}\atop
j^{\prime}\in\mathcal{O}_{j}}a_{i^{\prime}j^{\prime}}=b_{ij}=(\overline{\varepsilon}_{i},\overline{\varepsilon}_{j})_{\Gamma}$

for any $i,j\in\mathcal{I}$, (1) is obvious.

(2) Since the bilinear form $(-,-)_{Q}$ is $G$-invariant, we have

$S_{i}(\alpha)=\alpha-\sum_{i^{\prime}\in\mathcal{O}_{i}}(\alpha,\varepsilon_{i^{\prime}})_{Q}\varepsilon_{i^{\prime}}=\alpha-\sum_{i^{\prime}\in\mathcal{O}_{i}}\frac{1}{d_{i}}\big{(}\alpha,\sum_{j\in\mathcal{O}_{i}}\varepsilon_{j}\big{)}_{Q}\varepsilon_{i^{\prime}}=\alpha-\frac{1}{d_{i}}(f(\alpha),\overline{\varepsilon}_{i^{\prime}})_{\Gamma}\varepsilon^{i}$

by (1). We obtain that

$f(S_{i}(\alpha))=f(\alpha)-\frac{1}{d_{i}}(f(\alpha),\overline{\varepsilon}_{i^{\prime}})_{\Gamma}\overline{\varepsilon}_{i}=\gamma_{i}(f(\alpha)).$

(3) By the result of (2), it is easy to check that $\gamma_{i}$ and $S_{i}$
satisfy the same relations. Thus $\mathcal{W}(\Gamma)\cong
C_{G}(\mathcal{W}(Q))$. ∎

For a given $\alpha\in\mathbb{Z}I$, let $H_{\alpha}=\\{g\in G\mid
g(\alpha)=\alpha\\}$. Then $H_{\alpha}$ is a subgroup of $G$. We denote by
$G_{\alpha}$ a complete set of left coset representatives of $H_{\alpha}$ in
$G$, and let

$\Sigma(\alpha):=\sum_{g\in G_{\alpha}}g(\alpha).$

Obviously, $\Sigma(\alpha)\in(\mathbb{Z}I)^{G}$ and we have

######  Lemma 3.2.

The map $\alpha\mapsto f(\sigma(\alpha))$ induces a surjection
$\pi:\Delta_{Q}\rightarrow\Delta_{\Gamma}$. Moreover, if $f(\sigma(\alpha))$
is a real root, $\alpha$ has to be real and unique up to $G$-orbit.

###### Proof.

First, for any $\omega\in C_{G}(\mathcal{W}(Q))$, we have
$H_{\alpha}=H_{\omega(\alpha)}$ since the action of $C_{G}(\mathcal{W}(Q))$
and the action of $G$ on $\mathbb{Z}I$ is commutative. Thus we can take
$G_{\alpha}=G_{\omega(\alpha)}$ for any $\omega\in C_{G}(\mathcal{W}(Q))$.

We now consider $\beta:=\omega^{\prime}(f(\Sigma(\alpha)))$ with
$\omega^{\prime}\in\mathcal{W}(\Gamma)$. Let $\omega\in C_{G}(\mathcal{W}(Q))$
be the element corresponding to $\omega^{\prime}$ under the isomorphism in
Lemma 3.1(3). Then $\beta=f(\omega(\Sigma(\alpha)))=f(\Sigma(\omega(\alpha)))$
has connected support since the support of $\alpha$ is connected. It is either
positive or negative since $\Sigma$ preserves the partial order $\geq$. Denote
by $\mathcal{O}_{\beta}$ the orbit of $\beta$ under the action of
$\mathcal{W}(\Gamma)$. Then

  * •

if all elements in $\mathcal{O}_{\beta}$ are positive, the element in
$\mathcal{O}_{\beta}$ with minimal height lies in $F_{\Gamma}$;

  * •

if all elements in $\mathcal{O}_{\beta}$ are negative, the element in
$\mathcal{O}_{\beta}$ with maximal height lies in $-F_{\Gamma}$;

  * •

otherwise, there exists a positive number $m$ and $i\in\mathcal{I}$ such that
$m\overline{\varepsilon}_{i}\in\mathcal{O}_{\beta}$.

In the last case, we have $\omega(\alpha)=m\varepsilon_{i^{\prime}}$ for some
$\omega\in\mathcal{W}(Q)$, $i^{\prime}\in\mathcal{O}_{i}$. But
$\omega(\alpha)\in\Delta_{Q}$, we must have $m=1$ and so that
$\overline{\varepsilon}_{i}\in\mathcal{O}_{\beta}$. Thus $\beta$ is a root of
$\Gamma$ and $\pi:\Delta_{Q}\rightarrow\Delta_{\Gamma}$, $\alpha\mapsto
f(\Sigma(\alpha))$ is well-defined.

Now, we prove that the map $\pi$ is surjective. Here we only need to show that
$F_{\Gamma}$ lies in the image of $\pi$. For any $\beta\in F_{\Gamma}$,
$\gamma:=f^{-1}(\beta)$ satisfies

$0\geq(\beta,\overline{\varepsilon}_{i})_{\Gamma}=(\gamma,\Sigma(\varepsilon_{i^{\prime}}))_{Q}=\sum_{g\in
G_{\varepsilon_{i^{\prime}}}}(\gamma,g(\varepsilon_{i^{\prime}}))_{Q}=d_{i}(\gamma,\varepsilon_{i^{\prime}})_{Q}$

for any $i\in\mathcal{I}$ and $i^{\prime}\in\mathcal{O}_{i}$. Thus any
connected component $\alpha$ of $\gamma$ lies in $F_{Q}$ and
$\Sigma(\alpha)=\gamma$. By Lemma 3.1(3) we get the proof. ∎

For any $g\in G$, we have an additive autoequivalence functor

$\displaystyle F_{g}:\quad\mbox{{\bf mod}-}\mathbbm{k}Q$
$\displaystyle\longrightarrow\quad\mbox{{\bf mod}-}\mathbbm{k}Q$
$\displaystyle M$ $\displaystyle\mapsto\qquad{{}^{g}M}$

where the $\mathbbm{k}Q$-module ${{}^{g}M}$ is defined by taking the same
underlying vector space as $M$ with the action $m\cdot\lambda=mg(\lambda)$ for
$m\in M$ and $\lambda\in\mathbbm{k}Q$, and $F_{g}(\psi)=\psi$ for any
homomorphism $\psi:M\rightarrow N$. Let $(M_{i},~{}M_{\alpha})_{i\in
I,\alpha\in E}$ be the $Q$-representation corresponding to $M$. Then the
$Q$-representation ${}^{g}M$ is $(^{g}X_{i},{{}^{g}X}_{\alpha})_{i\in
I,\alpha\in E}$, where ${}^{g}X_{i}=X_{g(i)}$ and
${}^{g}X_{\alpha}=\sum_{\beta}\zeta_{\beta}X_{\beta}$ if
$g(\alpha)=\sum_{\beta}\zeta_{\beta}\beta$, $\beta\in E$,
$\zeta_{\beta}\in\mathbbm{k}$.

A $\mathbbm{k}Q$-module $M$ is said to be $G$-invariant if $F_{g}(M)\cong M$
for any $g\in G$, a $G$-invariant $\mathbbm{k}Q$-module $M$ is said to be
indecomposable $G$-invariant if $M$ is non-zero and $M$ cannot be written as a
direct sum of two non-zero $G$-invariant $\mathbbm{k}Q$-modules. It is known
that $\mathbbm{k}Q$-module $M$ has a $\mathbbm{k}Q\ast G$-module structure if
and only if $M$ is $G$-invariant, and the full subcategory of $\mbox{{\bf
mod}-}\mathbbm{k}Q$ generated by the $G$-invariant $\mathbbm{k}Q$-module is
also a Krull-Schmidt category (see [10]).

For a given $\mathbbm{k}Q$-module $M$, we let $H_{M}:=\\{g\in G\mid
F_{g}(M)\cong M\\}$ and $G_{M}$ be a complete set of left coset
representatives of $H_{M}$ in $G$. Then for each $\mathbbm{k}Q$-module $M$, we
define a $G$-invariant $\mathbbm{k}Q$-module

$\sum(M):=\bigoplus_{g\in G_{M}}{{}^{g}M}.$

It is easy to see that each $G$-invariant $\mathbbm{k}Q$-module has this form.
For each $\mathbbm{k}Q$-module $M$, we denote the dimension vector of $M$ by
the linear combination ${\bf dim}X:=\sum_{i\in
I}\mbox{dim}X_{i}\,\varepsilon_{i}\in\mathbb{Z}I$. It is easy to see that
${\bf dim}F_{g}(M)=g({\bf dim}M)$ for any $g\in G$ and $M\in\mbox{{\bf
mod}-}\mathbbm{k}Q$.

######  Proposition 3.3.

For any indecomposable $G$-invariant $\mathbbm{k}Q$-module $M$, $f({\bf
dim}M)$ is a root of $\Gamma$. Moreover, for any positive real root $\beta$ of
$\Gamma$, there is a unique (up to isomorphism) indecomposable $G$-invariant
$\mathbbm{k}Q$-module $M$ with $\frac{1}{2}({\bf dim}M,{\bf dim}M)_{Q}$
indecomposable summands $($as $\mathbbm{k}Q$-module$)$ such that $f({\bf
dim}M)=\beta$.

###### Proof.

Let $N$ be an indecomposable $\mathbbm{k}Q$-module and $\alpha:={\bf dim}N$.
Then $\sum(N)$ is an indecomposable $G$-invariant $\mathbbm{k}Q$-module with
dimension vector $\sum_{g\in G_{N}}g(\alpha)$. We claim that

$\sum_{g\in G_{N}}g(\alpha)=m\Sigma(\alpha)$

for some positive integer $m$. Indeed, since $H_{N}\subseteq H_{\alpha}$, we
have $|H_{\alpha}|=m|H_{N}|$ and so that $|G_{N}|=m|G_{\alpha}|$ for some
positive integer $m$. Note that

$\sum_{g\in G_{\alpha}}g(\alpha)=\frac{\sum_{g\in
G}g(\alpha)}{|H_{\alpha}|}\quad\hbox{ and }\quad\sum_{g\in
G_{N}}g(\alpha)=\frac{\sum_{g\in G}g(\alpha)}{|H_{N}|},$

we obtain that

${\bf dim}\sum(N)=\sum_{g\in G_{N}}g(\alpha)=m\sum_{g\in
G_{\alpha}}g(\alpha)=m\Sigma(\alpha).$

In particular, if $\alpha$ is a real root of $Q$, then $H_{N}=H_{\alpha}$ and
so that we take $G_{N}=G_{\alpha}$ in this case. Therefore, $f({\bf
dim}\sum(N))\in\Delta_{\Gamma}$. Note that for every indecomposable
$G$-invariant $\mathbbm{k}Q$-module $M$, there is an indecomposable
$\mathbbm{k}Q$-module $N$ such that $M\cong\sum(N)$, we get $f({\bf
dim}M)\in\Delta_{\Gamma}$.

If $\beta:=f({\bf dim}M)$ is a real root with $f({\bf
dim}M)=\omega^{\prime}(\overline{\varepsilon}_{i})$ for some
$\omega^{\prime}\in\mathcal{W}(\Gamma)$ and $i\in\mathcal{I}$, then ${\bf
dim}M=\omega(\Sigma(\varepsilon_{i^{\prime}}))=\Sigma(\omega(\varepsilon_{i^{\prime}}))$
for any $i^{\prime}\in\mathcal{O}_{i}$, where $\omega\in
C_{G}(\mathcal{W}(Q))$ corresponding to $\omega^{\prime}$, by the proof of
Lemma 3.2. Denote by $N$ the unique indecomposable $\mathbbm{k}Q$-module with
${\bf dim}N=\omega(\varepsilon_{i^{\prime}})$, then $M=\sum(N)$ is the unique
indecomposable $G$-invariant $\mathbbm{k}Q$-module satisfying ${\bf
dim}M=\omega(\Sigma(\varepsilon_{i^{\prime}}))$ and $M$ is independent on the
taking of $i^{\prime}\in\mathcal{O}_{i}$. Finally, note that

$\frac{1}{2}({\bf dim}M,{\bf
dim}M)_{Q}=\frac{1}{2}(\Sigma(\varepsilon_{i^{\prime}}),\Sigma(\varepsilon_{i^{\prime}}))_{Q}=d_{i}=|G_{\varepsilon_{i^{\prime}}}|=|G_{N}|,$

we are done. ∎

We suppose now that $G$ is abelian and let

$e:=\sum_{i\in\mathcal{I}}e_{i}\in\mathbbm{k}Q\subseteq\mathbbm{k}Q\ast G,$

where $e_{i}$ is the idempotent element of $\mathbbm{k}Q$ corresponding to
vertex $i\in I$. By the proof of [6, Theorem 1], we know that
$\mathbbm{k}Q\ast G$ is Morita equivalent to $e\mathbbm{k}Q\ast Ge$ and
$e\mathbbm{k}Q\ast Ge\cong\mathbbm{k}\widehat{Q}$. Thus we view the functor

$\displaystyle E:\quad\mbox{{\bf mod}-}\mathbbm{k}Q\ast G$
$\displaystyle\longrightarrow\quad\mbox{{\bf mod}-}\mathbbm{k}\widehat{Q}$
$\displaystyle M$ $\displaystyle\mapsto\qquad eM$

as the equivalence functor between ${\bf mod}$-$\mathbbm{k}Q\ast G$ and ${\bf
mod}$-$\mathbbm{k}\widehat{Q}$. Denote by

$\begin{array}[]{ll}F:=\mathbbm{k}Q\ast G\otimes_{\mathbbm{k}Q}-:&\mbox{{\bf
mod}-}\mathbbm{k}Q\quad\longrightarrow\quad\mbox{{\bf mod}-}\mathbbm{k}Q\ast
G\\\ H:=\mbox{Res}|_{\mathbbm{k}Q}:&\mbox{{\bf mod}-}\mathbbm{k}Q\ast
G\quad\longrightarrow\quad\mbox{{\bf mod}-}\mathbbm{k}Q\end{array}$

Following from [6, Theorem 1.1], $(H,F)$ and $(F,H)$ are adjoint pairs.

Moreover, for any $\mathbbm{k}\widehat{Q}$-module $X$, $HE^{-1}(X)$ is a
$G$-invariant $\mathbbm{k}Q$-module and there is a $\mathbbm{k}Q$-module $M$
such that $HE^{-1}(X)\cong\sum(M)$, where $E^{-1}$ is the quasi-inverse of
$E$. Identifying $X$ with a $\widehat{Q}$-representation
$(X_{i\rho},X_{\alpha})$, we have

$\sum_{\rho\in{\rm irr}G_{i}}X_{i\rho}\cong
e_{i}HE^{-1}(X)e_{i}\cong\bigoplus_{g\in G_{M}}(^{g}M)_{i}.$

Suppose ${\bf
dim}X:=\sum_{(i\rho)\in\widehat{I}}\alpha_{i\rho}\varepsilon_{i\rho}$, then

$\sum_{\rho\in{\rm irr}G_{i}}\alpha_{i\rho}=\sum_{g\in
G_{M}}\mbox{dim}(^{g}M)_{i}=f\Big{(}{\bf dim}\sum(M)\Big{)}_{i}.$

Therefore, the Moriat equivalence and the restriction functor induce a map

$h:\quad\mathbb{Z}\widehat{I}\longrightarrow\mathbb{Z}\mathcal{I}$

given by $h(\alpha)_{i}=\sum_{\rho\in{\rm
irr}G_{i}}\alpha_{i\rho}\overline{\varepsilon}_{i}$ for any
$\alpha=\sum_{(i\rho)\in\widehat{I}}\alpha_{i\rho}\varepsilon_{i\rho}\in\mathbb{Z}\widehat{I}$.
The restriction of $h$ to the root system $\Delta_{\widehat{Q}}$ is also
denoted by $h$. Then $h:\Delta_{\widehat{Q}}\rightarrow\Delta_{\Gamma}$ is
well-defined since $X$ is an indecomposable $\mathbbm{k}\widehat{Q}$-module if
and only if $M$ is an indecomposable $\mathbbm{k}Q$-module. By Proposition
3.3, we have

######  Corollary 3.4.

For any indecomposable $\widehat{Q}$-representation $X$, $h({\bf dim}X)$ is a
positive root of $\Gamma$.

Up to now, we have obtained the map
$h:\mathbb{Z}\widehat{I}\rightarrow\mathbb{Z}\mathcal{I}$ and have shown the
half of Theorem 1.1(1). Before completing the proof of Theorem 1.1, we should
define an action of $G$ on $\mathbbm{k}\widehat{Q}$ and give the dual between
$(Q,G)$ and $(\widehat{Q},G)$. In the following subsection, we first describe
the duality of $(Q,G)$.

3.2. We write the abelian group $G$ as the product of some finite cyclic
group, i.e.,

$G=\langle g_{1}\rangle\times\langle g_{2}\rangle\times\cdots\times\langle
g_{n}\rangle,$

where the order of $g_{i}$ is $m_{i}$ for $1\leq i\leq n$. Then
$|G|=m_{1}m_{2}\cdots m_{n}$.

We now define an action of $G$ on $\widehat{Q}$. Since $G$ is abelian, all the
characters $\chi$ of $G$ are linear. The set of all the characters of $G$ is
an abelian group with the multiplication

$\chi\chi^{\prime}(g)=\chi(g)\chi^{\prime}(g),$

for all $g\in G$. We denote this group by $\widetilde{G}$. Setting
$\varphi:G\rightarrow\widetilde{G}$ by

$\varphi(g)=\chi_{g},\quad\chi_{g}(g^{\prime})=\xi_{1}^{t_{1}s_{1}}\xi_{2}^{t_{2}s_{2}}\cdots\xi_{n}^{t_{n}s_{n}}$

if $g=g_{1}^{t_{1}}g_{2}^{t_{2}}\cdots g_{n}^{t_{n}}$ and
$g^{\prime}=g_{1}^{s_{1}}g_{2}^{s_{2}}\cdots g_{n}^{s_{n}}$, where $\xi_{i}$
is a primitive $m_{i}$-th root of unity. It is easy to see that $\varphi$ is a
group isomorphism. By [18], we define a linear action of $G$ on
$\mathbbm{k}Q\ast G$ by setting

$g(\lambda h)=\chi_{g}(h)\lambda h,$

for all $g\in G,\lambda h\in\mathbbm{k}Q\ast G$. Then
$G\subseteq\mbox{Aut}(\mathbbm{k}Q\ast G)$. By [18, Proposition 5.1], we have

######  Proposition 3.5.

The map $\psi:(\mathbbm{k}Q\ast G)\ast G\rightarrow{\rm
End}_{\mathbbm{k}Q}(\mathbbm{k}Q\ast G)$ defined by

$\psi(\lambda gh)(\mu h^{\prime})=\chi_{h}(h^{\prime})\lambda g\mu h^{\prime}$

is an algebra isomorphism. It follows that $(\mathbbm{k}Q\ast G)\ast G$ is
Morita equivalent to $\mathbbm{k}Q$.

Since $e\mathbbm{k}Q\ast Ge\cong\mathbbm{k}\widehat{Q}$ and the action of $G$
on $\mathbbm{k}Q\ast G$ stabilizes $e$, the action of $G$ on $\mathbbm{k}Q\ast
G$ naturally induces an action of $G$ on $\mathbbm{k}\widehat{Q}$ such that
$G\subseteq\mbox{Aut}(\mathbbm{k}\widehat{Q})$. Therefore, we get a skew group
algebra $\mathbbm{k}\widehat{Q}\ast G$ under this action. Let
$\widehat{\widehat{Q}}$ be the generalized Mckay quiver of $(\widehat{Q},G)$.
Then, there is a Morita equivalence between $\mathbbm{k}\widehat{Q}\ast G$ and
$\mathbbm{k}\widehat{\widehat{Q}}$ by Theorem 2.1.

######  Proposition 3.6.

Let $\widehat{Q}$ be the generalized McKay quiver of $(Q,G)$ under the action
of $G$ defined as above. Then the generalized McKay quiver
$\widehat{\widehat{Q}}$ of $(\widehat{Q},G)$ coincides with $Q$.

Thus we get the dual between $(Q,G)$ and $(\widehat{Q},G)$. Now, for the
relationship between quivers $Q$ and $\widehat{Q}$, and the action of $G$ on
$\widehat{Q}$, we give some more description. Note that the stabilizer $G_{i}$
of $i\in I$ has the form

$G_{i}=\langle g_{1}^{d_{i_{1}}}\rangle\times\langle
g_{2}^{d_{i_{2}}}\rangle\times\cdots\times\langle g_{n}^{d_{i_{n}}}\rangle,$

where

$\nu_{i_{j}}:=|\langle
g_{j}^{d_{i_{j}}}\rangle|=\frac{m_{i}}{d_{i_{j}}},\qquad 1\leq j\leq n,$

and so that

$d_{i}=|\mathcal{O}_{i}|=\frac{|G|}{|G_{i}|}=d_{i_{1}}\times
d_{i_{2}}\times\cdots\times d_{i_{n}}.$

We set

$\displaystyle e_{(i,s_{i_{1}},s_{i_{2}},\cdots,s_{i_{n}})}$
$\displaystyle\qquad=\frac{1}{|G_{i}|}\sum_{j_{1}=0}^{\nu_{i_{1}}-1}\sum_{j_{2}=0}^{\nu_{i_{2}}-1}\cdots\sum_{j_{n}=0}^{\nu_{i_{n}}-1}\xi_{1}^{d_{i_{1}}j_{1}s_{i_{1}}}\xi_{2}^{d_{i_{2}}j_{2}s_{i_{2}}}\cdots\xi_{n}^{d_{i_{n}}j_{n}s_{i_{n}}}g_{1}^{d_{i_{1}}j_{1}}g_{2}^{d_{i_{2}}j_{2}}\cdots
g_{n}^{d_{i_{n}}j_{n}}.$

Then one can check that
$\big{\\{}e_{(i,s_{i_{1}},s_{i_{2}},\cdots,s_{i_{n}})}\mid
s_{i_{j}}\in\mathbb{Z}/\nu_{i_{j}}\mathbb{Z}\mbox{ for all }1\leq j\leq
n\big{\\}}$ is a complete set of primitive orthogonal idempotents of
$\mathbbm{k}[G_{i}]$.

It is obvious that

$g_{j}(e_{(i,s_{i_{1}},s_{i_{2}},\cdots,s_{i_{n}})})=e_{(i,s_{i_{1}},\cdots,s_{i_{j-1}},s^{\prime}_{i_{j}},s_{i_{j+1}},\cdots,s_{i_{n}})}$

for any $1\leq j\leq n,$ where
$s^{\prime}_{i_{j}}\in\mathbb{Z}/\nu_{i_{j}}\mathbb{Z}$ and
$s^{\prime}_{i_{j}}\equiv s_{i_{j}}+1\mod{\nu_{i_{j}}}$. Since for each
idempotent $e_{(i,s_{i_{1}},s_{i_{2}},\cdots,s_{i_{n}})}$, there is a unique
corresponding one dimensional irreducible representation $\rho$ of $G_{i}$
defined by the group homomorphism $\phi_{\rho}:G_{i}\rightarrow\mathbbm{k}$,
$g_{j}^{d_{i_{j}}}\mapsto\xi^{d_{i_{j}}s_{i_{j}}}$, for $1\leq j\leq n$. Thus
we can index the vertices set $\widehat{I}$ by some sequences
$(i,s_{i_{1}},s_{i_{2}},\cdots,s_{i_{n}})$, i.e.,

$\widehat{I}=\left\\{(i,s_{i_{1}},s_{i_{2}},\cdots,s_{i_{n}})\mid
i\in\mathcal{I},s_{i_{j}}\in\mathbb{Z}/\nu_{i_{j}}\mathbb{Z}\mbox{ for all
}1\leq j\leq n\right\\}.$

Then the action of $G$ on $\widehat{I}$ is clearly and so that the orbit of
$(i,\rho)\in\widehat{I}$ has the form

$\left\\{(i,s_{i_{1}},s_{i_{2}},\cdots,s_{i_{n}})\mid
s_{i_{j}}\in\mathbb{Z}/\nu_{i_{j}}\mathbb{Z}\mbox{ for }1\leq j\leq
n\\}=\\{(i,\rho)\mid\rho\in\mbox{irr}G_{i}\right\\}$

for some $i\in\mathcal{I}$. Furthermore, it is easy to see that if the action
of $G$ on $\mathbbm{k}Q$ is admissible, then so is on
$\mathbbm{k}\widehat{Q}$.

For any $i,j\in I$, we consider the group $G_{ij}:=G_{i}\cap G_{j}=\langle
g_{1}^{t_{1}}\rangle\times\langle
g_{2}^{t_{2}}\rangle\times\cdots\times\langle g_{n}^{t_{n}}\rangle$, where
$t_{l}$ is the least common multiple of $d_{i_{l}}$ and $d_{j_{l}}$ for $1\leq
l\leq n$. Note that the vector space $E_{ij}$ spanned by arrows
$\alpha:i\rightarrow j$ in $Q$ is a $\mathbbm{k}[G_{ij}]$-bimodule, we can
find a basis of $E_{ij}$ such that the action of $G_{ij}$ is diagonal. That
is, if $g=g_{1}^{t_{1}}g_{2}^{t_{2}}\cdots
g_{n}^{t_{n}}\in\mathbbm{k}[G_{ij}]$, then for any basis element
$\alpha^{\prime}\in E_{ij}$,

$g(\alpha^{\prime})=\xi_{1}^{t_{1}r_{1}}\xi_{2}^{t_{2}r_{2}}\cdots\xi_{n}^{t_{n}r_{n}}\alpha^{\prime}$

for some $r_{1},r_{2},\cdots,r_{n}\in\mathbb{Z}$. Since $G_{ij}$ is abelian,
the number of the basis elements of $E_{ij}$ is just the number of arrows from
$i$ to $j$ in $Q$. Moreover, it is easy to see that the $t_{1}t_{2}\cdots
t_{n}$ elements

$\alpha^{\prime},~{}g_{1}(\alpha^{\prime}),\cdots,~{}g_{n}(\alpha^{\prime}),~{}g^{2}_{1}(\alpha^{\prime}),~{}g_{1}g_{2}(\alpha^{\prime}),\cdots,~{}g^{2}_{n}(\alpha^{\prime}),\cdots\cdots,~{}g^{t_{1}-1}_{1}g^{t_{2}-1}_{2}\cdots
g^{t_{n}-1}_{n}(\alpha^{\prime})$

are linearly independent. That is, for any arrow $\alpha:i\rightarrow j$ in
$Q$, there are $t_{1}t_{2}\cdots t_{n}$ arrows in its orbit.

On the other hand, we can calculate that

$\displaystyle
e_{(j,s_{j_{1}},s_{j_{2}},\cdots,s_{j_{n}})}\alpha^{\prime}e_{(i,s_{i_{1}},s_{i_{2}},\cdots,s_{i_{n}})}$
$\displaystyle\qquad=\frac{d_{i}d_{j}}{|G|^{2}}\sum_{p_{1}=0}^{\nu_{i_{1}}-1}\cdots\sum_{p_{n}=0}^{\nu_{i_{n}}-1}\sum_{q_{1}=0}^{\nu_{j_{1}}-1}\cdots\sum_{q_{n}=0}^{\nu_{j_{n}}-1}\xi_{1}^{d_{i_{1}}p_{1}s_{i_{1}}+d_{j_{1}}q_{1}s_{j_{1}}}\cdots\xi_{n}^{d_{i_{n}}p_{n}s_{i_{n}}+d_{j_{n}}q_{n}s_{j_{n}}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad
g_{1}^{d_{j_{1}}q_{1}}\cdots
g_{n}^{d_{j_{n}}q_{n}}(\alpha^{\prime})g_{1}^{d_{i_{1}}p_{1}+d_{j_{1}}q_{1}}\cdots
g_{n}^{d_{i_{n}}p_{n}+d_{j_{n}}q_{n}}.$

We write

$\displaystyle d_{i_{l}}p_{l}$ $\displaystyle=$ $\displaystyle
P_{l}t_{l}+d_{i_{l}}p^{\prime}_{l},\qquad\hbox{ where }0\leq
P_{l}<\frac{m_{l}}{t_{l}},\quad 0\leq p^{\prime}_{l}<\frac{t_{l}}{d_{i_{l}}},$
$\displaystyle d_{j_{l}}q_{l}$ $\displaystyle=$ $\displaystyle
P^{\prime}_{l}t_{l}+d_{j_{l}}q^{\prime}_{l},\qquad\hbox{ where }0\leq
P^{\prime}_{l}<\frac{m_{l}}{t_{l}},\quad 0\leq
q^{\prime}_{l}<\frac{t_{l}}{d_{j_{l}}},$ $\displaystyle d_{i_{l}}k_{l}$
$\displaystyle\equiv$
$\displaystyle(P_{l}+P^{\prime}_{l})t_{l}+d_{i_{l}}p^{\prime}_{l}\mod{m_{i}},\qquad\hbox{
where }0\leq k_{l}<\nu_{i_{l}},$

for all $0\leq l\leq n$. Then the right side of the equation becomes

$\displaystyle\frac{d_{i}d_{j}}{|G|^{2}}$
$\displaystyle\Bigg{(}\sum_{P^{\prime}_{1}=0}^{\frac{m_{1}}{t_{1}}-1}\xi_{1}^{P^{\prime}_{1}t_{1}(r_{1}+s_{j_{1}}-s_{i_{1}})}\Bigg{)}\cdots\Bigg{(}\sum_{P^{\prime}_{n}=0}^{\frac{m_{n}}{t_{n}}-1}\xi_{n}^{P^{\prime}_{n}t_{n}(r_{n}+s_{j_{n}}-s_{i_{n}})}\Bigg{)}$
$\displaystyle\quad\Bigg{(}\sum_{k_{1}=0}^{\nu_{i_{1}}-1}\cdots\sum_{k_{n}=0}^{\nu_{i_{n}}-1}\sum_{q^{\prime}_{1}=0}^{\frac{t_{1}}{d_{j_{1}}}-1}\cdots\sum_{q^{\prime}_{n}=0}^{\frac{t_{n}}{d_{j_{n}}}-1}\xi_{1}^{d_{i_{1}}k_{1}s_{i_{1}}+d_{j_{1}}q^{\prime}_{1}s_{j_{1}}}\cdots\xi_{n}^{d_{i_{n}}k_{n}s_{i_{n}}+d_{j_{n}}q^{\prime}_{n}s_{j_{n}}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad
g_{1}^{d_{j_{1}}q^{\prime}_{1}}\cdots
g_{n}^{d_{j_{n}}q^{\prime}_{n}}(\alpha^{\prime})g_{1}^{d_{i_{1}}k_{1}+d_{j_{1}}q^{\prime}_{1}}\cdots
g_{n}^{d_{i_{n}}k_{n}+d_{j_{n}}q^{\prime}_{n}}\Bigg{)}.$

Note that

$\displaystyle\Big{\\{}g_{1}^{d_{j_{1}}q^{\prime}_{1}}\cdots
g_{n}^{d_{j_{n}}q^{\prime}_{n}}(\alpha^{\prime})g_{1}^{d_{i_{1}}k_{1}+d_{j_{1}}q^{\prime}_{1}}$
$\displaystyle\cdots g_{n}^{d_{i_{n}}k_{n}+d_{j_{n}}q^{\prime}_{n}}$
$\displaystyle\mid 0\leq k_{l}<\nu_{i_{l}},~{}0\leq
q^{\prime}_{l}<\frac{t_{l}}{d_{j_{l}}}\mbox{ for }1\leq l\leq n\Big{\\}}$

is a linearly independent set. We obtain that
$e_{(j,s_{j_{1}},s_{j_{2}},\cdots,s_{j_{n}})}\alpha^{\prime}e_{(i,s_{i_{1}},s_{i_{2}},\cdots,s_{i_{n}})}\neq
0$ if and only if $s_{i_{l}}\equiv s_{j_{l}}+r_{l}\mod{\frac{m_{l}}{t_{l}}}$
for all $0\leq l\leq n$. It follows that there are $\frac{t_{1}\cdots
t_{n}|G|}{d_{i}d_{j}}$ arrows in $\widehat{Q}$ for each arrow
$\alpha:i\rightarrow j$ in $Q$.

Denote by $\widehat{A}=(a_{(i\rho)(j\sigma)})_{\widehat{I}\times\widehat{I}}$
the Cartan matrix of $\widehat{Q}$, by $\widehat{\Gamma}$ the valued quiver
corresponding to $(\widehat{Q},G)$ and by
$\widehat{C}=(\widehat{c}_{ij})_{\mathcal{I}\times\mathcal{I}}=\widehat{D}^{-1}\widehat{B}$
the generalized Cartan matrix of $\widehat{\Gamma}$, where
$\widehat{B}=(\widehat{b}_{ij})_{\mathcal{I}\times\mathcal{I}}$ is symmetric,
$\widehat{D}=\mbox{diag}(\widehat{d}_{i})$ is diagonal. Then

$\frac{1}{t_{1}\cdots t_{n}}\sum_{i^{\prime}\in\mathcal{O}_{i}\atop
j^{\prime}\in\mathcal{O}_{j}}a_{i^{\prime}j^{\prime}}=\frac{d_{i}d_{j}}{t_{1}\cdots
t_{n}|G|}\sum_{\rho\in{\rm irr}G_{i}\atop\sigma\in{\rm
irr}G_{j}}a_{(i\rho)(j\sigma)}.$

It follows that $\widehat{b}_{ij}=\frac{|G|}{d_{i}d_{j}}b_{ij}$,
$\widehat{D}=|G|D^{-1}$, $\widehat{B}=|G|D^{-1}BD^{-1}$ and
$\widehat{C}=(\widehat{D})^{-1}\widehat{B}=BD^{-1}=C^{T}$, the transpose of
$C$. Therefore $\Gamma$ and $\widehat{\Gamma}$ are dual valued graph in the
sense of [13].

######  Remark 3.7.

If $G\subseteq\mbox{Aut}(\mathbbm{k}Q)$ is a finite abelian group, we have
given the dual of $(Q,G)$ and $(\widehat{Q},G)$ (see Proposition 3.6).
However, for a non-abelian group $G\subseteq\mbox{Aut}(\mathbbm{k}Q)$, the
conclusion does not hold in general. For example, Let $Q$ be the quiver

It is well-known that the quiver automorphism group of $Q$ is the group
$S_{3}$. Accordingly, we obtain the generalized McKay quiver $\widehat{Q}$ of
$(Q,S_{3})$ as follows

One can check that there does not exist a subgroup $G^{\prime}$ of
$\mbox{Aut}(\mathbbm{k}\widehat{Q})$ such that the generalized McKay quiver of
$(\widehat{Q},G^{\prime})$ is $Q$.

But if the action of $G$ is “good”, there exists the duality still. For
example, we consider the finite non-abelian group

$G=\left\langle a,b\mid a^{3}=b^{2},b^{4}=1,aba=b\right\rangle$

and the quiver $Q$:

$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\begin{picture}(50.0,32.0)\put(24.0,1.0){$\bullet$}
\put(24.0,10.0){$\bullet$} \put(37.0,24.0){$\bullet$}
\put(10.0,24.0){$\bullet$} \put(44.0,30.0){$\bullet$}
\put(2.0,30.0){$\bullet$} \put(25.0,9.0){\vector(0,-1){6.0}}
\put(26.0,13.0){\vector(1,1){10.0}} \put(22.5,13.0){\vector(-1,1){10.0}}
\put(13.0,24.0){\vector(1,-1){10.0}} \put(37.0,22.5){\vector(-1,-1){10.0}}
\put(14.5,26.0){\vector(1,0){20.0}} \put(34.5,25.0){\vector(-1,0){20.0}}
\put(9.5,26.0){\vector(-4,3){5.0}} \put(39.0,26.0){\vector(4,3){5.0}}
\put(22.0,26.0){$\alpha^{\ast}$} \put(26.0,23.0){$\alpha$}
\put(18.0,20.0){$\beta$} \put(16.0,14.0){$\beta^{\ast}$}
\put(28.0,17.0){$\gamma$} \put(34.0,17.0){$\gamma^{\ast}$}
\put(39.0,22.0){\small$1$} \put(46.0,28.0){\small$1^{\prime}$}
\put(8.0,22.0){\small$2$} \put(0.0,28.0){\small$2^{\prime}$}
\put(27.0,9.0){\small$3$} \put(27.0,0.0){\small$3^{\prime}$}
\put(21.0,6.0){$\sigma_{3}$} \put(8.0,28.0){$\sigma_{2}$}
\put(38.0,28.0){$\sigma_{1}$} \end{picture}.$

The action of $G$ is given by

| $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{1^{\prime}}$ | $e_{2^{\prime}}$ | $e_{3^{\prime}}$ | $\alpha$ | $\alpha^{\ast}$
---|---|---|---|---|---|---|---|---
$a$ | $e_{2}$ | $e_{3}$ | $e_{1}$ | $e_{2^{\prime}}$ | $e_{3^{\prime}}$ | $e_{1^{\prime}}$ | $\beta$ | $\beta^{\ast}$
$b$ | $e_{1}$ | $e_{3}$ | $e_{2}$ | $e_{1^{\prime}}$ | $e_{2^{\prime}}$ | $e_{2^{\prime}}$ | $-\gamma^{\ast}$ | $\gamma$
| $\beta$ | $\beta^{\ast}$ | $\gamma$ | $\gamma^{\ast}$ | $\sigma_{1}$ | $\sigma_{2}$ | $\sigma_{3}$
---|---|---|---|---|---|---|---
$a$ | $\gamma$ | $\gamma^{\ast}$ | $\alpha$ | $\alpha^{\ast}$ | $\sigma_{2}$ | $\sigma_{3}$ | $\sigma_{1}$
$b$ | $-\beta^{\ast}$ | $\beta$ | $-\alpha^{\ast}$ | $\alpha$ | $\sigma_{1}$ | $\sigma_{3}$ | $\sigma_{2}$

where $e_{i}$ is the idempotent element of $\mathbbm{k}Q$ corresponding to
vertex $i$, $i\in\\{1,2,3,1^{\prime},2^{\prime},$ $3^{\prime}\\}$. By direct
calculation, one see that the generalized McKay quiver of $(Q,G)$ is as
follows.

Now, we define an action of $G$ on $\mathbbm{k}\widehat{Q}$ by setting

| $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{1^{\prime}}$ | $e_{2^{\prime}}$ | $e_{3^{\prime}}$ | $e_{4^{\prime}}$ | $\alpha_{1}$ | $\alpha_{2}$ | $\alpha_{3}$ | $\alpha_{4}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$a$ | $e_{3}$ | $e_{4}$ | $e_{1}$ | $e_{2}$ | $e_{3^{\prime}}$ | $e_{4^{\prime}}$ | $e_{1^{\prime}}$ | $e_{4^{\prime}}$ | $\xi^{2}\alpha_{3}$ | $\xi^{4}\alpha_{4}$ | $\xi^{2}\alpha_{1}$ | $\xi^{4}\alpha_{2}$
$b$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{1}$ | $e_{2^{\prime}}$ | $e_{3^{\prime}}$ | $e_{4^{\prime}}$ | $e_{1^{\prime}}$ | $\alpha_{2}$ | $\alpha_{3}$ | $\alpha_{4}$ | $\alpha_{1}$
| $\alpha^{\ast}_{1}$ | $\alpha^{\ast}_{2}$ | $\alpha^{\ast}_{3}$ | $\alpha^{\ast}_{4}$ | $\sigma_{1}$ | $\sigma_{2}$ | $\sigma_{3}$ | $\sigma_{4}$
---|---|---|---|---|---|---|---|---
$a$ | $\xi^{2}\alpha^{\ast}_{3}$ | $\xi^{4}\alpha^{\ast}_{4}$ | $\xi^{2}\alpha^{\ast}_{1}$ | $\xi^{4}\alpha^{\ast}_{2}$ | $\sigma_{3}$ | $\sigma_{4}$ | $\sigma_{1}$ | $\sigma_{2}$
$b$ | $\alpha^{\ast}_{2}$ | $\alpha^{\ast}_{3}$ | $\alpha^{\ast}_{4}$ | $\alpha^{\ast}_{1}$ | $\sigma_{2}$ | $\sigma_{3}$ | $\sigma_{4}$ | $\sigma_{1}$

where $\xi$ is a primitive $6$-th root of unity. Then, one can check that
$\widehat{\widehat{Q}}=Q$.

3.3. Consider the admissible action of finite abelian group $G$ on
$\mathbbm{k}\widehat{Q}$ induced from the action of $G$ on $\mathbbm{k}Q$ as
the discussion above, we set

$\begin{array}[]{ll}F^{\prime}:=(\mathbbm{k}Q\ast G)\ast
G\otimes_{\mathbbm{k}Q\ast G}-:&\mbox{{\bf mod}-}\mathbbm{k}Q\ast
G\longrightarrow\mbox{{\bf mod}-}(\mathbbm{k}Q\ast G)\ast G\\\
H^{\prime}:=\mbox{Res}|_{\mathbbm{k}Q\ast G}:&\mbox{{\bf
mod}-}(\mathbbm{k}Q\ast G)\ast G\longrightarrow\mbox{{\bf
mod}-}\mathbbm{k}Q\ast G\end{array}$

Similar to the functors $F$ and $H$, one can check that
$(H^{\prime},F^{\prime})$ and $(F^{\prime},H^{\prime})$ are adjoint pairs.
Note that the Morita equivalence $\mbox{{\bf
mod}-}\mathbbm{k}Q\rightarrow\mbox{{\bf mod}-}(\mathbbm{k}Q\ast G)\ast G$ is
given by $\mathcal{M}:={{}_{(\mathbbm{k}Q\ast G)\ast G}\mathbbm{k}Q\ast
G}\otimes_{\mathbbm{k}Q}-$, we have

######  Lemma 3.8.

There are natural isomorphisms

$F\cong H^{\prime}\mathcal{M}\qquad\mbox{ and }\qquad
F^{\prime}\cong\mathcal{M}H.$

###### Proof.

First, $H^{\prime}\mathcal{M}={{}_{\mathbbm{k}Q\ast G}\mathbbm{k}Q\ast
G}\otimes_{\mathbbm{k}Q}-=F$ is clear. Next, since $(H^{\prime},F^{\prime})$
is an adjoint pair, for any $\mathbbm{k}Q$-module $X$ and $\mathbbm{k}Q\ast
G$-module $Y$, we have

$\displaystyle\mbox{Hom}_{\mathbbm{k}Q}(X,\mathcal{M}^{-1}F^{\prime}(Y))$
$\displaystyle\cong\mbox{Hom}_{(\mathbbm{k}Q\ast G)\ast
G}(\mathcal{M}(X),F^{\prime}(Y))$
$\displaystyle\cong\mbox{Hom}_{\mathbbm{k}Q\ast
G}(H^{\prime}\mathcal{M}(X),Y)\cong\mbox{Hom}_{\mathbbm{k}Q\ast G}(F(X),Y).$

This implies that $(F,\mathcal{M}^{-1}F^{\prime})$ is an adjoint pair and so
that $H\cong\mathcal{M}^{-1}F^{\prime}$, $F^{\prime}\cong\mathcal{M}H$. ∎

By Lemma 3.8 and [18, Proposition 1.8], we have the following proposition
immediately.

######  Proposition 3.9.

Let $X$ and $Y$ be indecomposable $\mathbbm{k}Q\ast G$-modules. Then

(1) $FH(X)\cong H^{\prime}F^{\prime}(X)\cong\bigoplus_{g\in G}{{}^{g}X}$;

(2) $H(X)\cong H(Y)$ if and only if $F^{\prime}(X)\cong F^{\prime}(Y)$, if and
only if $Y\cong{{}^{g}X}$ for some $g\in G$;

(3) $H(X)$ (or $F^{\prime}(X)$) has exactly $|H_{X}|$ indecomposable summands.

######  Remark 3.10.

Consider the action of $G$ on $\mathbbm{k}Q\ast G$, we denote by

$H_{X}:=\\{g\in G\mid F_{g}(X)\cong X\\}$

and by $G_{X}$ a complete set of left coset representatives of $H_{X}$ in $G$,
for any $X\in\mbox{{\bf mod}-}\mathbbm{k}Q\ast G$. In [10], we have shown that
the number of indecomposable summands of $F^{\prime}(X)$ is just $|H_{X}|$
whenever $G$ is abelian (see [10, Theorem 1.2]). This means that $H(X)$ has
$|H_{X}|$ indecomposable summands. Note that $H(X)$ is an indecomposable
$G$-invariant $\mathbbm{k}Q$-module, there exists a unique indecomposable
$\mathbbm{k}Q$-module $M$ such that $H(X)\cong\sum(M)$. Therefore, we have
$|H_{X}|=|G_{M}|$ and $|G_{X}|=|H_{M}|$. Following from Proposition 3.9(2),
for an indecomposable $\mathbbm{k}Q$-module $M$, there are $|G_{X}|=|H_{M}|$
non-isomorphic indecomposable $\mathbbm{k}Q\ast G$-module structures on
$\sum(M)$. This coincides with the result in [10].

For the generalized McKay quiver $\widehat{Q}$, we denote by
$(-,-)_{\widehat{Q}}$ the bilinear form on $\mathbb{Z}\widehat{I}$ determined
by $\widehat{A}$, by $\Delta_{\widehat{Q}}$ the root system of $\widehat{Q}$
with simple roots $\varepsilon_{i\rho}$, $(i,\rho)\in\widehat{I}$, and by
$\mathcal{W}(\widehat{Q})$ the Weyl group of $\widehat{Q}$ with simple
reflections $r_{i\rho}$, $(i,\rho)\in\widehat{I}$. Consider the map
$h:\mathbb{Z}\widehat{I}\rightarrow\mathbb{Z}\mathcal{I}$ defined above, we
have

######  Lemma 3.11.

Let $\widehat{S}_{i}:=\prod_{\rho\in{\rm irr}G_{i}}r_{i\rho}$ for
$i\in\mathcal{I}$. Then, for each $i\in\mathcal{I}$ and
$\beta=\sum_{(i,\rho)\in\widehat{I}}\beta_{i\rho}\varepsilon_{i\rho}\in\mathbb{Z}\widehat{I}$,
we have

(1) $(h(\beta),\overline{\varepsilon}_{i})_{\Gamma}=d_{i}\sum_{\rho\in{\rm
irr}G_{i}}(\beta,\varepsilon_{i\rho})_{\widehat{Q}}$;

(2) $h(\widehat{S}_{i}(\beta))=\gamma_{i}(h(\beta))$;

(3) the map $\gamma_{i}\mapsto\widehat{S}_{i}$ induces an isomorphism
$\mathcal{W}(\Gamma)\stackrel{{\scriptstyle\simeq}}{{\longrightarrow}}C_{G}(\mathcal{W}(\widehat{Q}))$,
the set of elements in $\mathcal{W}(\widehat{Q})$ commuting with the action of
$G$.

###### Proof.

(1) By the dual between $(Q,G)$ and $(\widehat{Q},G)$, we obtain that

$b_{ij}=\sum_{i^{\prime}\in\mathcal{O}_{i}\atop
j^{\prime}\in\mathcal{O}_{j}}a_{i^{\prime}j^{\prime}}=\frac{d_{i}d_{j}}{|G|}\sum_{\rho\in{\rm
irr}G_{i}\atop\sigma\in{\rm irr}G_{j}}a_{(i\rho)(j\sigma)},$

and so that

$b_{ij}=d_{i}\sum_{\rho\in{\rm irr}G_{i}}a_{(i\rho)(j\sigma)}$

for any $\sigma\in{\rm irr}G_{j}$. Therefore, we get

$(h(\beta),\overline{\epsilon}_{i})_{\Gamma}=\sum_{i,j\in\mathcal{I}}b_{ij}h(\beta)_{j}=d_{i}\sum_{\rho\in{\rm
irr}G_{i}\atop\sigma\in{\rm
irr}G_{j}}a_{(i\rho)(j\sigma)}\beta_{j\sigma}=d_{i}\sum_{\rho\in{\rm
irr}G_{i}}(\beta,\varepsilon_{i\rho})_{\widehat{Q}}.$

(2) Firstly, $\widehat{S}_{i}$ is well-defined since the action of $G$ on
$\widehat{Q}$ is admissible. Secondly, it is easy to check that the bilinear
form $(-,-)_{\widehat{Q}}$ is $G$-invariant and $\widehat{S}_{i}$ commutes
with the action of $G$. Thus, we have

$h(\widehat{S}_{i}(\beta))=h(\beta)-\sum_{\rho\in{\rm
irr}G_{i}}(\beta,\varepsilon_{i\rho})_{\widehat{Q}}\overline{\varepsilon}_{i}=h(\beta)-\frac{1}{d_{i}}(h(\beta),\overline{\varepsilon}_{i})_{\Gamma}\overline{\varepsilon}_{i}=\gamma_{i}(h(\beta)).$

(3) By induction on the length, one can check that
$C_{G}(\mathcal{W}(\widehat{Q}))$ is generated by $\widehat{S}_{i}$,
$i\in\mathcal{I}$. Following from (2), we get $\gamma_{i}\mapsto S_{i}$
induces an isomorphism. ∎

We are in a position to complete the proof of Theorem 1.1. We have shown that
for any positive root $\alpha\in\Delta_{\Gamma}$, there exists an
indecomposable $\widehat{Q}$-representation X such that $h({\bf
dim}X)=\alpha$. Moreover, if $\alpha$ is real, the number of $X$ (up to
isomorphism) can be determined. Applying the technique in [12, Proposition
15], we have

######  Proposition 3.12.

The map $h:\Delta_{\widehat{Q}}\rightarrow\Delta_{\Gamma}$ is a surjection. If
$\alpha\in\Delta_{\Gamma}$ is a positive real root, then there is a unique
$G$-orbit of roots mapping to $\alpha$, and all of which are real.

###### Proof.

Firstly, by Corollary 3.4, the map
$h:\Delta_{\widehat{Q}}\rightarrow\Delta_{\Gamma}$ is well-defined. To show
the surjectivity, we need to fine the preimages of all the fundamental roots
in $\Delta_{\widehat{Q}}$.

We suppose that $\Gamma$ is connected. Then, for any $\alpha\in F_{\Gamma}$,
we consider the set

${\mathcal{R}}:=\\{\beta\in\Delta_{\widehat{Q}}\mid\beta\mbox{ is positive and
}h(\beta)\leq\alpha\\}.$

Since ${\mathcal{R}}$ is finite and non-empty, we take an element $\beta$ with
maximal height. Suppose that $h(\beta)_{i}<\alpha_{i}$ for all
$i\in\mathcal{I}$, then for any $\rho\in\mbox{irr}G_{i}$,
$h(\beta+\varepsilon_{i\rho})=h(\beta)+\overline{\varepsilon}_{i}\leq\alpha$.
By the maximality of $\beta$, $\beta+\varepsilon_{i\rho}$ is not a root and so
that $(\beta,\varepsilon_{i\rho})_{\widehat{Q}}\geq 0$. Thus
$(h(\beta),\overline{\varepsilon}_{i})_{\Gamma}\geq 0$ for all
$i\in\mathcal{I}$. We conclude that $h(\beta)$ and $\alpha$ have the same
support, for otherwise, we can find such a vertex $(i,\rho)$ adjacent to the
support of $\beta$ such that $(\beta,\varepsilon_{i\rho})_{\widehat{Q}}<0$.

We take $\alpha\in F_{\Gamma}$ such that the support of $\alpha$ is
${\mathcal{I}}$, and set

$\Phi:=\\{i\in\mathcal{I}\mid h(\beta)_{i}=\alpha_{i}\\}.$

If $\Phi$ is the empty set, then $\beta+\varepsilon_{i\rho}$ is not a root for
any vertex $(i,\rho)\in\widehat{Q}$, and so that the connected component of
$\widehat{Q}$ which $\beta$ lies in is Dynkin (see [13, Proposition 4.9]).
Therefore, $\widehat{Q}$ must be a disjoint union of copies of this Dynkin
quiver, all in a single $G$-orbit. Thus $\widehat{Q}$ and $Q$ are
representation finite [18], $\Gamma$ is a connected Dynkin diagram. This
contradicts to that $\alpha$ is a imaginary root.

It follows that $\Phi$ is non-empty. We denote by $\widetilde{\Phi}$ the full
subgraph of $\Gamma$ determined by $\Phi$. Let $T$ be a non-empty connected
component of $\Gamma-\widetilde{\Phi}$, and let $\widetilde{\beta}$ be the
restriction of $h(\beta)$ to $T$. If $T\neq\emptyset$, then for all vertices
$j\in T$, we have
$(\widetilde{\beta},\overline{\varepsilon}_{j})_{T}\geq(h(\beta),\overline{\varepsilon}_{j})_{\Gamma}\geq
0$, where $(-,-)_{T}$ is the restriction of $(-,-)_{\Gamma}$ on $T$. Moreover,
note that there exists a vertex $j\in T$ adjacent to $\widetilde{\Phi}$, we
have $(\widetilde{\beta},\overline{\varepsilon}_{j})_{T}>0$. Therefore, $T$ is
a Dynkin diagram [13, Corollary 4.9]. On the other hand, let
$\widetilde{\beta}^{\prime}$ be the restriction of $\alpha-h(\beta)$ to $T$.
Then $\widetilde{\beta}^{\prime}$ has the support $T$, and for any vertex
$j\in T$,

$(\widetilde{\beta}^{\prime},\overline{\varepsilon}_{j})_{T}=(\alpha-h(\beta),\overline{\varepsilon}_{j})_{\Gamma}=(\alpha,\overline{\varepsilon}_{j})_{\Gamma}-(h(\beta),\overline{\varepsilon}_{j})_{\Gamma}\leq
0.$

Hence $T$ is not Dynkin. This is a contradiction. Therefore, $T$ is empty,
$\widetilde{\Phi}=\Gamma$ and so that $h(\beta)=\alpha$. Thus, we have shown
that $h$ is surjective by Lemma 3.11(3).

In general, assume that $\Gamma$ is non-connected. In this case,
$F_{\Gamma}=\bigcup F_{\Gamma^{\prime}}$, where $\Gamma^{\prime}$ run over all
connected components of $\Gamma$. By the discussion above, we see that any
element $\alpha\in F_{\Gamma}$, there exists an element
$\beta\in\Delta_{\widehat{Q}}$ such that $h(\beta)=\alpha$. Hence, $h$ is also
surjective.

Finally, for any real root $\alpha\in\Delta_{\Gamma}$, we let
$\beta\in\Delta_{\widehat{Q}}$ be the element such that $h(\beta)=\alpha$.
Then, there is an element $\omega^{\prime}\in\mathcal{W}(\Gamma)$ and
$i\in\mathcal{I}$ such that
$\omega^{\prime}(\alpha)=\overline{\varepsilon}_{i}$. Let $\omega$ be the
element in $C_{G}(\mathcal{W}(\widehat{Q}))$ corresponding to
$\omega^{\prime}$. It follows that $\omega(\beta)$ must also be a simple root
$\varepsilon_{i\rho}$ for some $\rho\in\mbox{irr}G_{i}$. Therefore $\beta$ is
real and uniquely determined up to a $G$-orbit. ∎

Consider the action of $G$ on $\mathbbm{k}\widehat{Q}$, any $g\in G$ also
induces an additive autoequivalence functor $F_{g}:\mbox{{\bf
mod}-}\mathbbm{k}\widehat{Q}\rightarrow\mbox{{\bf
mod}-}\mathbbm{k}\widehat{Q}$, $X\mapsto{{}^{g}X}$. Here we also denote by
$G_{X}$ a complete set of left coset representatives of $H_{X}:=\\{g\in G\mid
F_{g}(X)\cong X\\}$ in $G$, for any $X\in\mbox{{\bf
mod}-}\mathbbm{k}\widehat{Q}$. Following from Kac Theorem, for any positive
real root $\beta\in\Delta_{\widehat{Q}}$, there exists a unique
$\widehat{Q}$-representation $X$ such that ${\bf dim}X=\beta$ and
$H_{X}=H_{\beta}$. By Proposition 3.12, there are $|G_{X}|$ indecomposable
$\widehat{Q}$-representations (up to isomorphism) such that the image of their
dimension vector under the map $h$ are $\alpha$, if $h({\bf dim}X)=\alpha$.
Thus the proof of Theorem 1.1 is completed.

##  4\. Proof of Theorem 1.2

Assume that $G\subseteq\mbox{Aut}(\mathbbm{k}Q)$ is a finite abelian group. In
this section, we lift $G$ to $\overline{G}\subseteq\mbox{Aut}(\mathfrak{g})$
such that the Kac-Moody algebra $\mathfrak{g}(\Gamma)$ can be embedded into
the fixed point algebra $\mathfrak{g}^{\overline{G}}$. In this case,
$\mathfrak{g}^{\overline{G}}$ is integrable as a
$\mathfrak{g}(\Gamma)$-module.

Firstly, we recall some notations of Kac-Moody algebras. For a symmetricable
generalized Cartan matrix $C=(c_{ij})$ of size $n$ and rank $l$, there exist a
diagonal matrix $D=\mbox{diag}(d_{1},\cdots,d_{n})$ and a symmetric matrix
$B=(b_{ij})$ such that $C=D^{-1}B$. In fact, $d_{i}(1\leq i\leq n)$ may be
chosen to be positive integers. Let $\mathfrak{h}$ be a $2n-l$ dimension
$\mathbbm{k}$-vector space. Choose linearly independent sets
$\left\\{H_{i}\in\mathfrak{h}|1\leq i\leq n\right\\}$ and
$\left\\{\varepsilon_{i}\in\mathfrak{h}^{\ast}|1\leq i\leq n\right\\}$ such
that $\varepsilon_{j}(H_{i})=c_{ij}$. Then the triple
$\left(\mathfrak{h},\\{\varepsilon_{i}\\},\\{H_{i}\\}\right)_{1\leq i\leq n}$
is called a (minimal) realization of $C$. Since any two realizations of $C$
are isomorphic, there is a unique (up to isomorphism) Kac-Moody algebra
$\mathfrak{g}(C)$ generated by $\mathfrak{h}$, $E_{i}$, $F_{i}$, $1\leq i\leq
n$, with relations

$\begin{array}[]{llll}\quad[H,H^{\prime}]=0,&[H,E_{j}]=\varepsilon_{j}(H)E_{j},&(\mbox{ad}E_{i})^{1-c_{ij}}E_{j}=0,\\\
\quad[E_{i},F_{j}]=\delta_{ij}H_{i},&[H,F_{j}]=-\varepsilon_{j}(H)F_{j},&(\mbox{ad}F_{i})^{1-c_{ij}}F_{j}=0.\end{array}$

for any $H,H^{\prime}\in\mathfrak{h}$, where $\delta_{ij}$ is the Kronecker
sign. Moreover, the center $\mathfrak{c}$ of $\mathfrak{g}(C)$ is given by

$\\{H\in\mathfrak{h}\mid\varepsilon_{i}(H)=0\mbox{ for all }1\leq i\leq
n\\}\subseteq[\mathfrak{g}(C),\mathfrak{g}(C)].$

For the details one can see [13].

For the pair $(Q,G)$, we have obtained the valued graph $\Gamma$ with
symmetricable generalized Cartan matrix $C=(c_{ij})$ of size $|\mathcal{I}|$
and the generalized McKay quiver $\widehat{Q}$ with symmetric generalized
Cartan matrix $\widehat{A}=(a_{(i\rho)(j\sigma)})$ of size $|\widehat{I}|$,
see Section 2. Therefore we have Kac-Moody algebras
$\mathfrak{g}(\Gamma):=\mathfrak{g}(C)$ corresponding to the realization
$\big{(}\mathfrak{h}(\Gamma),\\{\overline{\varepsilon}_{i}\\},\\{\overline{H}_{i}\\}\big{)}$
of $C$ and $\mathfrak{g}:=\mathfrak{g}(\widehat{Q})=\mathfrak{g}(\widehat{A})$
corresponding to the realization
$\big{(}\mathfrak{h},\\{\varepsilon_{i\rho}\\},\\{H_{i\rho}\\}\big{)}$ of
$\widehat{A}$. Denote by $r$ and $s$ the coranks of $C$ and $\widehat{A}$,
then $\mbox{dim}_{\mathbbm{k}}\mathfrak{h}(\Gamma)=|\mathcal{I}|+r$ and
$\mbox{dim}_{\mathbbm{k}}\mathfrak{h}=|\widehat{I}|+s$.

We suppose that $\mathfrak{g}(\Gamma)$ generated by $\mathfrak{h}(\Gamma)$ and
$\overline{E}_{i},\overline{F}_{i}$, $i\in\mathcal{I}$. There is a symmetric
bilinear form $(-,-)_{\Gamma}$ on $\mathfrak{h}(\Gamma)$ such that

$(\overline{H}_{i},\overline{H})_{\Gamma}=\frac{1}{d_{i}}\overline{\varepsilon}_{i}(\overline{H})$

for all $\overline{H}\in\mathfrak{h}(\Gamma)$. Then we can extend it uniquely
to an invariant non-degenerate symmetric bilinear form on
$\mathfrak{g}(\Gamma)$ such that

$(\overline{E}_{i},\overline{F}_{i})_{\Gamma}=\frac{1}{d_{i}}.$

Moreover, $(-,-)_{\Gamma}$ determines a bijection
$\nu:\mathfrak{h}(\Gamma)\rightarrow\mathfrak{h}^{\ast}(\Gamma)$ sending
$\overline{H}_{i}$ to $\frac{1}{d_{i}}\overline{\varepsilon}_{i}$, and hence
induces a bilinear form on $\mathfrak{h}^{\ast}(\Gamma)$. We also denote this
bilinear form by $(-,-)_{\Gamma}$. Note that
$(\overline{\varepsilon}_{i},\overline{\varepsilon}_{i})_{\Gamma}=b_{ij}$. It
recovers the bilinear form defined in Section 2.3 for the root lattice
$\mathbb{Z}\mathcal{I}$. Similarly, there is a symmetric bilinear form on
$\mathfrak{h}^{\ast}=\mathfrak{h}^{\ast}(\widehat{Q})$ with
$(\varepsilon_{i\rho},\varepsilon_{j\sigma})_{\widehat{Q}}=a_{(i\rho)(j\sigma)}.$

We now consider the action of $G$ on the quiver $\widehat{Q}$ defined in
Section 3.2. Recall that the derived algebra $\mathfrak{g}^{\prime}$ of
$\mathfrak{g}$ is generated by $H_{i\rho}$, $E_{i\rho}$, $F_{i\rho}$,
$(i,\rho)\in\widehat{I}$ and the action of $G$ on $\widehat{Q}$ satisfies

$a_{(i\rho)(j\sigma)}=a_{(i\rho^{\prime})(j\sigma^{\prime})},\quad\hbox{ if
}(i,\rho^{\prime})=g(i,\rho)\hbox{ and }(j,\sigma^{\prime})=g(j,\sigma)$

for some $g\in G$. Then, there is a natural action of $G$ on
$\mathfrak{g}^{\prime}$ given by

$g(H_{i\rho})=H_{i\rho^{\prime}},\quad g(E_{i\rho})=E_{i\rho^{\prime}},\quad
g(F_{i\rho})=F_{i\rho^{\prime}}$

for any $g\in G$. Denote by $\mathfrak{h}^{\prime}(\Gamma)$ and
$\mathfrak{h}^{\prime}$ the Cartan subalgebra of
$\mathfrak{g}^{\prime}(\Gamma):=[\mathfrak{g}(\Gamma),\mathfrak{g}(\Gamma)]$
and $\mathfrak{g}^{\prime}$ respectively. It is easy to see that the map

$\phi:\quad\mathfrak{h}^{\prime}(\Gamma)\rightarrow(\mathfrak{h}^{\prime})^{G}$

given by $\phi(\overline{H}_{i})=\sum_{\rho\in{\rm irr}G_{i}}H_{i\rho}$ is an
isomorphism and

$(\overline{H},\overline{H}^{\prime})_{\Gamma}=\frac{1}{|G|}(\phi(\overline{H}),\phi(\overline{H}^{\prime}))_{\widehat{Q}}$

for $\overline{H},\overline{H}\in\mathfrak{h}^{\prime}(\Gamma)$. In
particular, the fixed point subalgebra $\mathfrak{c}^{G}$ of the center of
$\mathfrak{g}(\widehat{Q})$ is isomorphic to the center $\mathfrak{c}(\Gamma)$
of $\mathfrak{g}(\Gamma)$.

We wish to extend the action of $G$ on $\mathfrak{g}^{\prime}$ to the whole
Lie algebra $\mathfrak{g}$. Let $\mbox{Aut}(\widehat{A})$ denote the set of
permutations $g$ of $\widehat{I}$ satisfying

$a_{(i\rho)(j\sigma)}=a_{(l\rho^{\prime})(k\sigma^{\prime})}\quad\hbox{ if
}(l,\rho^{\prime})=g(i,\rho)\hbox{ and }(k,\sigma^{\prime})=g(j,\sigma).$

Let $\mbox{DAut}(\mathfrak{g})$ denote the subgroup of
$\mbox{Aut}(\mathfrak{g})$ consisting of the automorphisms preserving each of
the sets $\mathfrak{h}$, $\\{E_{i\rho}\\}$ and $\\{F_{i\rho}\\}$.

######  Proposition 4.1.

(see [14, Section 4.19]) There is a short exact sequence

$0\rightarrow{\rm
Hom}_{\mathbbm{k}}(\mathfrak{h}/\mathfrak{h}^{\prime},\mathfrak{c})\longrightarrow{\rm
DAut}(\mathfrak{g})\longrightarrow{\rm Aut}(\widehat{A})\rightarrow 0.$

###### Proof.

It is easy to see that $\overline{g}(H_{i\rho})=H_{j\sigma}$,
$\overline{g}(E_{i\rho})=E_{j\sigma}$ and
$\overline{g}(F_{i\rho})=F_{j\sigma}$ for any
$\overline{g}\in\mbox{DAut}(\mathfrak{g})$. Thus, there exists a unique
permutation $g\in\mbox{Aut}(\widehat{A})$ corresponding to $\bar{g}$ such that
$(j,\sigma)=g(i,\rho)$. Moreover, each $g\in\mbox{Aut}(\widehat{A})$ can be
obtained in this way.

Let $\Lambda:=\mathbbm{k}\widehat{I}$ be the subspace of $\mathfrak{h}^{\ast}$
spanned by $\\{\varepsilon_{i\rho}\mid(i,\rho)\in\widehat{I}\\}$. Then there
is an natural action of $\mbox{Aut}(\widehat{A})$ on $\Lambda$:
$g(\varepsilon_{i\rho})=\varepsilon_{j\sigma}$, where $(j,\sigma)=g(i,\rho)$,
$g\in G$, and it induces an action of $\mbox{Aut}(\widehat{A})$ on the
quotient space $\mathfrak{h}/\mathfrak{c}$ since $\mathfrak{h}/\mathfrak{c}$
is dual to $\Lambda$. It maps $H_{i\rho}\mod{\mathfrak{c}}$ to
$H_{j\sigma}\mod{\mathfrak{c}}$, and so that
$\mathfrak{h}^{\prime}/\mathfrak{c}$ is $\mbox{Aut}(\widehat{A})$-stable.
Since $\mbox{Aut}(\widehat{A})$ is finite, there exists
$\mathfrak{h}^{\prime\prime}$ such that
$\mathfrak{h}=\mathfrak{h}^{\prime}\oplus\mathfrak{h}^{\prime\prime}$ and
$(\mathfrak{h}^{\prime\prime}+\mathfrak{c})/\mathfrak{c}$ is
$\mbox{Aut}(\widehat{A})$-stable. For any $g\in\mbox{Aut}(\widehat{A})$, we
can define an automorphism $\overline{g}\in\mbox{DAut}(\mathfrak{g})$ by

$\overline{g}(H_{i\rho})=H_{j\sigma},\quad\overline{g}(E_{i\rho})=E_{j\sigma}\quad\hbox{
and }\quad\overline{g}(F_{i\rho})=F_{j\sigma},$

and $\overline{g}|_{\mathfrak{h}^{\prime\prime}}$ is the pull-back of $g$ on
$(\mathfrak{h}^{\prime\prime}+\mathfrak{c})/\mathfrak{c}$.

Obviously, the kernel of the map
$\mbox{DAut}(\mathfrak{g})\rightarrow\mbox{Aut}(\widehat{A})$ is the subgroup
$\mbox{Aut}(\mathfrak{g};\mathfrak{g}^{\prime})$ consisting of all
automorphisms acting trivially on $\mathfrak{g}^{\prime}$. One can check that
an automorphism $\alpha\in\mbox{Aut}(\mathfrak{g};\mathfrak{g}^{\prime})$ if
and only if there exists a map
$\varphi:\mathfrak{h}^{\prime\prime}\rightarrow\mathfrak{c}$ such that
$\alpha(H)=H+\varphi(H)$ for all $H\in\mathfrak{h}^{\prime\prime}$. Thus,
there are isomorphisms
$\mbox{Aut}(\mathfrak{g};\mathfrak{g}^{\prime})\cong\mbox{Hom}_{\mathbbm{k}}(\mathfrak{h}^{\prime\prime},\mathfrak{c})\cong\mbox{Hom}_{\mathbbm{k}}(\mathfrak{h}/\mathfrak{h}^{\prime},\mathfrak{c})$.
∎

Therefore, for each $\alpha\in\mbox{Aut}(\mathfrak{g};\mathfrak{g}^{\prime})$
and $g\in\mbox{Aut}(\widehat{A})$, we have an element
$\overline{g}\in\mbox{DAut}(\mathfrak{g})$ by setting
$\overline{g}|_{\mathfrak{g}^{\prime}}=g$ and
$\overline{g}|_{\mathfrak{h}^{\prime\prime}}=\alpha$. Moreover, for any
$\alpha\in\mbox{Aut}(\mathfrak{g};\mathfrak{g}^{\prime})$ corresponding to
$\varphi:\mathfrak{h}^{\prime\prime}\rightarrow\mathfrak{c}$, it is easy to
see that $\alpha^{t}(H)=H+t\varphi(H)$ for any $t\in\mathbb{Z}$ and
$H\in\mathfrak{h}^{\prime\prime}$. That is to say, an automorphism
$\alpha\in\mbox{Aut}(\mathfrak{g};\mathfrak{g}^{\prime})$ has finite order if
and only if the corresponding map
$\varphi:\mathfrak{h}^{\prime\prime}\rightarrow\mathfrak{c}$ is zero.

We now fix $\Omega=\\{g_{1},g_{2},\cdots,g_{n}\\}$ a set of generators of $G$.
We can view $G$ as a finite abelian subgroup of $\mbox{Aut}(\widehat{A})$. By
Proposition 4.1, we can lift $G$ to an automorphism group
$\overline{G}=\\{\overline{g}\mid g\in G\\}$ of $\mathfrak{g}$ corresponding
to a set of linear maps
$\\{\varphi_{i}=\varphi_{g_{i}}:\mathfrak{h}^{\prime\prime}\rightarrow\mathfrak{c}\mid
g_{i}\in\Omega\\}$. It is easy to see that for any $H\in\mathfrak{h}$, we have
$\varepsilon_{i\rho^{\prime}}(\overline{g}(H))=\varepsilon_{i\rho}(H)$ if
$(i,\rho^{\prime})=g(i,\rho)$. Let

$\mathcal{S}:=\mbox{span}\\{\varepsilon_{i\rho}-\varepsilon_{i\rho^{\prime}}\mid
i\in\mathcal{I},~{}\rho,\rho^{\prime}\in\mbox{irr}G_{i}\\}\subseteq\mathfrak{h}^{\ast}$

and

$\mathcal{H}:=\\{H\in\mathfrak{h}\mid\varepsilon_{i\rho}(H)=\varepsilon_{i\rho^{\prime}}(H)\mbox{
for all }\rho,\rho^{\prime}\in\mbox{irr}G_{i},\mbox{ and
}i\in\mathcal{I}\\}=\mbox{ann}_{\mathfrak{h}}\mathcal{S}.$

Then $\mathcal{H}$ contains the center $\mathfrak{c}$,
$\mathcal{H}/\mathfrak{c}=(\mathfrak{h}/\mathfrak{c})^{G}$ and so that, for
any lifting $\overline{G}$ of $G$,
$\mathcal{H}^{\overline{G}}=\mathfrak{h}^{\overline{G}}.$

######  Lemma 4.2.

$\mathcal{H}$ has $\mathbbm{k}$-dimension $|\mathcal{I}|+s$,
$\mathcal{H}\cap\mathfrak{h}^{\prime}$ has $\mathbbm{k}$-dimension
$|\mathcal{I}|+s-r$ and therefore $\mathcal{H}\cap\mathfrak{h}^{\prime\prime}$
has $\mathbbm{k}$-dimension $r$.

###### Proof.

Note that

$\\{\varepsilon_{i\rho}-\varepsilon_{i\rho^{\prime}}\mid
i\in\mathcal{I},~{}\rho^{\prime}\in\mbox{irr}G_{i}\setminus\rho\\}$

is a basis of $\mathcal{S}$, we obtain that
$\mbox{dim}_{\mathbbm{k}}\mathcal{H}=\mbox{dim}_{\mathbbm{k}}\mathfrak{h}-\mbox{dim}_{\mathbbm{k}}\mathcal{S}=|\mathcal{I}|+s$.
Since
$(\mathcal{H}\cap\mathfrak{h}^{\prime})/\mathfrak{c}=(\mathfrak{h}^{\prime}/\mathfrak{c})^{G}$
is isomorphic to $(\mathfrak{h}^{\prime})^{G}/\mathfrak{c}^{G}$,
$\mbox{dim}_{\mathbbm{k}}(\mathfrak{h}^{\prime})^{G}=|\mathcal{I}|$ and
$\mbox{dim}_{\mathbbm{k}}\mathfrak{c}^{G}=\mbox{dim}_{\mathbbm{k}}\mathfrak{c}(\Gamma)=r$,
$\mathcal{H}\cap\mathfrak{h}^{\prime}$ has $\mathbbm{k}$-dimension
$|\mathcal{I}|+s-r$ and so that $\mathcal{H}\cap\mathfrak{h}^{\prime\prime}$
has $\mathbbm{k}$-dimension $r$. ∎

######  Proposition 4.3.

Let $\overline{G}$ be a lifting of $G$ to $\mathfrak{g}$ corresponding to
$\\{\varphi_{i}:\mathfrak{h}^{\prime\prime}\rightarrow\mathfrak{c}\mid 1\leq
i\leq n\\}$. Then

$\bigg{(}\mathcal{H}^{\overline{G}},\bigg{\\{}\frac{d_{i}}{|G|}\sum_{\rho\in{\rm
irr}G_{i}}\varepsilon_{i\rho}\bigg{\\}},\bigg{\\{}\sum_{\rho\in{\rm
irr}G_{i}}H_{i\rho}\bigg{\\}}\bigg{)}$

is a realization of $C$ if and only if
$\varphi_{i}(\mathcal{H}\cap\mathfrak{h}^{\prime\prime})=0$ for all $1\leq
i\leq n$.

###### Proof.

We denote by

$H_{i}:=\sum_{\rho\in{\rm irr}G_{i}}H_{i\rho}\quad\hbox{ and
}\quad\epsilon_{i}:=\frac{d_{i}}{|G|}\sum_{\rho\in{\rm
irr}G_{i}}\varepsilon_{i\rho}$

for all $i\in\mathcal{I}$. Since $\\{H_{i}\mid i\in\mathcal{I}\\}$ is a basis
of $(\mathcal{H}\cap\mathfrak{h}^{\prime})^{G}$, $\mathcal{H}^{\overline{G}}$
has dimension $|\mathcal{I}|+r$ if and only if there are
$h^{\prime}_{1},h^{\prime}_{2},\cdots,h^{\prime}_{r}\in\mathcal{H}^{\overline{G}}$
spanning the complementary space of
$(\mathcal{H}\cap\mathfrak{h}^{\prime})^{G}$ in $\mathcal{H}^{\overline{G}}$.

Since $(\mathfrak{h}^{\prime\prime}+\mathfrak{c})/\mathfrak{c}$ is $G$-stable,
$\big{(}(\mathfrak{h}^{\prime\prime}+\mathfrak{c})/\mathfrak{c}\big{)}^{G}$
has $\mathbbm{k}$-dimension $r$ by Lemma 4.2. We can find linearly independent
elements
$h^{\prime\prime}_{1},h^{\prime\prime}_{2},\cdots,h^{\prime\prime}_{r}\in\mathcal{H}\cap\mathfrak{h}^{\prime\prime}$
such that $h^{\prime\prime}_{i}\mod{\mathfrak{c}}$ are fixed by $G$. Since
$\varphi_{i}(\mathcal{H}\cap\mathfrak{h}^{\prime\prime})=0$ for all $i$,
$h^{\prime\prime}_{1},h^{\prime\prime}_{2},\cdots,h^{\prime\prime}_{r}$ are
$G$-stable and form a basis of $\mathcal{H}\cap\mathfrak{h}^{\prime\prime}$.
Therefore, we take $h^{\prime}_{i}=h^{\prime\prime}_{i}$ for all $1\leq i\leq
r$. On the other hand, if we can find such elements
$h^{\prime}_{1},h^{\prime}_{2},\cdots,h^{\prime}_{r}$, then each
$h^{\prime\prime}_{i}$ has the form

$h^{\prime\prime}_{i}=\sum_{j=1}^{s}p_{ij}h^{\prime}_{j}-\sum_{(j,\sigma)\in\widehat{I}}q_{i(j\sigma)}H_{j\sigma}$

for some $p_{ij},q_{i(j\sigma)}\in\mathbbm{k}$, and

$\displaystyle\varphi_{l}(h^{\prime\prime}_{i})$
$\displaystyle=\overline{g}_{l}\Big{(}\sum_{j=1}^{s}p_{ij}h^{\prime}_{j}-\sum_{(j,\sigma)\in\widehat{I}}q_{i(j\sigma)}H_{j\sigma}\Big{)}-\sum_{j=1}^{s}p_{ij}h^{\prime}_{j}+\sum_{(j,\sigma)\in\widehat{I}}q_{i(j\sigma)}H_{j\sigma}$
$\displaystyle=\sum_{(j,\sigma)\in\widehat{I}}q_{i(j\sigma)}(H_{j\sigma}-H_{j\sigma^{1}}),$

where $(j,\sigma^{1})=g_{l}(j,\sigma)$. It follows that

$t\varphi_{l}(h^{\prime\prime}_{i})=\sum_{(j,\sigma)\in\widehat{I}}q_{i(j\sigma)}(H_{j\sigma}-H_{j\sigma^{t}})$

for any $t\in\mathbb{Z}$, where $(j,\sigma^{t})=g_{l}^{t}(j,\sigma)$. Note
that $\widehat{I}$ is a finite set, there exist some $t\in\mathbb{Z}$ such
that $g_{l}^{t}(j,\sigma)=(j,\sigma)$ for all $(j,\sigma)\in\widehat{I}$, and
so that $t\varphi_{l}(h^{\prime\prime}_{i})=0$,
$\varphi_{l}(h^{\prime\prime}_{i})=0$ for all $i$ and $l$. Thus
$\varphi_{i}(\mathcal{H}\cap\mathfrak{h}^{\prime\prime})=0$ for any $1\leq
i\leq n$.

Since

$\epsilon_{j}(H_{i})=\frac{d_{i}}{|G|}\sum_{\rho\in{\rm
irr}G_{i}\atop\sigma\in{\rm
irr}G_{j}}\varepsilon_{j\sigma}(H_{i\rho})=\frac{d_{i}}{|G|}\sum_{\rho\in{\rm
irr}G_{i}\atop\sigma\in{\rm irr}G_{j}}a_{(i\rho)(j\sigma)}=c_{ij}$

and $H_{i}$ $(i\in\mathcal{I})$ are linearly independent, it remains to show
$\epsilon_{i}$, $i\in\mathcal{I}$ are linearly independent modulo
$\mbox{ann}_{\mathfrak{h}^{\ast}}({\mathcal{H}}^{\overline{G}})$. Let

$\epsilon:=\sum_{j\in\mathcal{I}}\mu_{j}\epsilon_{j}\in\mbox{ann}_{\mathfrak{h}^{\ast}}({\mathcal{H}}^{\overline{G}}),\quad\mu_{j}\in\mathbbm{k}.$

Then

$0=\epsilon(H_{i})=\sum_{j\in\mathcal{I}}\mu_{j}\epsilon_{j}(H_{i})=\sum_{j\in\mathcal{I}}c_{ij}\mu_{j}$

for all $i\in\mathcal{I}$, and so that

$\epsilon(H_{i\rho})=\sum_{j\in\mathcal{I}}\mu_{j}\epsilon_{j}(H_{i\rho})=\frac{1}{|G|}\sum_{j\in\mathcal{I}}b_{ij}\mu_{j}=\frac{d_{i}}{|G|}\sum_{j\in\mathcal{I}}c_{ij}\mu_{j}=0$

for all $(i,\rho)\in\widehat{I}$. Therefore,

$\epsilon\in\mbox{ann}_{\mathfrak{h}^{\ast}}({\mathcal{H}}^{\overline{G}}+\mathfrak{h}^{\prime})=\mbox{ann}_{\mathfrak{h}^{\ast}}({\mathcal{H}}+\mathfrak{h}^{\prime})\subseteq\mbox{ann}_{\mathfrak{h}^{\ast}}({\mathcal{H}})=\mathcal{S},$

and

$g(\epsilon)=g\bigg{(}\sum_{j\in\mathcal{I}}\mu_{j}\epsilon_{j}\bigg{)}=g\bigg{(}\sum_{j\in\mathcal{I}}\frac{d_{j}\mu_{j}}{|G|}\sum_{\sigma\in{\rm
irr}G_{j}}\varepsilon_{j\sigma}\bigg{)}=\epsilon$

for any $g\in G$. It concludes that
$\epsilon=\sum_{j\in\mathcal{I}}\mu_{j}\epsilon_{j}=0$ by the construction of
$\mathcal{S}$. Therefore, $\mu_{j}=0$ for all $j\in\mathcal{I}$, and so that
$\epsilon_{j}$ are linearly independent in $\mathfrak{h}^{\ast}$.

The proof is completed. ∎

######  Remark 4.4.

Since

$\mbox{Hom}_{\mathbbm{k}}(\mathfrak{h}^{\prime\prime},\mathfrak{c})\cong\mbox{Hom}_{\mathbbm{k}}(\mathfrak{h}/\mathfrak{h}^{\prime},\mathfrak{c}),$

for any lifting $\overline{G}$ of $G$, there exist a family of maps
$\\{\psi_{i}=\psi_{g_{i}}:\mathfrak{h}/\mathfrak{h}^{\prime}\rightarrow\mathfrak{c}\mid
g_{i}\in\Omega\\}$ corresponding to it. Moreover, it is easy to see that the
condition $\varphi_{i}(\mathcal{H}\cap\mathfrak{h}^{\prime\prime})=0$ is
equivalent to
$\psi_{i}((\mathcal{H}+\mathfrak{h}^{\prime})/\mathfrak{h}^{\prime})=0$.

Now we can prove the main results of this section.

######  Proposition 4.5.

There is a monomorphism
$\mathfrak{g}^{\prime}(\Gamma)\rightarrow(\mathfrak{g}^{\prime})^{G}$, and for
the lifting $\overline{G}$ of $G$ corresponding to
$\\{\varphi_{i}:\mathfrak{h}^{\prime\prime}\rightarrow\mathfrak{c}\mid 1\leq
i\leq n\\}$ with $\varphi_{i}(\mathcal{H}\cap\mathfrak{h}^{\prime\prime})=0$,
we can extend this monomorphism to the whole Lie algebra such that

$\mathfrak{g}(\Gamma)\rightarrow\mathfrak{g}^{\overline{G}}$

is also a monomorphism.

###### Proof.

We set

$H_{i}:=\sum_{\rho\in{\rm irr}G_{i}}H_{i\rho},\quad E_{i}:=\sum_{\rho\in{\rm
irr}G_{i}}E_{i\rho},\quad F_{i}:=\sum_{\rho\in{\rm irr}G_{i}}F_{i\rho}$

for all $i\in\mathcal{I}$. Then
$H_{i},E_{i},F_{i}\in(\mathfrak{g}^{\prime})^{G}$ and

$\displaystyle[H_{i},H_{j}]=0,$ $\displaystyle[E_{i},F_{j}]=\sum_{\rho\in{\rm
irr}G_{i}\atop\sigma\in{\rm
irr}G_{j}}[E_{i\rho},F_{j\sigma}]=\delta_{ij}\sum_{\rho\in{\rm
irr}G_{i}}H_{i\rho}=\delta_{ij}H_{i},$
$\displaystyle[H_{i},E_{j}]=\sum_{\rho\in{\rm irr}G_{i}\atop\sigma\in{\rm
irr}G_{j}}[H_{i\rho},E_{j\sigma}]=\sum_{\rho\in{\rm
irr}G_{i}\atop\sigma\in{\rm
irr}G_{j}}a_{(i\rho)(j\sigma)}E_{j\sigma}=c_{ij}\sum_{\sigma\in{\rm
irr}G_{j}}E_{j\sigma}=c_{ij}E_{j}.$

Similarly, we have $[H_{i},F_{j}]=c_{ij}F_{j}$ for any $i,j\in\mathcal{I}$.
Note that $\mbox{ad}E_{i\rho}$ and $\mbox{ad}E_{i\rho^{\prime}}$ commute for
any $\rho,\rho^{\prime}\in\mbox{irr}G_{i}$, we have

$(\mbox{ad}E_{i})^{n}=\sum_{\lambda}\Phi_{\lambda}^{n}\prod_{\rho\in{\rm
irr}G_{i}}(\mbox{ad}E_{i\rho})^{\lambda_{\rho}}$

for any positive integer $n$, where $\lambda$ takes thought all the sequence
$\lambda=(\lambda_{\rho})_{\rho\in{\rm irr}G_{i}}$ satisfying

$\sum_{\rho\in{\rm irr}G_{i}}\lambda_{\rho}=n,$

and

$\Phi_{\lambda}^{n}=\left({\begin{array}[]{*{20}c}n\\\ \rho_{1}\\\
\end{array}}\right)\left({\begin{array}[]{*{20}c}n-\rho_{1}\\\ \rho_{2}\\\
\end{array}}\right)\cdots\left({\begin{array}[]{*{20}c}n-\rho_{1}-\cdots-\rho_{|{\rm
irr}G_{i}|-1}\\\ \rho_{|{\rm irr}G_{i}|}\\\ \end{array}}\right)$

for any $\lambda=(\rho_{1},\rho_{2},\cdots,\rho_{|{\rm irr}G_{i}|})$. In
particular, if $n=1-c_{ij}$, then $\lambda_{\rho}>1-a_{(i\rho)(j\sigma)}$ for
some $\rho\in\mbox{irr}G_{i}$ and so that

$(\mbox{ad}E_{i\rho})^{\lambda_{\rho}}E_{j\sigma}=0,\qquad(\mbox{ad}E_{i})^{1-c_{ij}}E_{j}=0.$

Similarly, $(\mbox{ad}F_{i})^{1-c_{ij}}F_{j}=0$ for any $i,j\in\mathcal{I}$.
Therefore, there exists a non-zero homomorphism
$\mathfrak{g}^{\prime}(\Gamma)\rightarrow(\mathfrak{g}^{\prime})^{G}$.

Since
$\big{(}\mathcal{H}^{\overline{G}},\\{\frac{d_{i}}{|G|}\phi(\varepsilon_{i})\\},\\{\phi(H_{i})\\}\big{)}$
is a realization of $C$ by Proposition 4.3, there is an isomorphism
$\mathfrak{h}(\Gamma)\rightarrow\mathcal{H}^{\overline{G}}$,
$\overline{H}_{i}\rightarrow H_{i}$. Therefore we can get a homomorphism
$\mathfrak{g}(\Gamma)\rightarrow\mathfrak{g}^{\overline{G}}$ by compositing
the homomorphisms
$\mathfrak{g}^{\prime}(\Gamma)\rightarrow(\mathfrak{g}^{\prime})^{G}$ and
$\mathfrak{h}(\Gamma)\rightarrow\mathcal{H}^{\overline{G}}$. By [13,
Proposition 1.7(b)],
$\mathfrak{g}(\Gamma)\rightarrow\mathfrak{g}^{\overline{G}}$ and
$\mathfrak{g}^{\prime}(\Gamma)\rightarrow(\mathfrak{g}^{\prime})^{G}$ are
monomorphisms. ∎

Now, we can identify $\mathfrak{g}(\Gamma)$ with a subalgebra of
$\mathfrak{g}^{\overline{G}}$. Following from Section 3.1, the map

$h:\quad\mathbb{Z}\widehat{I}\rightarrow\mathbb{Z}\mathcal{I},\qquad\beta\mapsto
h(\beta),\quad h(\beta)_{i}=\sum_{\rho\in{\rm irr}G_{i}}\beta_{i\rho},$

satisfies

$d_{i}\bigg{(}\beta,\sum_{\rho\in{\rm
irr}G_{i}}\varepsilon_{i\rho}\bigg{)}_{\widehat{Q}}=(h(\beta),\overline{\varepsilon}_{i})_{\Gamma}$

for all
$\beta=\sum_{(i,\rho)\in\widehat{I}}\beta_{i\rho}\varepsilon_{i\rho}\in\mathbb{Z}\widehat{I}$
and $h(\Delta_{\widehat{Q}})=\Delta_{\Gamma}$ by Proposition 3.12.

######  Proposition 4.6.

The monomorphism $\mathfrak{g}(\Gamma)\rightarrow\mathfrak{g}^{\overline{G}}$
endows $\mathfrak{g}^{\overline{G}}$ with an integrable
$\mathfrak{g}(\Gamma)$-module structure under the adjoint action of
$\mathfrak{g}(\Gamma)$.

###### Proof.

Firstly, we identity the realization
$\big{(}\mathfrak{h}(\Gamma),\\{\overline{\varepsilon}_{i}\\},\\{\overline{H}_{i}\\}\big{)}$
with
$\big{(}\mathcal{H}^{\overline{G}},\\{\epsilon_{i}\\},\\{H_{i}\\}\big{)}$. For
any non-zero
$\beta=\sum_{(i,\rho)\in\widehat{I}}\beta_{i\rho}\varepsilon_{i\rho}\in\Delta_{\widehat{Q}}$
and $H\in\mathcal{H}^{\overline{G}}$, we have

$\varepsilon_{i\rho}(H)=\frac{d_{i}}{|G|}\sum_{\rho\in{\rm
irr}G_{i}}\varepsilon_{i\rho}(H)=\overline{\varepsilon}_{i}(H)$

and

$\beta(H)=\sum_{(i,\rho)\in\widehat{I}}\beta_{i\rho}\varepsilon_{i\rho}(H)=\sum_{i\in\mathcal{I}}\Big{(}\sum_{\rho\in{\rm
irr}G_{i}}\beta_{i\rho}\Big{)}\overline{\varepsilon}_{i}(H)=\sum_{i\in\mathcal{I}}h(\beta)_{i}\overline{\varepsilon}_{i}(H)=h(\beta)(H).$

Denote by $H_{\beta}=\\{g\in G\mid g(\beta)=\beta\\}$ and $G_{\beta}$ a
complete set of left coset representatives of $H_{\beta}$ in $G$. Then
$H_{\beta}$ acts on the root space $\mathfrak{g}_{\beta}$. Suppose that
$x\in\mathfrak{g}_{\beta}$ satisfies $g(x)=x$ for any $g\in H_{\beta}$. Let

$\Sigma(x):=\sum_{g\in G_{\beta}}g(x).$

It is easy to see that $\Sigma(x)\in\mathfrak{g}^{\overline{G}}$ and

$[H,\Sigma(x)]=\sum_{g\in G_{\beta}}[H,g(x)]=\sum_{g\in
G_{\beta}}g(\beta)(H)g(x)=h(\beta)(H)\sum_{g\in
G_{\beta}}g(x)=h(\beta)(H)\Sigma(x)$

for all $H\in\mathcal{H}^{\overline{G}}$, since $h(g(\beta))=h(\beta)$ for any
$g\in G$. It follows that $\Sigma(x)$ lies in the weight space
$(\mathfrak{g}^{\overline{G}})_{h(\beta)}$. Note that each element in
$\mathfrak{g}^{\overline{G}}$ can be written as a sum of some $\Sigma(x)$ with
$x\in\mathfrak{g}_{\beta}$, $\beta\in\Delta_{\widehat{Q}}$, we obtain that
$\mathfrak{g}^{\overline{G}}$ is $\mathfrak{h}(\Gamma)$-diagonalisable.

Secondly, it is easy to see that the non-zero weights of
$\mathfrak{g}^{\overline{G}}$ must be roots of $\Gamma$ since
$h(\Delta_{\widehat{Q}})=\Delta_{\Gamma}$. On the other hand, every root of
$\Gamma$ is also a weight of $\mathfrak{g}^{\overline{G}}$ under the adjoint
action by the monomorphism
$\mathfrak{g}(\Gamma)\rightarrow\mathfrak{g}^{\overline{G}}$.

Finally, for any $\beta\in\Delta_{\Gamma}$, the set
$\\{\beta+k\overline{\varepsilon}_{i}\mid
k\in\mathbb{Z}\\}\cap\Delta_{\Gamma}$ is finite. Thus the action of
$\overline{E}_{i}$ and $\overline{F}_{i}$ are local nilpotent on
$\mathfrak{g}^{\overline{G}}$. The proof is completed. ∎

Following from the proof of Proposition 4.6,
$(\mathfrak{g}^{\overline{G}})_{\alpha}$ is spanned by the elements
$\Sigma(x)=\sum_{g\in G_{\beta}}g(x)$, where $x\in\mathfrak{g}_{\beta}$
satisfies $g(x)=x$ for any $g\in H_{\beta}$, and
$\beta\in\Delta_{\widehat{Q}}$ satisfies $h(\beta)=\alpha$. Thus, by the
action of $G$ on $\\{E_{i\rho}\\}$ and $\\{F_{i\rho}\\}$, $\mbox{ad}H_{\beta}$
acts on $\mathfrak{g}_{\beta}$ is identity and so that

$\mbox{dim}_{\mathbbm{k}}(\mathfrak{g}^{\overline{G}})_{h(\beta)}=1$

for any simple root $\beta$. That is,
$\mbox{dim}_{\mathbbm{k}}(\mathfrak{g}^{\overline{G}})_{\alpha}=1$ for all
simple root $\alpha\in\Delta_{\Gamma}$. Moreover, we have the following claim.

######  Claim 4.7.

${\rm dim}_{\mathbbm{k}}(\mathfrak{g}^{\overline{G}})_{\alpha}=1$ for any real
root $\alpha\in\Delta_{\Gamma}$.

###### Proof.

We consider the automorphism

$\overline{r}_{i\rho}:=\mbox{exp}(\mbox{ad}F_{i\rho})\mbox{exp}(-\mbox{ad}E_{i\rho})\mbox{exp}(\mbox{ad}F_{i\rho})$

of $\mathfrak{g}$. Then
$\overline{r}_{i\rho}(\mathfrak{g}_{\beta})=\mathfrak{g}_{r_{i\rho}(\beta)}$
and $\overline{r}_{i\rho}(H)=H-\varepsilon_{i\rho}(H)H_{i\rho}$ for any
$H\in\mathfrak{h}$ (see [13, Lemma 3.8]). Note that $\overline{r}_{i\rho}$ and
$\overline{r}_{i\rho^{\prime}}$ commute for any
$\rho,\rho^{\prime}\in\mbox{irr}G_{i}$, we let

$\overline{S}_{i}:=\prod_{\rho\in{\rm irr}G_{i}}\overline{r}_{i\rho}.$

Then, for any $H\in\mathcal{H}^{\overline{G}}$, we have

$\overline{S}_{i}(H)=H-\sum_{\rho\in{\rm
irr}G_{i}}\varepsilon_{i\rho}(H)H_{i\rho}=H-\epsilon_{i}(H)\sum_{\rho\in{\rm
irr}G_{i}}H_{i\rho}=H-\epsilon_{i}(H)H_{i},$

and $\overline{S}_{i}(H)\in\mathcal{H}^{\overline{G}}$. Note that
$\overline{S}_{i}$ and $G$ commute on $\mathfrak{g}^{\prime}$, it deduces that
$\overline{S}_{i}$ can define an automorphism of $\mathfrak{g}^{\overline{G}}$
such that

$\overline{S}_{i}\big{(}(\mathfrak{g}^{\overline{G}})_{\alpha}\big{)}=(\mathfrak{g}^{\overline{G}})_{\widehat{S}_{i}(\alpha)}.$

Therefore, $\overline{S}_{i}$ is an extension of the automorphism
$\mbox{exp}(\mbox{ad}\overline{F}_{i})\mbox{exp}(-\mbox{ad}\overline{E}_{i})\mbox{exp}(\mbox{ad}\overline{F}_{i})$
of $\mathfrak{g}(\Gamma)$.

Let $\alpha\in\Delta_{\Gamma}$ be a real root. By Lemma 3.11 and Proposition
3.12, there exist a real root $\beta\in\Delta_{\widehat{Q}}$ and $\omega\in
C_{G}(\mathcal{W}(\widehat{Q}))$ such that $h(\beta)=\alpha$, $\omega(\beta)$
is a simple root and $H_{\omega(\beta)}=H_{\beta}$. Let
$\omega=\widehat{S}_{i_{1}}\widehat{S}_{i_{2}}\cdots\widehat{S}_{i_{r}}$ and
$\overline{\omega}=\overline{S}_{i_{1}}\overline{S}_{i_{2}}\cdots\overline{S}_{i_{r}}$,
then $\overline{\omega}(\mathfrak{g}_{\beta})=\mathfrak{g}_{\omega(\beta)}$
and hence $\mathfrak{g}_{\beta}$ is fixed by $H_{\beta}$. Finally, note that
all these $\beta$ are in the same $G$-orbit, we have
$\mbox{dim}_{\mathbbm{k}}(\mathfrak{g}^{\overline{G}})_{\alpha}=1$. ∎

In particular, if $Q$ is a finite union of Dynkin quivers, then $\mathfrak{g}$
is a direct sum of simple Lie algebras and all roots of $\Gamma$ are real. By
the claim, we have

######  Corollary 4.8.

If $Q$ is a finite union of Dynkin quivers and $G\subseteq{\rm
Aut}(\mathbbm{k}Q)$ is finite abelian, then there is a Lie algebra isomorphism
$\mathfrak{g}(\Gamma)\cong\mathfrak{g}^{\overline{G}}$.

##  5\. Examples

In this section, we give two examples to elucidate our results.

######  Example 5.1.

Let $Q=(I,E)$ be the quiver

The action of $G=\langle g\rangle\cong\mathbb{Z}/6\mathbb{Z}$ on
$\mathbbm{k}Q$ given by

| $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $\alpha$ | $\beta$ | $\gamma$
---|---|---|---|---|---|---|---
$g$ | $e_{1}$ | $e_{3}$ | $e_{4}$ | $e_{2}$ | $-\beta$ | $-\gamma$ | $-\alpha$

where $e_{i}$ is the idempotent element of $\mathbbm{k}Q$ corresponding to
vertex $i$, $i\in\\{1,2,3,4\\}$. Then the Cartan matrix of $Q$ is

$A=(a_{ij})={\small\left(\begin{array}[]{cccc}2&-1&-1&-1\\\ -1&2&0&0\\\
-1&0&2&0\\\ -1&0&0&2\\\ \end{array}\right)}.$

Let $\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4}$ be all
the simple roots of the symmetric Kac-Moody algebra $\mathfrak{g}(Q)$. We
endow the root lattice $\mathbb{Z}I$ with a symmetric bilinear form
$(-,-)_{Q}$ via $(\varepsilon_{i},\varepsilon_{j})_{Q}=a_{ij}$ and define
reflection
$r_{i}:\alpha\mapsto\alpha-(\alpha,\varepsilon_{i})_{Q}\varepsilon_{i}$ for
each vertex $i\in I$. Then, it is well-knowen that Weyl group
$\mathcal{W}(Q)\cong(\mathbb{Z}/2\mathbb{Z})^{3}\rtimes S_{4}$, and one can
check that
$\Delta_{Q}=\pm\\{\varepsilon_{1},\varepsilon_{2},\varepsilon_{3},\varepsilon_{4},\varepsilon_{1}+\varepsilon_{2},\varepsilon_{1}+\varepsilon_{3},\varepsilon_{1}+\varepsilon_{4},\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3},\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{4},\varepsilon_{1}+\varepsilon_{3}+\varepsilon_{4},\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4},2\varepsilon_{1}+\varepsilon_{2}+\varepsilon_{3}+\varepsilon_{4}\\}$
is the root system of $\mathfrak{g}(Q)$.

We get the generalized McKay quiver $\widehat{Q}=(\widehat{I},\widehat{E})$ of
$(Q,G)$ as follows.

where $\rho_{i}$ is the irreducible representation of $G=\langle
g\rangle\cong\mathbb{Z}/6\mathbb{Z}$ defined by

$a\cdot g=\xi^{i}a,\qquad a\in\rho_{i},$

$\sigma_{j}$ is the irreducible representation of $\langle
g^{3}\rangle\cong\mathbb{Z}/2\mathbb{Z}$ defined by

$b\cdot g^{3}=\xi^{3j}b,\qquad b\in\sigma_{j},$

and $\xi$ is a primitive $6$-th root of unity. As we have discussed in Section
3.2, by the group isomorphism

$\varphi:G\rightarrow\widetilde{G},\qquad\varphi(g^{i})=\chi_{g^{i}},\qquad\chi_{g^{i}}(g^{j})=\xi^{ij},$

we define the action of $G$ on $\mathbbm{k}Q\ast G$ by setting

$g^{i}(\lambda g^{j})=\xi^{ij}\lambda g^{j}$

for any $g^{i}\in G$, $\lambda g^{j}\in\mathbbm{k}Q\ast G$. This induces an
action of $G=\langle g\rangle\cong\mathbb{Z}/6\mathbb{Z}$ on
$\mathbbm{k}\widehat{Q}$ given by

| $e_{0}$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e^{\prime}_{0}$ | $e^{\prime}_{1}$ | $\alpha_{0}$ | $\alpha_{1}$ | $\alpha_{2}$ | $\alpha_{3}$ | $\alpha_{4}$ | $\alpha_{5}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$g$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{0}$ | $e^{\prime}_{1}$ | $e^{\prime}_{0}$ | $\xi_{0}\alpha_{1}$ | $\xi_{1}\alpha_{2}$ | $\xi_{2}\alpha_{3}$ | $\xi_{3}\alpha_{4}$ | $\xi_{4}\alpha_{5}$ | $\xi_{5}\alpha_{0}$

where idempotent elements $e_{i}$, $e^{\prime}_{i}$ are corresponding to the
vertex $(1,\rho_{i})$, $(2,\sigma_{i})$ respectively, and
$\xi_{i}\in\mathbbm{k}$ satisfying $\xi_{0}\xi_{1}\cdots\xi_{5}=1$. One can
check that the generalized McKay quiver of $(\widehat{Q},G)$ is just the
quiver $Q$.

By the definition given in Section 2.3, we obtain the symmetrisable
generalized Cartan matrix $C$ corresponding to $(Q,G)$, i.e.,

$C={\small\left(\begin{array}[]{cc}2&-1\\\ -3&2\\\ \end{array}\right)}.$

Then the valued graph $\Gamma$ corresponding to $C$ is

Let $\overline{\varepsilon}_{1},\overline{\varepsilon}_{2}$ be all the simple
roots of $\Gamma$. Then the Weyl group

$\mathcal{W}(\Gamma)\cong D_{6}=\langle a,b\mid
a^{2}=1,b^{3}=1,ab=b^{-1}a\rangle$

and root system
$\Delta_{\Gamma}=\\{\overline{\varepsilon}_{1},\overline{\varepsilon}_{2},\overline{\varepsilon}_{1}+\overline{\varepsilon}_{2},2\overline{\varepsilon}_{1}+\overline{\varepsilon}_{2},3\overline{\varepsilon}_{1}+\overline{\varepsilon}_{2},3\overline{\varepsilon}_{1}+2\overline{\varepsilon}_{2}\\}$.
See Section 2.3 for detail.

We consider the map

$h:\quad\mathbb{Z}\widehat{I}\longrightarrow\mathbb{Z}\mathcal{I},\qquad
h(\alpha)_{i}=\sum_{\rho\in{\rm irr}G_{i}}\alpha_{i\rho}$

for any
$\alpha=\sum_{(i,\rho)\in\widehat{I}}\alpha_{i\rho}\varepsilon_{(i\rho)\in\widehat{I}}\in\mathbb{Z}\widehat{I}$.
The restriction of $h:\Delta_{\widehat{Q}}\rightarrow\Delta_{\Gamma}$ is a
surjective, this means that for any positive root $\beta$ of $\Gamma$, there
exists an indecomposable $\widehat{Q}$-representation $X$ such that $h({\bf
dim}X)=\beta$. For example, we consider the positive root
$\overline{\varepsilon}_{1}+\overline{\varepsilon}_{2}\in\Delta_{\Gamma}$.
Then, we have the following indecomposable $\widehat{Q}$-representation
$X_{(\rho_{3}\sigma_{0})}$:

and obviously, $h({\bf
dim}X_{(\rho_{3}\sigma_{0})})=\overline{\varepsilon}_{1}+\overline{\varepsilon}_{2}$.
Furthermore, for any $0\leq l\leq 5$, $0\leq j\leq 1$ and $l\not\equiv
j\mod{2}$, we define the $\widehat{Q}$-representation
$X_{(\rho_{l}\sigma_{j})}=(X_{i\rho},X_{\alpha})$ by

$X_{i\rho}=\left\\{\begin{array}[]{ll}\mathbbm{k},&\mbox{ if
}(i,\rho)=(1,\rho_{l})\mbox{ or }(2,\sigma_{j});\\\ 0,&\mbox{ otherwise.
}\end{array}\right.\qquad X_{\alpha}=\left\\{\begin{array}[]{ll}1,&\mbox{ if
}\alpha=\alpha_{l};\\\ 0,&\mbox{ otherwise. }\end{array}\right.$

Then, it is easy to see that the set of all indecomposable
$\widehat{Q}$-representations with $h({\bf
dim}X)=\overline{\varepsilon}_{1}+\overline{\varepsilon}_{2}$ is the set

$\left\\{X_{(\rho_{l}\sigma_{j})}\mid 0\leq l\leq 5,~{}0\leq j\leq 1\mbox{ and
}l\not\equiv j\mod{2}\right\\},$

and which is just the orbit of $X_{(\rho_{1}\sigma_{0})}$ under that action of
$G$. Similarly, for any positive real root
$\beta=h(\alpha)\in\Delta_{\Gamma}$, there are $|H_{\alpha}|$ (up to
isomorphism) indecomposable $\widehat{Q}$-representations $X$ such that
$h({\bf dim}X)=\beta.$

######  Example 5.2.

Let $Q=(I,E)$ be the quiver

and $G=\langle a\rangle\times\langle
b\rangle\cong\mathbb{Z}/2\mathbb{Z}\times\mathbb{Z}/2\mathbb{Z}$. The action
of $G$ on $\mathbbm{k}Q$ is given as follows

| $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$ | $e_{1^{\prime}}$ | $e_{2^{\prime}}$ | $e_{3^{\prime}}$ | $e_{4^{\prime}}$ | $e_{5^{\prime}}$
---|---|---|---|---|---|---|---|---|---|---
$a$ | $e_{5}$ | $e_{4}$ | $e_{3}$ | $e_{2}$ | $e_{1}$ | $e_{5^{\prime}}$ | $e_{4^{\prime}}$ | $e_{3^{\prime}}$ | $e_{2^{\prime}}$ | $e_{1^{\prime}}$
$b$ | $e_{1^{\prime}}$ | $e_{2^{\prime}}$ | $e_{3^{\prime}}$ | $e_{4^{\prime}}$ | $e_{5^{\prime}}$ | $e_{1}$ | $e_{2}$ | $e_{3}$ | $e_{4}$ | $e_{5}$
| $\alpha_{1}$ | $\alpha_{2}$ | $\alpha_{3}$ | $\alpha_{4}$ | $\alpha^{\prime}_{1}$ | $\alpha^{\prime}_{2}$ | $\alpha^{\prime}_{3}$ | $\alpha^{\prime}_{4}$
---|---|---|---|---|---|---|---|---
$a$ | $\alpha_{4}$ | $\alpha_{3}$ | $\alpha_{2}$ | $\alpha_{1}$ | $\alpha^{\prime}_{4}$ | $\alpha^{\prime}_{3}$ | $\alpha^{\prime}_{2}$ | $\alpha^{\prime}_{1}$
$b$ | $\alpha^{\prime}_{1}$ | $\alpha^{\prime}_{2}$ | $\alpha^{\prime}_{3}$ | $\alpha^{\prime}_{4}$ | $\alpha_{1}$ | $\alpha_{2}$ | $\alpha_{3}$ | $\alpha_{4}$

where $e_{i}$ is the idempotent element of $\mathbbm{k}Q$ corresponding to the
vertex $i$. Taken $\mathcal{I}=\\{1,2,3\\}$, then the generalized McKay quiver
of $(Q,G)$ is

where $\rho_{0}$, $\rho_{1}$ are the non-isomorphism irreducible
representations of $G_{3}=\langle a\rangle\cong\mathbb{Z}/2\mathbb{Z}$.
Reindexing the vertex set $\widehat{I}=\\{1,2,(3,\rho_{0}),(3,\rho_{1})\\}$ by
$\\{1,2,3,4\\},$ the Cartan matrix of $\widehat{Q}$ is

$A=(a_{ij})={\small\left(\begin{array}[]{cccc}2&-1&0&0\\\ -1&2&-1&-1\\\
0&-1&2&0\\\ 0&-1&0&2\\\ \end{array}\right)}.$

The Lie algebra $\mathfrak{g}:=\mathfrak{g}(\widehat{Q})$ is generated by
$\\{x_{i},y_{i},h_{i}\mid 1\leq i\leq 4\\}$ satisfying the relations

$\begin{array}[]{llll}\quad[h_{i},h_{j}]=0,&[x_{i},y_{j}]=\delta_{ij}h_{i};\\\
\quad[h_{i},x_{j}]=a_{ij}x_{j},&[h_{i},y_{j}]=-a_{ij}y_{j};\\\
\quad(\mbox{ad}x_{i})^{1-a_{ij}}(x_{j})=0,&(\mbox{ad}y_{i})^{1-a_{ij}}(y_{j})=0,\qquad
i\neq j.\end{array}$

In this case, the valued graph $\Gamma$ of $(Q,G)$ is

with the Cartan Matrix

$C={\small\left(\begin{array}[]{ccc}2&-1&0\\\ -1&2&-1\\\ 0&-2&2\\\
\end{array}\right)}.$

The Lie algebra $\mathfrak{g}(\Gamma)$ is generated by
$\\{X_{i},Y_{i},H_{i}\mid 1\leq i\leq 3\\}$ satisfying the relations

(5.4)
$\displaystyle\begin{array}[]{llll}\quad[H_{i},H_{j}]=0,&[X_{i},Y_{j}]=\delta_{ij}H_{i};\\\
\quad[H_{i},X_{j}]=c_{ij}X_{j},&[H_{i},Y_{j}]=-c_{ij}Y_{j};\\\
\quad(\mbox{ad}X_{i})^{1-c_{ij}}(X_{j})=0,&(\mbox{ad}Y_{i})^{1-c_{ij}}(Y_{j})=0,\qquad
i\neq j.\end{array}$

As discussed in Section 3.2, we see that the vertices $(3,\rho_{0})$ and
$(3,\rho_{1})$ of $\widehat{Q}$ are in the same $G$-orbit. Therefore, the Lie
algebra $\mathfrak{g}^{\overline{G}}$ is generated by

$\\{\overline{x}_{i},\overline{y}_{i},\overline{h}_{i}\mid 1\leq i\leq 3\\},$

where $\overline{x}_{i}=x_{i}$, $\overline{y}_{i}=y_{i}$,
$\overline{h}_{i}=h_{i}$ for $i=1,2$, and $\overline{x}_{3}=x_{3}+x_{4}$,
$\overline{y}_{3}=y_{3}+y_{4}$, $\overline{h}_{3}=h_{3}+h_{4}$, satisfying the
relations (5.4). Then, it is easy to see that the map

$\Phi:\quad\mathfrak{g}(\Gamma)\longrightarrow\mathfrak{g}^{\overline{G}}$

given by

$\Phi(X_{i})=\overline{x}_{i},\quad\Phi(Y_{i})=\overline{y}_{i},\quad\Phi(H_{i})=\overline{h}_{i}$

is an Lie algebra isomorphism.

At last, we consider quivers of $A$-type and $D$-type,

They have the same quiver isomorphism group $G=\mathbb{Z}/2\mathbb{Z}$. In
these cases, we have

$Q$ |  $G$ |  $\Gamma$ |  $\widehat{Q}$ |  $\widehat{\Gamma}$ |  Conclusion
---|---|---|---|---|---
$A_{2n+1}$ |  $\mathbb{Z}/2\mathbb{Z}$ |  $C_{n+1}$ |  $D_{n+2}$ |  $B_{n+1}$ | $\mathfrak{g}(C_{n+1})\cong\mathfrak{g}(D_{n+2})^{\mathbb{Z}/2\mathbb{Z}}$
$D_{n}$ |  $\mathbb{Z}/2\mathbb{Z}$ |  $B_{n-1}$ |  $A_{2n-1}$ |  $C_{n-1}$ | $\mathfrak{g}(B_{n-1})\cong\mathfrak{g}(A_{2n-1})^{\mathbb{Z}/2\mathbb{Z}}$

where $C_{n}$ and $B_{n}$ is the $C$-type and $B$-type Dynkin diagram,
respectively.

## References

  * [1] I. Assem, D. Simson, A. SKowroński, Elements of representation theory of associative algebras, Cambridge University Press, 2006.
  * [2] M. Auslander, Rational singularities and almost split sequences, Trans. Amer. Math. Soc. 293(1986) 511-531.
  * [3] M. Auslander, I. Reiten, S.O. Smalo, Representation theory of Artin algebras. Cambridge Stud. Adv. Math.,Vol.36, Cambridge University Press, 1995.
  * [4] E. Brieskorn, Rationale singularitäten komplexer Fliichen, Invent. Math. 4(1968) 336-358.
  * [5] W. Crawley-Boevey, M.P. Holland, Noncommutative deformations of Kleinan singularities. Duke Math. J. 92(1998) 605-635.
  * [6] L. Demonet, Skew group algebras of path algebras and preprojective algebras, J. Algebra 323(2010) 1052-1059.
  * [7] V. Dlab and C. M. Ringel, Indecomposable representations of graphs and algebras, Mem. Amer. Math. Soc. 173, 1976.
  * [8] J. Guo, On the McKay quivers and $m$-Cartan Matrices, Science in China (Series A: Mathematics), 52(2009) 513-518.
  * [9] J. Guo, Martiínez-Villa, Algebra pairs associated to McKay quivers. Comm. in Algebra 30(2002) 1017-1032.
  * [10] B. Hou, S. Yang, Skew group algebras of deformed preprojective algebras, J. Algebra (2011), 10.1016/j.jalgebra.2011.02.007. see also: arXiv: 1003.1797.
  * [11] A. Hubery, Representations of quiver respecting a quiver automorphism and a of Kac, Ph. D. thesis, Leeds Univeraity 2002.
  * [12] A. Hubery, Quiver representations respecting a quiver automorphism: a generalisation of a theorem of Kac, J. London Math. Soc. 69(2004) 79-96.
  * [13] V.G. Kac, Infinite dimensional Lie algebras, 3rd edn, Cambridge University Press, Cambridge, 1990.
  * [14] V.G. Kac, S.P. Wang, On automorphisms of Kac-Moody algebras and groups, Adv. in Math. 92(1992) 129-195.
  * [15] G. Lusztig, Affine quivers and canonical bases, Publ. Math. Inst. Hautes Études Sci. 76(1992) 111 C163.
  * [16] J. McKay, Graphs, singularities and finite groups, Proc. Sympos. Pure Math., vol. 37, Amer. Math. Soc., Providence, R. I., 1980, pp. 183-186.
  * [17] M. Reid, McKay correspondance, Preprint, math.AG/9702016.
  * [18] I. Reiten, C. Riedtmann, Skew group algebras in the representation theory of Artin algebras, J. Algebra 92(1)(1985) 224-282.