2.1. Finite $G$-categories

Let $\mathcal {C}$ be a finite category, that is, a category with only finitely many morphisms. As any object is determined by its identity endomorphism, the finite category $\mathcal {C}$ necessarily has only finitely many objects. Denote by ${\rm Obj}(\mathcal {C})$ (resp. ${\rm Mor}(\mathcal {C})$) the finite set of objects (resp. morphisms) in $\mathcal {C}$.

Recall that an automorphism on $\mathcal {C}$ is a fully faithful endofunctor which is bijective on objects. Therefore, such an automorphism has a genuine inverse. We denote by ${\rm Aut}(\mathcal {C})$ the automorphism group of $\mathcal {C}$, whose elements are automorphisms on $\mathcal {C}$ and whose multiplication is given by the composition of automorphisms.

Let $G$ be a finite group with its unit $1_G$. A finite $G$-category $\mathcal {C}$ is a finite category equipped with a group homomorphism

\[ \rho\colon G\longrightarrow {\rm Aut}(\mathcal{C}). \]

To simplify the notation, the following convention will be used: for $g\in G$ and $x\in {\rm Obj}(\mathcal {C})$, we write $g(x)=\rho (g)(x)$; for $\alpha \in {\rm Mor}(\mathcal {C})$, we write $g(\alpha )=\rho (g)(\alpha )$.

For a finite $G$-category $\mathcal {C}$, we will recall the skew group category $\mathcal {C}\rtimes G$; compare [Reference Reiten and Riedtmann26, Subsection 3.1] and [Reference Cibils and Marcos5, Definition 2.3]. It has the same objects as $\mathcal {C}$; for two objects $x$ and $y$, the corresponding Hom set is defined to be

\[ {\rm Hom}_{\mathcal{C}\rtimes G}(x,y)=\{(\alpha,g) \mid g\in G, \alpha\in {\rm Hom}_{\mathcal{C}}(g(x),y)\}. \]

For any morphisms $(\alpha,\,g)\in {\rm Hom}_{\mathcal {C}\rtimes G}(x,\,y)$ and $(\beta,\,h)\in {\rm Hom}_{\mathcal {C}\rtimes G}(y,\,z)$, the composition is defined by

(2.1)\begin{equation} (\beta,h)\circ (\alpha,g)=(\beta\circ h(\alpha), hg). \end{equation}

We observe that the identity endomorphism of $x$ in $\mathcal {C}\rtimes G$ is given by $({\rm Id}_x,\, 1_G)$, where ${\rm Id}_x$ is the identity endomorphism of $x$ in $\mathcal {C}$. We mention that the formation of a skew group category might be viewed as a very special case of the Grothendieck construction; compare [Reference Grothendieck15, VI.8], [Reference Webb32, Section 7] and [Reference Xu35, Subsection 2.1].

Let $\mathbb {K}$ be a field and $\mathcal {C}$ be a finite category. The category algebra $\mathbb {K}\mathcal {C}$ of $\mathcal {C}$ is a finite dimensional $\mathbb {K}$-algebra defined as follows. As a $\mathbb {K}$-vector space, $\mathbb {K}\mathcal {C}=\bigoplus \limits _{\alpha \in {\rm Mor}(\mathcal {C})} \mathbb {K}\alpha$, and the product between the basis elements is given by the following rule:

\[ \alpha \beta=\left\{\begin{array}{@{}ll} \alpha\circ\beta, & \text{ if } \alpha\text{ and }\beta\text{ can be composed in }\mathcal{C}; \\ 0, & \text{otherwise.} \end{array}\right. \]

The unit of $\mathbb {K}\mathcal {C}$ is given by $1_{\mathbb {K}\mathcal {C}}=\sum \limits _{x \in {\rm Obj}(\mathcal {C})}{\rm Id}_x$.

Denote by $(\mathbb {K}\mbox {-mod})^\mathcal {C}$ the category of covariant functors from $\mathcal {C}$ to $\mathbb {K}\mbox {-mod}$. There is a canonical equivalence

(2.2)\begin{equation} {\rm can}\colon \mathbb{K}\mathcal{C}\mbox{-mod} \stackrel{\sim}\longrightarrow (\mathbb{K}\mbox{-mod})^\mathcal{C}, \end{equation}

sending a $\mathbb {K}\mathcal {C}$-module $M$ to the functor ${\rm can}(M)\colon \mathcal {C}\rightarrow \mathbb {K}\mbox {-mod}$ described as follows: ${\rm can}(M)(x)={\rm Id}_x.M$ for each object $x$ in $\mathcal {C}$; for any morphism $\alpha \colon x\rightarrow y$, we have

\[ {\rm can}(M)(\alpha)\colon {\rm can}(M)(x)\longrightarrow {\rm can}(M)(y),\; m\mapsto \alpha.m. \]

For details, we refer to [Reference Webb32, Proposition 2.1].

Denote by ${\rm Aut}(\mathbb {K}\mathcal {C})$ the group of algebra automorphisms on $\mathbb {K}\mathcal {C}$. Each categorical automorphism on $\mathcal {C}$ induces uniquely an algebra automorphism on $\mathbb {K}\mathcal {C}$. Therefore, there is a canonical embedding of groups

\[ {\rm Aut}(\mathcal{C})\hookrightarrow {\rm Aut}(\mathbb{K}\mathcal{C}). \]

Assume that $\mathcal {C}$ is a finite $G$-category. The group homomorphism $\rho \colon G\rightarrow {\rm Aut}(\mathcal {C})$ induces a group homomorphism $\rho '\colon G\rightarrow {\rm Aut}(\mathbb {K}\mathcal {C})$. In other words, the group $G$ acts on the algebra $\mathbb {K}\mathcal {C}$ by algebra automorphisms. We denote by $\mathbb {K}\mathcal {C}\# G$ the corresponding skew group algebra. Here, we recall that $\mathbb {K}\mathcal {C}\# G=\mathbb {K}\mathcal {C}\otimes \mathbb {K}G$ as a $\mathbb {K}$-vector space, where the tensor product $\alpha \otimes g$ is written as $\alpha \#g$. The multiplication is given by

\[ (\beta\# h)(\alpha \# g)=\beta h(\alpha)\# hg \]

for any $\alpha,\, \beta \in {\rm Mor}(\mathcal {C})$ and $g,\, h\in G$. We emphasize that on the right hand side, $\beta h(\alpha )$ means the product of $\beta$ and $h(\alpha )$ in $\mathbb {K}\mathcal {C}$, namely, the composition $\beta \circ h(\alpha )$ in $\mathcal {C}$.

The following easy observation, extending [Reference Xu36, Lemma 2.3.2], justifies the terminology ‘skew group category’.

Proposition 2.1 Let $\mathcal {C}$ be a finite $G$-category. Then there is an isomorphism of algebras

\[ \mathbb{K}(\mathcal{C}\rtimes G)\overset{\sim}{\longrightarrow} \mathbb{K}\mathcal{C}\# G, \]

sending a morphism $(\alpha,\, g)$ in $\mathcal {C}\rtimes G$ to the element $\alpha \# g$ in $\mathbb {K}\mathcal {C}\#G$.

In the following lemma, we collect elementary facts on skew group categories.

2.2. The EI property

Let $\mathcal {C}$ be a finite $G$-category as above. For each object $x$ in $\mathcal {C}$, we denote by $G_x=\{g\in G \mid g(x)=x\}$ its stabilizer. We observe that $G_x$ acts on the monoid ${\rm Hom}_\mathcal {C}(x,\, x)$ by monoid automorphisms. Denote by ${\rm Hom}_\mathcal {C}(x,\, x)\rtimes G_x$ the corresponding semi-direct product. There is an inclusion between monoids

\[ {\rm inc}_x\colon {\rm Hom}_\mathcal{C}(x, x)\rtimes G_x\hookrightarrow {\rm Hom}_{\mathcal{C}\rtimes G}(x, x), \quad (\alpha, g)\mapsto (\alpha, g). \]

The following terminology is inspired by [Reference Lusztig23, Subsection 12.1.1].

Recall from [Reference Webb32] that a finite category $\mathcal {C}$ is EI if every endomorphism is an isomorphism. Therefore, for each object $x$, ${\rm Hom}_{\mathcal {C}}(x,\,x)={\rm Aut}_{\mathcal {C}}(x)$ is a finite group. Finite EI categories are of interest from many different perspectives; for example, see [Reference Webb33, Reference Xu36].

Proof. For the ‘if’ part, we assume that $\mathcal {C}\rtimes G$ is an EI category. For any $\alpha \in {\rm Hom}_{\mathcal {C}}(x,\,x)$, $(\alpha,\,1_G)$ is an endomorphism of $x$ in $\mathcal {C}\rtimes G$. Since $\mathcal {C}\rtimes G$ is EI, $(\alpha,\, 1_G)$ is an isomorphism. By lemma 2.2(1), the endomorphism $\alpha$ is an isomorphism in $\mathcal {C}$, as required.

For the ‘only if’ part, we assume that $\mathcal {C}$ is an EI category. Any endomorphism of $x$ in $\mathcal {C}\rtimes G$ is of the form $(\alpha,\, g)$, where $\alpha \colon g(x)\rightarrow x$ is a morphism in $\mathcal {C}$. Assume that $g^d=1_G$ for some $d\geq 1$. Then we have a chain

\[ x=g^d(x)\stackrel{g^{d-1}(\alpha)} \longrightarrow g^{d-1}(x) \longrightarrow \cdots \longrightarrow g^2(x)\stackrel{g(\alpha)}\longrightarrow g(x)\stackrel{\alpha}\longrightarrow x \]

of morphisms in $\mathcal {C}$. Since $\mathcal {C}$ is EI, it follows that all the morphisms in the chain are isomorphisms. In particular, the morphism $\alpha$ is an isomorphism. Applying lemma 2.2(1), we infer that the endomorphism $(\alpha,\, g)$ is an isomorphism, proving that $\mathcal {C}\rtimes G$ is an EI category.

The following corollary follows immediately from lemma 2.4.

Corollary 2.6 Let $\mathcal {C}$ be a finite admissible $G$-category. Assume that $\mathcal {C}$ is EI. Then for each object $x$, we have an identification of groups

\[ {\rm Aut}_\mathcal{C}(x)\rtimes G_x={\rm Aut}_{\mathcal{C}\rtimes G}(x). \]

4.1. Categories associated to finite EI quivers

Let $Q=(Q_0,\, Q_1; s,\,t)$ be a finite quiver, where $Q_0$ and $Q_1$ are the finite sets of vertices and arrows, respectively. The maps $s,\, t\colon Q_1\rightarrow Q_0$ assign to each arrow $\alpha$ its starting vertex $s(\alpha )$ and terminating vertex $t(\alpha )$, respectively.

A path $p=\alpha _n\cdots \alpha _2\alpha _1$ of length $n$ in $Q$ consists of arrows $\alpha _i$ satisfying $t(\alpha _i)=s(\alpha _{i+1})$ for each $1\leq i\leq n-1$. Here, we write concatenation from right to left. We set $s(p)=s(\alpha _1)$ and $t(p)=t(\alpha _n)$. An arrow is identified with a path of length one. To each vertex $i\in Q_0$, we associate a trivial path $e_i$ of length zero, satisfying $s(e_i)=i=t(e_i)$.

A finite quiver $Q$ is said to be acyclic, provided that there is no oriented cycle in $Q$, that is, there is no nontrivial path with the same starting and terminating vertex. This is equivalent to the condition that there are only finitely many paths in $Q$.

Let $H$ be a finite group, and let $X$ be a right $H$-set, that is, $H$ acts on $X$ on the right. Let $Y$ be a left $H$-set. The biset product $X\times _H Y$ is defined to be the set

\[ X\times Y/{\sim} \]

of equivalence classes with respect to the equivalence relation $\sim$ given by $(x.h,\, y)\sim (x,\, h.y)$ for $x\in X,\, h\in H$ and $y\in Y$. By abuse of notation, the elements in $X\times Y/\sim$ are still denoted by $(x,\,y)$ for $x\in X$ and $y\in Y$.

Let $G$ and $K$ be finite groups. By a $(G,\, H)$-biset $X$, we mean a set $X$ which is a left $G$-set and a right $H$-set satisfying $(g.x).h=g.(x.h)$ for any $g\in G$, $x\in X$ and $h\in H$. Here, we use the dot to denote the group actions. Let $Y$ be a $(H,\, K)$-biset. Then the biset product $X\times _H Y$ is naturally a $(G,\, K)$-biset.

Recall from [Reference Li21, Definition 2.1] that a finite EI quiver $(Q,\, U)$ consists of a finite acyclic quiver $Q$ and an assignment $U=(U(i),\, U(\alpha ))_{i\in Q_0, \alpha \in Q_1}$. In more details, for each vertex $i\in Q_0$, $U(i)$ is a finite group, and for each arrow $\alpha \in Q_1$, $U(\alpha )$ is a finite $(U(t\alpha ),\, U(s\alpha ))$-biset. Here, we emphasize that each $U(\alpha )$ is nonempty.

For any path $p=\alpha _n\cdots \alpha _2\alpha _1$ in $Q$, we define

\[ U(p)=U({\alpha_n}) \times_{U({t\alpha_{n-1}})} U({\alpha_{n-1}}) \times_{U({t\alpha_{n-2}})} \cdots \times_{U({t\alpha_2})} U({\alpha_2})\times_{U({t\alpha_1})} U({\alpha_1}) . \]

Then $U(p)$ is naturally a $(U(tp),\, U(sp))$-biset. A typical element in $U(p)$ will be denoted by $(u_n,\, \cdots,\, u_2,\, u_1)$ with each $u_i\in U(\alpha _i)$. For each vertex $i\in Q_0$, we identify $U(e_i)$ with $U(i)$.

For two paths $p,\, q$ satisfying $s(p)=t(q)$, we have a natural isomorphism of $(U(tp),\, U(sq))$-bisets

(4.1)\begin{equation} U(p)\times_{U(tq)} U(q)\stackrel{\sim}\longrightarrow U(pq), \end{equation}

sending $((u_m',\,\cdots,\, u_1'),\, (u_n,\, \cdots,\, u_1))$ to $(u_m',\, \cdots,\, u_1',\, u_n,\, \cdots,\, u_1)$, where $pq$ denotes the concatenation of paths.

Each finite EI quiver $(Q,\, U)$ gives rise to a finite EI category $\mathcal {C}(Q,\, U)$; see [Reference Li21, Section 2]. The objects of $\mathcal {C}(Q,\, U)$ coincide with the vertices of $Q$. For two objects $i$ and $j$, we have a disjoint union

\[ {\rm Hom}_{\mathcal{C}(Q, U)}(i, j)=\bigsqcup_{\{p {\rm{ \; paths \; in\; }} Q {\rm{\; with \; }} s(p)=i {\rm{\; and \; }} t(p)=j\}} U(p). \]

The composition of morphisms is induced by the concatenation of paths and the isomorphism (4.1). Since $Q$ has only finitely many paths, we infer that $\mathcal {C}(Q,\, U)$ is a finite category. As $e_i$ is the only path starting and terminating at $i$, we infer that

(4.2)\begin{equation} {\rm Hom}_{\mathcal{C}(Q, U)}(i, i)=U(e_i)=U(i), \end{equation}

which is a finite group. We conclude that the category $\mathcal {C}(Q,\, U)$ is indeed finite EI. We mention the following immediate fact

(4.3)\begin{equation} {\rm Hom}^0_{\mathcal{C}(Q, U)}(i, j)=\bigsqcup_{\{\alpha\in Q_1\mid s(\alpha)=i, \quad t(\alpha)=j\}} U(\alpha). \end{equation}

By [Reference Li21, Proposition 2.8], the EI category $\mathcal {C}(Q,\, U)$ is free. Moreover, a finite EI category is free if and only if it is equivalent to $\mathcal {C}(Q,\, U)$ for some finite EI quiver $(Q,\, U)$.

4.2. A universal property

The free EI category $\mathcal {C}(Q,\, U)$ enjoys a certain universal property; compare [Reference Li21, Proposition 2.9].

Before giving the proof, we leave two comments to clarify the statements. The $(U(t\alpha ),\, U(s\alpha ))$-biset structure on ${\rm Hom}_\mathcal {D}(\phi (s\alpha ),\, \phi (t\alpha ))$ is given as follows: for a morphism $f\colon \phi (s\alpha )\rightarrow \phi (t\alpha )$ in $\mathcal {D}$, $x\in U(t\alpha )$ and $x'\in U(s\alpha )$, we have

\[ x.f.x'=\psi_{t(\alpha)}(x)\circ f\circ \psi_{s(\alpha)}(x'). \]

In the condition (E4), the domain of the bijection is a disjoint union; moreover, it implies that $\psi _\alpha (u)$ is unfactorizable in $\mathcal {D}$ for any $u\in U(\alpha )$.

Proof. Set $\mathcal {C}=\mathcal {C}(Q,\, U)$. For any path $p=\alpha _n\cdots \alpha _2\alpha _1$ in $Q$ and an element $(u_n,\, \cdots,\, u_2,\, u_1)\in U(p)$, we define

\[ \Phi(u_n, \cdots, u_2, u_1)=\psi_{\alpha_n}(u_n)\circ \cdots \circ \psi_{\alpha_2}(u_2)\circ \psi_{\alpha_1}(u_1). \]

We claim that this is independent of the choice of the representatives; compare [Reference Li21, the proof of Proposition 2.9].

Assume that $(u_n,\, \cdots,\, u_2,\, u_1)=(v_n,\, \cdots,\, v_2,\, v_1)$ in $U(p)$. This means that there are elements $x_i\in U(t\alpha _i)$ for each $1\leq i\leq n-1$, such that the following identities hold:

\[ v_n=u_n.x_{n-1}, \; v_{i}=x_i^{{-}1}.u_i.x_{i-1}, \mbox{ and } v_1=x_1^{{-}1}.u_1. \]

Since each $\psi _{\alpha _i}$ is a map of bisets, we have

\begin{align*} \psi_{\alpha_n}(v_n)& =\psi_{\alpha_n}(u_n)\circ \psi_{t(\alpha_{n-1})}(x_{n-1}),\; \psi_{\alpha_i}(v_{i})\\ & =\psi_{t(\alpha_i)}(x_i)^{{-}1}\circ \psi_{\alpha_i}(u_i)\circ \psi_{t(\alpha_{i-1})}(x_{i-1}), \end{align*}

and

\[ \psi_{\alpha_1}(v_1)=\psi_{t(\alpha_1)}(x_1)^{{-}1}\circ \psi_{\alpha_1}(u_1). \]

Then the following identity follows immediately.

\begin{align*} \Phi(v_n, \cdots, v_2,v_1)& =\psi_{\alpha_n}(v_n)\circ \cdots \circ \psi_{\alpha_2}(v_2)\circ \psi_{\alpha_1}(v_1)\\ & =\psi_{\alpha_n}(u_n) \psi_{t(\alpha_{n-1})}(x_{n-1})\\ & \quad \circ \cdots \circ \psi_{t(\alpha_2)}(x_2)^{{-}1} \psi_{\alpha_2}(u_2)\psi_{t(\alpha_{1})}(x_{1})\circ \psi_{t(\alpha_1)}(x_1)^{{-}1} \psi_{\alpha_1}(u_1)\\ & =\psi_{\alpha_n}(u_n)\circ \cdots \circ \psi_{\alpha_2}(u_2)\circ \psi_{\alpha_1}(u_1)=\Phi(u_n, \cdots, u_2, u_1). \end{align*}

The above claim yields a well-defined functor $\Phi$. The uniqueness of $\Phi$ is clear, as $(u_n,\, \cdots,\, u_2,\, u_1)$ might be viewed as the composition $u_n\circ \cdots \circ u_2\circ u_1$ in $\mathcal {C}$.

For the ‘only if’ part of the second statement, we assume that $\Phi$ is an equivalence. Then (E1) is clear, since $\mathcal {C}$ is free. Since $\mathcal {C}$ is skeletal and $\Phi$ respects isomorphism classes, (E2) follows immediately. The condition (E3) is just the denseness of $\Phi$. For (E4), we observe that the equivalence $\Phi$ necessarily induces isomorphisms

\[ {\rm Aut}_\mathcal{C}(i)\simeq {\rm Aut}_\mathcal{D}(\phi(i)) \]

of groups and bijections

\[ {\rm Hom}_\mathcal{C}^0(i, j)\simeq {\rm Hom}_\mathcal{D}^0(\phi(i), \phi(j)) \]

between the sets of unfactorizable morphisms. Then we apply (4.2) and (4.3).

For the ‘if’ part, we assume the conditions (E1)-(E4). By (E3), the functor $\Phi$ is dense. It suffices to prove that for any $i,\, j\in Q_0$, the following map

\[ \Phi_{i,j}\colon {\rm Hom}_\mathcal{C}(i, j)\longrightarrow {\rm Hom}_\mathcal{D}(\phi(i), \phi(j)),\quad f\mapsto \Phi(f) \]

is bijective.

By (E4), each $\psi _i$ is an isomorphism, and then the case $i=j$ follows. We now assume that $i\neq j$. Then by (E2), $\phi (i)$ and $\phi (j)$ are not isomorphic.

Recall from [Reference Li21, Proposition 2.6] that each morphism in $\mathcal {D}$ has a factorization into unfactorizable morphisms. Since $\Phi$ is dense, any morphism $g\colon \phi (i)\rightarrow \phi (j)$ admits a factorization

\[ \phi(i)\stackrel{g_0}\longrightarrow \phi(i_1) \stackrel{g_1}\longrightarrow \phi(i_2) \longrightarrow \cdots \longrightarrow \phi(i_{n-1}) \stackrel{g_{n-1}} \longrightarrow \phi(j) \]

with each $g_k$ unfactorizable. By (E4), each $g_k$ belongs to the image of $\Phi$. It follows that there is a morphism $f\colon i\rightarrow j$ in $\mathcal {C}$ satisfying $\Phi (f)=g$. This proves that $\Phi _{i,j}$ is surjective.

It remains to show that $\Phi _{i,j}$ is injective. Assume that $p=\alpha _n\cdots \alpha _2\alpha _1$ and $q=\beta _m\cdots \beta _2\beta _1$ are two paths from $i$ to $j$, and that $(u_n,\, \cdots,\, u_2,\, u_1)\in U(p)$ and $(v_m,\, \cdots,\, v_2,\, v_1)\in U(q)$ satisfy

\[ \Phi(u_n, \cdots, u_2, u_1)=\Phi(v_m, \cdots, v_2, v_1)=g'. \]

We claim that $p=q$ and $(u_n,\, \cdots,\, u_2,\, u_1)=(v_m,\, \cdots,\, v_2,\, v_1)$. Then we are done.

For the claim, we observe that the morphism $g'$ admits two factorizations:

\[ \phi(i)\stackrel{\psi_{\alpha_1}(u_1)}\longrightarrow \phi(i_1) \stackrel{\psi_{\alpha_2}(u_2)}\longrightarrow \phi(i_2) \longrightarrow \cdots \longrightarrow \phi(i_{n-1}) \stackrel{\psi_{\alpha_n}(u_n)}\longrightarrow \phi(j) \]

and

\[ \phi(i)\stackrel{\psi_{\beta_1}(v_1)}\longrightarrow \phi(j_1) \stackrel{\psi_{\beta_2}(v_2)}\longrightarrow \phi(j_2) \longrightarrow \cdots \longrightarrow \phi(j_{m-1}) \stackrel{\psi_{\beta_m}(v_m)}\longrightarrow \phi(j). \]

Here, $i_k=t(\alpha _k)$ and $j_k=t(\beta _k)$. By (E4), all the morphisms appearing in the two factorizations are unfactorizable. By (E1), the EI category $\mathcal {D}$ is free, that is, any morphism satisfies the UFP. Consequently, $m=n$ and there are isomorphisms $g'_k\colon \phi (i_k)\rightarrow \phi (j_k)$ making the following diagram commute.

By (E2), we infer that $i_k=j_k$; moreover, by (E4) we obtain automorphisms $a_k\in {\rm Aut}_\mathcal {C}(i_k)=U(i_k)$ satisfying $\psi _{i_k}(a_k)=g'_k$. The commutativity yields

\[ \psi_{\beta_{k+1}}(v_{k+1})\circ \psi_{i_k}(a_k)=\psi_{i_{k+1}}(a_{k+1})\circ \psi_{\alpha_{k+1}}(u_{k+1}) \]

for each $0\leq k\leq n-1$. Here, $a_0$ and $a_n$ are the identity elements in $U(i)$ and $U(j)$, respectively. The above identity is equivalent to

\[ \psi_{\beta_{k+1}}(v_{k+1}.a_k)=\psi_{\alpha_{k+1}}(a_{k+1}.u_{k+1}). \]

By the bijection in (E4), we infer that $\beta _{k+1}=\alpha _{k+1}$ and that

\[ v_{k+1}.a_k=a_{k+1}.u_{k+1} \]

for each $0\leq k\leq n-1$. It follows that $p=q$; moreover, in view of the definition of $U(p)$ via biset products, we infer that $(u_n,\, \cdots,\, u_2,\, u_1)=(v_n,\, \cdots,\, v_2,\, v_1)$, proving the claim.

4.3. $G$-actions on finite EI quivers

Let $(Q,\,U)$ be a finite EI quiver. An automorphism $\sigma =(\sigma ^0,\,\sigma ^1)$ of $(Q,\,U)$ consists of an automorphism $\sigma ^0\colon Q\rightarrow Q$ of the acyclic quiver $Q$ and an assignment $\sigma ^1=(\sigma ^1_i,\,\sigma ^1_\alpha )_{i\in Q_0, \alpha \in Q_1}$ of isomorphisms. More precisely, for each $i\in Q_0$,

\[ \sigma^1_i\colon U(i)\overset{\sim}{\longrightarrow} U(\sigma^0(i)) \]

is an isomorphism of groups; for each arrow $\alpha \in Q_1$,

\[ \sigma^1_\alpha \colon U(\alpha)\overset{\sim}{\longrightarrow} U(\sigma^0(\alpha)) \]

is an isomorphism of $(U(t\alpha ),\, U(s\alpha ))$-bisets. Here, the $(U(t\alpha ),\, U(s\alpha ))$-biset structure on $U(\sigma ^0(\alpha ))$ is induced by the group isomorphisms $\sigma ^1_{t(\alpha )}$ and $\sigma ^1_{s(\alpha )}$.

The composition of two automorphisms $\sigma =(\sigma ^0,\,\sigma ^1)$ and $\theta =(\theta ^0,\,\theta ^1)$ on $(Q,\,U)$ is given by

\[ \theta\circ \sigma =(\theta^0\circ \sigma^0,\theta^1\star\sigma^1), \]

where the assignment $\theta ^1\star \sigma ^1$ is given by

\[ (\theta^1\star\sigma^1)_i=\theta^1_{\sigma^0(i)}\circ \sigma^1_i \mbox{ and } (\theta^1\star\sigma^1)_\alpha=\theta^1_{\sigma^0(\alpha)} \circ \sigma^1_\alpha. \]

We denote by ${\rm Aut}(Q,\,U)$ the group of automorphisms of $(Q,\,U)$, whose multiplication is given by the composition of automorphisms.

We observe that each automorphism $\sigma =(\sigma ^0,\, \sigma ^1)$ on $(Q,\, U)$ induces an automorphism $\tilde {\sigma }$ on $\mathcal {C}(Q,\, U)$ in the following natural manner: the action of $\tilde {\sigma }$ on objects is given by $\sigma ^0$; for $u\in U(i)$, we have $\tilde {\sigma }(u)=\sigma ^1_i(u)\in U(\sigma ^0(i))$; for a path $p=\alpha _n\cdots \alpha _2\alpha _1$ and a morphism $(u_n,\, \cdots,\, u_2,\, u_1)\in U(p)$, we have

\[ \tilde{\sigma}(u_n, \cdots, u_2, u_1)=(\sigma^1_{\alpha_n}(u_n), \cdots, \sigma^1_{\alpha_2}(u_2), \sigma^1_{\alpha_1}(u_1))\in U(\sigma^0(p)). \]

This actually gives rise to an injective group homomorphism

(4.4)\begin{equation} {\rm Aut}(Q, U)\hookrightarrow {\rm Aut}(\mathcal{C}(Q, U)), \quad \sigma\mapsto \tilde{\sigma}. \end{equation}

Let $G$ be a finite group. By a $G$-action on a finite EI quiver $(Q,\,U)$, we mean a group homomorphism

\[ \rho\colon G\longrightarrow {\rm Aut}(Q,U),\quad g\mapsto \rho(g)=(\rho(g)^0,\rho(g)^1). \]

Composing $\rho$ with (4.4), the $G$-action makes $\mathcal {C}(Q,\, U)$ into a $G$-category.

The following convention for the $G$-action $\rho$ will simplify the notation. For $g\in G$ and $i\in Q_0={\rm Obj}(\mathcal {C}(Q,\,U))$, we write

(4.5)\begin{equation} g(i)=\rho(g)^0(i)\in Q_0. \end{equation}

Similarly, for $\alpha \in Q_1$, we write $g(\alpha )=\rho (g)^0(\alpha )\in Q_1$. For $a\in U(i)={\rm Aut}_{\mathcal {C}(Q,\,U)}(i)$ with $i\in Q_0$, we write

(4.6)\begin{equation} g(a)=\rho(g)^1_i(a)\in U(g(i)). \end{equation}

For $u\in U(\alpha )\subseteq {\rm Hom}^0_{\mathcal {C}(Q,\, U)}(i,\, j)$ with an arrow $\alpha \in Q_1$ from $i$ to $j$, we write

(4.7)\begin{equation} g(u)=\rho(g)^1_\alpha(u)\in U(g(\alpha)). \end{equation}

For admissible $G$-categories, we refer to definition 2.3.

Proof. Let $g\in G$ and $i\in {\rm Obj}(\mathcal {C}(Q,\,U))=Q_0$ such that $g(i)\neq i$. Recall that

\[ {\rm Hom}_{\mathcal{C}(Q,U)}(g(i),i)=\bigsqcup_{\{p {\rm{ \; paths \; in\; }} Q {\rm{\; with \; }} s(p)=g(i) {\rm{\; and \; }} t(p)=i\}} U(p). \]

Since $Q$ is acyclic, there is no path $p$ satisfying $s(p)=g(i)$ and $t(p)=i$. Therefore, the set ${\rm Hom}_{\mathcal {C}(Q,\,U)}(g(i),\,i)$ is actually empty, proving that the $G$-category $\mathcal {C}(Q,\, U)$ is admissible.

5.1. The construction of $(\overline {Q},\, \overline {U})$

The finite quiver $\overline {Q}=(\overline {Q}_0,\, \overline {Q}_1; s,\, t)$ is just the quotient quiver of $Q$ by $G$. In more details, $\overline {Q}_0=Q_0/G$ and $\overline {Q}_1=Q_1/G$ are the corresponding sets of $G$-orbits, and the maps $s$ and $t$ are induced by the ones of $Q$. Since $Q$ is acyclic, we infer that the finite quiver $\overline {Q}$ is acyclic. By definition, we have the canonical projections

\[ \pi_0\colon Q_0\longrightarrow \overline{Q}_0 \mbox{ and } \pi_1\colon Q_1\longrightarrow \overline{Q}_1. \]

The vertices and arrows in $\overline {Q}$ are written in the bold form. For example, the vertices are usually denoted by $\boldsymbol {i}$ and $\boldsymbol {j}$.

To define the assignment $\overline {U}$, we have to fix three maps

(5.1)\begin{equation} \iota_0\colon \overline{Q}_0\longrightarrow Q_0,\quad \iota_1\colon \overline{Q}_1\longrightarrow Q_1,\quad \mbox{and } g_{(-)}\colon \overline{Q}_1\longrightarrow G \end{equation}

satisfying the following conditions: $\pi _0\circ \iota _0={\rm Id}_{\overline {Q}_0}$, $\pi _1\circ \iota _1={\rm Id}_{\overline {Q}_1}$,

(5.2)\begin{equation} t(\iota_1(\boldsymbol{\alpha}))=\iota_0(t(\boldsymbol{\alpha})) \mbox{ and } s(\iota_1(\boldsymbol{\alpha}))=g_{\boldsymbol{\alpha}}(\iota_0(s\boldsymbol{\alpha})) \end{equation}

for each arrow $\boldsymbol {\alpha }\in \overline {Q}_1$. Here, we use the convention (4.5) for $g_{\boldsymbol {\alpha }}(\iota _0(s\boldsymbol {\alpha }))$.

The inclusion $G_{\iota _1(\boldsymbol {\alpha })}\subseteq G_{\iota _0(t\boldsymbol {\alpha })}$ makes $G_{\iota _0(t\boldsymbol {\alpha })}$ a right $G_{\iota _1(\boldsymbol {\alpha })}$-set. The injective group homomorphism

\[ G_{\iota_1(\boldsymbol{\alpha})}\subseteq G_{s(\iota_1(\boldsymbol{\alpha}))} \overset{\sim}{\longrightarrow} G_{\iota_0(s \boldsymbol{\alpha})}, \quad k\mapsto g_{\boldsymbol{\alpha}}^{{-}1}kg_{\boldsymbol{\alpha}} \]

makes $G_{\iota _0(s\boldsymbol {\alpha })}$ a left $G_{\iota _1(\boldsymbol {\alpha })}$-set. Therefore, we have the biset product

\[ G_{\iota_0(t\boldsymbol{\alpha})} \times_{G_{\iota_1(\boldsymbol{\alpha})}} G_{\iota_0(s \boldsymbol{\alpha})}, \]

which is naturally a $(G_{\iota _0(t\boldsymbol {\alpha })},\,G_{\iota _0(s\boldsymbol {\alpha })})$-biset. A typical element in the above biset product is written as $(h,\, g)$ with $h\in G_{\iota _0(t\boldsymbol {\alpha })}$ and $g\in G_{\iota _0(s\boldsymbol {\alpha })}$. By definition, we have

(5.3)\begin{equation} (hk, g)=(h, g_{\boldsymbol{\alpha}}^{{-}1}kg_{\boldsymbol{\alpha}}g) \end{equation}

for each $k\in G_{\iota _1(\boldsymbol {\alpha })}$.

Finally, we choose a right coset decomposition

(5.4)\begin{equation} G_{\iota_0(t\boldsymbol{\alpha})} = \bigsqcup_{r=1}^{m_{\boldsymbol{\alpha}}} h_{\boldsymbol{\alpha},r} G_{\iota_1(\boldsymbol{\alpha})}. \end{equation}

Consequently, any element in $G_{\iota _0(t\boldsymbol {\alpha })} \times _{G_{\iota _1(\boldsymbol {\alpha })}} G_{\iota _0(s\boldsymbol {\alpha })}$ is uniquely written as $(h_{\boldsymbol {\alpha },\, r},\,k)$ for $1\leq r\leq m_{\boldsymbol {\alpha }}$ and $k\in G_{\iota _0(s\boldsymbol {\alpha })}$.

The construction of the assignment $\overline {U}$ is as follows. For each vertex $\boldsymbol {i}$ of $\overline {Q}$, we set

\[ \overline{U}(\boldsymbol{i})=U(\iota_0(\boldsymbol{i}))\rtimes G_{\iota_0(\boldsymbol{i})}. \]

Here, we note that $G_{\iota _0(\boldsymbol {i})}$ acts on $U(\iota _0(\boldsymbol {i}))$ by group automorphisms. Therefore, the semi-direct product is well defined. For each arrow $\boldsymbol {\alpha }\colon \boldsymbol {i}\rightarrow \boldsymbol {j}$ in $\overline {Q}$, we set

\begin{align*} \overline{U}(\boldsymbol{\alpha})& =U(\iota_1(\boldsymbol{\alpha})) \times (G_{\iota_0(t\boldsymbol{\alpha})} \times_{G_{\iota_1(\boldsymbol{\alpha})}} G_{\iota_0(s\boldsymbol{\alpha})})\\ & =U(\iota_1(\boldsymbol{\alpha})) \times (G_{\iota_0(\boldsymbol{j})} \times_{G_{\iota_1(\boldsymbol{\alpha})}} G_{\iota_0(\boldsymbol{i})}). \end{align*}

A typical element in $\overline {U}(\boldsymbol {\alpha })$ is denoted by $(u,\, (h_{\boldsymbol {\alpha },\, r},\, k))$. The right $\overline {U}(\boldsymbol {i})$-action is given by

\[ (u,(h_{\boldsymbol{\alpha}, r},k)).(a, g)=(u.(g_{\boldsymbol{\alpha}}k(a)), (h_{\boldsymbol{\alpha}, r},kg)) \]

for any $(a,\, g)\in \overline {U}(\boldsymbol {i})=U(\iota _0(\boldsymbol {i}))\rtimes G_{\iota _0(\boldsymbol {i})}$. Here, using the convention (4.6), we observe that $g_{\boldsymbol {\alpha }}k(a)$ lies in $U(g_{\boldsymbol {\alpha }} \iota _0(\boldsymbol {i}))=U(s\iota _1(\boldsymbol {\alpha }))$, and that $u.(g_{\boldsymbol {\alpha }}k(a))$ means the right $U(s\iota _1(\boldsymbol {\alpha }))$-action on $U(\iota _1(\boldsymbol {\alpha }))$.

To describe the left $\overline {U}(\boldsymbol {j})$-action, we take an arbitrary element $(b,\, h)\in \overline {U}(\boldsymbol {j})=U(\iota _0(\boldsymbol {j}))\rtimes G_{\iota _0(\boldsymbol {j})}$. Assume that

\[ h h_{\boldsymbol{\alpha}, r}=h_{\boldsymbol{\alpha}, p} k' \]

for some $1\leq p\leq m_{\boldsymbol {\alpha }}$ and $k'\in G_{\iota _1(\boldsymbol {\alpha })}$. The left $\overline {U}(\boldsymbol {j})$-action is given by

\begin{align*} (b, h).(u,(h_{\boldsymbol{\alpha}, r},k))& =(h^{{-}1}_{\boldsymbol{\alpha}, p}(b).k'(u), (h_{\boldsymbol{\alpha},p}k', k))\\ & =(h^{{-}1}_{\boldsymbol{\alpha}, p}(b).k'(u), (h_{\boldsymbol{\alpha},p}, g^{{-}1}_{\boldsymbol{\alpha}}k'g_{\boldsymbol{\alpha}}k)). \end{align*}

Here, we observe that $h^{-1}_{\boldsymbol {\alpha },\, p}(b)$ lies in $U(\iota _0(\boldsymbol {j}))=U(t\iota _1(\boldsymbol {\alpha }))$, and that $k'(u)$ lies in $U(\iota _1(\boldsymbol {\alpha }))$. The convention (4.7) is used for $k'(u)$. Finally, $h^{-1}_{\boldsymbol {\alpha },\, p}(b).k'(u)$ denotes the left $U(t\iota _1(\boldsymbol {\alpha }))$-action on $U(\iota _1(\boldsymbol {\alpha }))$.

The above actions make $\overline {U}(\boldsymbol {\alpha })$ a $(\overline {U}(\boldsymbol {j}),\, \overline {U}(\boldsymbol {i}))$-biset. In summary, we have defined the finite EI quiver $(\overline {Q},\, \overline {U})$.

The above construction of the quotient EI quiver $(\overline {Q},\, \overline {U})$ is rather general. In what follows, we impose conditions which will simplify the construction.

Remark 5.1 Assume that the $G$-action $\rho$ on $(Q,\, U)$ satisfies the following triviality conditions:

1. (1) for each $i\in Q_0$, $g\in G_i$ and $a\in U(i)$, we have $g(a)=a$;

2. (2) for each $\alpha \in Q_1$, $g\in G_\alpha$ and $u\in U(\alpha )$, we have $g(u)=u$.

Then the quotient EI quiver $(\overline {Q},\, \overline {U})$ is described as follows:

\[ \overline{U}(\boldsymbol{i})=U(\iota_0(\boldsymbol{i}))\times G_{\iota_0(\boldsymbol{i})} \]

is the direct product; for each arrow $\boldsymbol {\alpha }\colon \boldsymbol {i}\rightarrow \boldsymbol {j}$ in $\overline {Q}$, we have

\[ \overline{U}(\boldsymbol{\alpha})=U(\iota_1(\boldsymbol{\alpha})) \times (G_{\iota_0(\boldsymbol{j})} \times_{G_{\iota_1(\boldsymbol{\alpha})}} G_{\iota_0(\boldsymbol{i})}). \]

Its typical element is denoted by $(u,\, (h,\, k))$ for $u\in U(\iota _1(\boldsymbol {\alpha }))$, $h\in G_{\iota _0(\boldsymbol {j})}$ and $k\in G_{\iota _0(\boldsymbol {i})}$. The right $\overline {U}(\boldsymbol {i})$-action is given by

\[ (u,(h,k)).(a, g)=(u.g_{\boldsymbol{\alpha}}(a), (h, kg)). \]

The left $\overline {U}(\boldsymbol {j})$-action is given by

\[ (b, g').(u,(h,k))=(b.u, (g'h, k)). \]

Here, $u.g_{\boldsymbol {\alpha }}(a)$ and $b.u$ mean the right $U(s\iota _1(\boldsymbol {\alpha }))$-action and the left $U(t\iota _1(\boldsymbol {\alpha }))$-action on $U(\iota _1(\boldsymbol {\alpha }))$, respectively.

The group action in the following example does not satisfy the triviality condition (1) in remark 5.1.

Example 5.2 Let $Q$ be the following Kronecker quiver.

We endow it with the following assignment $U$: for each vertex $i=1,\, 2$, we set

\[ U(i)={<}a_i\mid a_i^3=1> \]

to be a cyclic group of order $3$ generated by $a_i$; for each arrow $\alpha _i$, we set $U(\alpha _i)=\{u_i\}$ to be a trivial $(U(2),\,U(1))$-biset with a single element.

Let $G=\{1_G,\, g\}$ be a cyclic group of order $2$. We define a $G$-action on the EI quiver $(Q,\, U)$ as follows: $g$ acts on $Q$ by interchanging the arrows $\alpha _1$ and $\alpha _2$; its action on the assignment $U$ is given by

\[ g(a_1)=a_1^2, \; g(a_2)=a_2, \; \mbox{and } g(u_i)=u_i, \; i=1,2. \]

The quotient quiver $\overline {Q}$ is of type $A_2$.

To describe the assignment $\overline {U}$ on $\overline {Q}$, we set $\iota _1({\boldsymbol {\alpha }})=\alpha _1$ and $g_{\boldsymbol {\alpha }}=1_G$. The stabilizers are given by $G_{\alpha _1}=\{1_G\}$ and $G_1=G=G_2$. Therefore, we have

\[ \overline{U}(\boldsymbol{1})=U(1)\rtimes G \mbox{ and }\overline{U}(\boldsymbol{2})=U(2)\times G. \]

We observe that $\overline {U}(\boldsymbol {1})$ is isomorphic to the symmetric group $S_3$, and that $\overline {U}(\boldsymbol {2})$ is isomorphic to a cyclic group of order $6$. For the unique arrow $\boldsymbol {\alpha }$, we have

\[ \overline{U}(\boldsymbol{\alpha})=\{u_1\}\times (G\times G)\simeq G\times G \]

as a set. The left $\overline {U}(\boldsymbol {2})$-action on $\overline {U}(\boldsymbol {\alpha })$ is give by

\[ (a_2^m,g^n).(x,y)=(g^nx,y). \]

for any $0\leq m\leq 2$, $0\leq n\leq 1$ and $(x,\,y)\in G\times G$. The right $\overline {U}(\boldsymbol {1})$-action on $\overline {U}(\boldsymbol {\alpha })$ is given by

\[ (x,y).(a_1^m, g^n)=(x, yg^n). \]

5.2. An equivalence of categories

The EI quiver $(\overline {Q},\, \overline {U})$ might be viewed as a certain ‘orbifold’ quotient of $(Q,\, U)$. We mention a similar construction in the classic work [Reference Bass2, Section 3] on graphs of groups. The EI quiver $(\overline {Q},\, \overline {U})$ depends on the choices in (5.1) and (5.4).

The following result justifies the quotient construction.

Theorem 5.3 Let $(Q,\, U)$ be a finite EI quiver with a $G$-action $\rho$, and let $(\overline {Q},\, \overline {U})$ be its quotient as above. Then there is an equivalence of categories

\[ \mathcal{C}(\overline{Q}, \overline{U})\simeq \mathcal{C}(Q, U)\rtimes G. \]

Proof. We write $\mathcal {C}=\mathcal {C}(Q,\, U)$ in this proof. We will apply proposition 4.2 to deduce the equivalence. We use the map $\iota _0\colon \overline {Q}_0\rightarrow Q_0={\rm Obj}(\mathcal {C}\rtimes G)$, and the identification

\[ \overline{U}(\boldsymbol{i})=U(\iota_0(\boldsymbol{i}))\rtimes G_{\iota_0(\boldsymbol{i})}={\rm Aut}_{\mathcal{C}\rtimes G}(\iota_0(\boldsymbol{i})), \]

where the right equality follows by combining lemma 4.3 and corollary 2.6.

To apply proposition 4.2, it remains to construct for each arrow $\boldsymbol {\alpha }$ in $\overline {Q}$, a map between bisets

\[ \overline{U}(\boldsymbol{\alpha})\longrightarrow {\rm Hom}_{\mathcal{C}\rtimes G}(\iota_0(s\boldsymbol{\alpha}), \iota_0(t\boldsymbol{\alpha})). \]

We will see in the following construction that these maps between bisets yield the required bijection in (E4).

By corollary 3.4, the category $\mathcal {C}\rtimes G$ is EI free, therefore the condition (E1) is satisfied. For two different vertices $\boldsymbol {i}$ and $\boldsymbol {j}$, the vertices $\iota _0(\boldsymbol {i})$ and $\iota _0(\boldsymbol {j})$ are not in the same $G$-orbit. By lemma 2.2(2), $\iota _0(\boldsymbol {i})$ and $\iota _0(\boldsymbol {j})$ are not isomorphic in $\mathcal {C}\rtimes G$, proving the condition (E2). For each vertex $i\in Q_0$, the corresponding object $i$ is isomorphic to $\iota _0(\pi _0(i))$ in $\mathcal {C}\rtimes G$, proving (E3). Once we construct the above maps between bisets, we will infer by proposition 4.2 the required equivalence of categories.

To construct the required maps, we take arbitrary vertices $\boldsymbol {i}$ and $\boldsymbol {j}$ in $\overline {Q}$. By lemma 3.2, we have

\begin{align*} {\rm Hom}^0_{\mathcal{C}\rtimes G}(\iota_0(\boldsymbol{i}),\iota_0(\boldsymbol{j}))& =\{(\theta,g)\mid g\in G, \theta\in {\rm Hom}^0_{\mathcal{C}}(g(\iota_0(\boldsymbol{i})),\iota_0(\boldsymbol{j}))\}\\ & =\bigsqcup_{g\in G} {\rm Hom}^0_{\mathcal{C}}(g(\iota_0(\boldsymbol{i})),\iota_0(\boldsymbol{j}))\times \{g\}. \\ & =\bigsqcup_{g\in G} \; \; \bigsqcup_{\{\alpha\in Q_1\mid s(\alpha)=g(\iota_0(\boldsymbol{i})), \; t(\alpha)=\iota_0(\boldsymbol{j})\}} U(\alpha)\times \{g\}. \end{align*}

Here, for the last equality we use (4.3).

For each arrow $\boldsymbol {\alpha }\colon \boldsymbol {i}\rightarrow \boldsymbol {j}$, we define the following subset of ${\rm Hom}^0_{\mathcal {C}\rtimes G}(\iota _0(\boldsymbol {i}),\,\iota _0(\boldsymbol {j}))$

\[ S(\boldsymbol{\alpha})=\bigsqcup_{\{\alpha\in Q_1\mid \pi_1(\alpha)=\boldsymbol{\alpha}, \; t(\alpha)=\iota_0(\boldsymbol{j})\}} \; \bigsqcup_{\{g\in G\mid s(\alpha)=g(\iota_0(\boldsymbol{i})\}} U(\alpha)\times \{g\}. \]

Recall from example 4.1 that the $(\overline {U}(\boldsymbol {j}),\, \overline {U}(\boldsymbol {i}))$-biset structure on ${\rm Hom}^0_{\mathcal {C}\rtimes G}(\iota _0(\boldsymbol {i}), \iota _0(\boldsymbol {j}))$ is induced by composition (2.1) of morphisms in $\mathcal {C}\rtimes G$. Then we infer that $S(\boldsymbol {\alpha })$ is a $(\overline {U}(\boldsymbol {j}),\, \overline {U}(\boldsymbol {i}))$-sub-biset. Now, we have the following disjoint union

(5.5)\begin{equation} {\rm Hom}^0_{\mathcal{C}\rtimes G}(\iota_0(\boldsymbol{i}),\iota_0(\boldsymbol{j}))=\bigsqcup_{\{\boldsymbol{\alpha}\in \overline{Q}_1\mid s(\boldsymbol{\alpha})=\boldsymbol{i},\; t(\boldsymbol{\alpha})=\boldsymbol{j}\}} S(\boldsymbol{\alpha}). \end{equation}

We will complete the proof by establishing an isomorphism of $(\overline {U}(\boldsymbol {j}),\, \overline {U}(\boldsymbol {i}))$-bisets

\[ \overline{U}(\boldsymbol{\alpha})\simeq S(\boldsymbol{\alpha}) \]

for any arrow $\boldsymbol {\alpha }\colon \boldsymbol {i}\rightarrow \boldsymbol {j}$. Indeed, in view of (5.5), the isomorphism yields the required bijection in (E4). Then we are done.

To analyse $S(\boldsymbol {\alpha })$, we observe that in the index set of the outer disjoint union, the arrows $\alpha$ are of the form $h(\iota _1(\boldsymbol {\alpha }))$ for some $h\in G_{\iota _0(\boldsymbol {j})}$. By the coset decomposition (5.4), there is a unique $1\leq r\leq m_{\boldsymbol {\alpha }}$ satisfying

\[ \alpha=h_{\boldsymbol{\alpha}, r}(\iota_1(\boldsymbol{\alpha})). \]

For simplicity, we write $h_r$ for $h_{\boldsymbol {\alpha },\, r}$. In the inner disjoint union, we have

\[ g(\iota_0(\boldsymbol{i}))=s(\alpha)=h_r(s \iota_1(\boldsymbol{\alpha}))=h_rg_{\boldsymbol{\alpha}} (\iota_0(\boldsymbol{i})). \]

Consequently, there is a unique $k\in G_{\iota _0(\boldsymbol {i})}$ satisfying

\[ g=h_rg_{\boldsymbol{\alpha}}k. \]

The above analysis implies that

\[ S(\boldsymbol{\alpha})=\bigsqcup_{r=1}^{m_{\boldsymbol{\alpha}}} \bigsqcup_{k\in G_{\iota_0(\boldsymbol{i})}} U(h_r(\iota_1(\boldsymbol{\alpha})))\times \{h_rg_{\boldsymbol{\alpha}} k\}. \]

We observe that $U(\iota _1(\boldsymbol {\alpha }))$ is bijective to each $U(h_r(\iota _1(\boldsymbol {\alpha })))$ via $\rho (h_r)^1_{\iota _1(\boldsymbol {\alpha })}$, which sends $u\in U(\iota _1(\boldsymbol {\alpha }))$ to $h_r(u)\in U(h_r(\iota _1(\boldsymbol {\alpha })))$. Moreover, the biset product

\[ G_{\iota_0(\boldsymbol{j})} \times_{G_{\iota_1(\boldsymbol{\alpha})}} G_{\iota_0(\boldsymbol{i})} \]

is bijective to

\[ \{1,2, \cdots, m_{\boldsymbol{\alpha}}\}\times G_{\iota_0(\boldsymbol{i})}, \]

which is further bijective to the following disjoint union

\[ \bigsqcup_{r=1}^{m_{\boldsymbol{\alpha}}} \bigsqcup_{k\in G_{\iota_0(\boldsymbol{i})}} \{h_r g_{\boldsymbol{\alpha}}k\}. \]

Using these bijections, we infer that the following map

\[ \overline{U}(\boldsymbol{\alpha})= U(\iota_1(\boldsymbol{\alpha})) \times (G_{\iota_0(\boldsymbol{j})} \times_{G_{\iota_1(\boldsymbol{\alpha})}} G_{\iota_0(\boldsymbol{i})})\longrightarrow S(\boldsymbol{\alpha}), \;(u, (h_r, k))\mapsto (h_r(u), h_rg_{\boldsymbol{\alpha}} k) \]

is a bijection. We omit the routine verification that this explicit bijection is indeed a map of $(\overline {U}(\boldsymbol {j}),\, \overline {U}(\boldsymbol {i}))$-bisets. This is the required isomorphism of bisets.

Let $\Delta =(\Delta _0,\, \Delta _1; s,\, t)$ be a finite acyclic quiver. Recall that the path category $\mathcal {P}_\Delta$ is defined as follows: ${\rm Obj}(\mathcal {P}_\Delta )=\Delta _0$ and ${\rm Hom}_{\mathcal {P}_\Delta }(i,\, j)$ consists of all paths from $i$ to $j$; the composition is given by concatenation of paths.

Denote by $(\Delta,\, U_{\rm tr})$ the EI quiver with trivial assignment $U_{\rm tr}$, that is, each group $U_{\rm tr}(i)$ is trivial and each biset $U_{\rm tr}(\alpha )$ has only one element. We observe

(5.6)\begin{equation} \mathcal{C}(\Delta, U_{\rm tr})=\mathcal{P}_\Delta. \end{equation}

Let $G$ be a finite group which acts on $\Delta$ by quiver automorphisms. Then $G$ acts on the associated EI quiver $(\Delta,\, U_{\rm tr})$. Denote by $(\overline {\Delta },\, \overline {U}_{\rm tr})$ the quotient EI quiver. Therefore, $\overline {\Delta }$ is the quotient quiver of $\Delta$ by $G$. Fix the choices (5.1). In view of remark 5.1, the assignment $\overline {U}_{\rm tr}$ is described as follows: for each vertex $\boldsymbol {i}$ of $\overline {\Delta }$, we have

\[ \overline{U}_{\rm tr}(\boldsymbol{i})=G_{\iota_0(\boldsymbol{i})}; \]

for each arrow $\boldsymbol {\alpha }\colon \boldsymbol {i}\rightarrow \boldsymbol {j}$ in $\overline {\Delta }$, we have

\[ \overline{U}_{\rm tr}(\boldsymbol{\alpha})=G_{\iota_0(\boldsymbol{i})}\times_{G_{\iota_1(\boldsymbol{\alpha})}} G_{\iota_0(\boldsymbol{j})}, \]

whose $(G_{\iota _0(\boldsymbol {i})},\, G_{\iota _0(\boldsymbol {j})})$-biset structure is given by the multiplication of $G_{\iota _0(\boldsymbol {i})}$ from the left, and of $G_{\iota _0(\boldsymbol {j})})$ from the right.

In view of (5.6), we have the following special case of theorem 5.3.

Corollary 5.4 Let $\Delta$ be a finite acyclic quiver with a $G$-action. Keep the notation as above. Then there is an equivalence of categories

\[ \hskip 130pt \mathcal{C}(\overline{\Delta}, \overline{U}_{\rm tr})\simeq \mathcal{P}_\Delta\rtimes G.\]

6.1. Cartan triples

Let $n\geq 1$ be a positive integer. An $n\times n$ matrix $C=(c_{ij})$ with integer coefficients is called a symmetrizable generalized Cartan matrix, provided that the following conditions are satisfied:

1. (C1) $c_{ii}=2$ for all $i$;

2. (C2) $c_{ij}\leq 0$ for all $i\neq j$, and $c_{ij}<0$ if and only if $c_{ji}<0$;

3. (C3) There is a diagonal matrix $D={\rm diag}(c_1,\,\cdots,\,c_n)$ with $c_i\in \mathbb {Z}_{\geq 1}$ for all $i$ such that the product matrix $DC$ is symmetric.

The matrix $D$ appearing in (C3) is called a symmetrizer of $C$. For brevity, a symmetrizable generalized Cartan matrix is called a Cartan matrix.

Let $C=(c_{ij})$ be a Cartan matrix. An (acyclic) orientation of $C$ is a subset $\Omega \subset \{1,\,2,\,\cdots,\,n\}\times \{1,\,2,\,\cdots,\,n\}$ such that the following conditions are satisfied:

1. (O1) $\{(i,\,j),\,(j,\,i)\}\cap \Omega \neq \emptyset$ if and only if $c_{ij}<0$;

2. (O2) for each sequence $((i_1,\,i_2),\,(i_2,\,i_3),\,\cdots,\,(i_t,\,i_{t+1}))$ with $t\geq 1$ and $(i_s,\,i_{s+1})\in \Omega$ for all $1\leq s\leq t$, we have $i_1\neq i_{t+1}$.

Following [Reference Chen and Wang4], we will call $(C,\, D,\, \Omega )$ a Cartan triple, where $C$ is a Cartan matrix, $D$ its symmetrizer and $\Omega$ an orientation of $C$.

In what follows, we recall that, associated to each Cartan triple, there are a finite free EI category $\mathcal {C}(C,\, D,\, \Omega )$ and a finite dimensional algebra $H(C,\, D,\, \Omega )$.

Let $Q=Q(C,\,\Omega )$ be the finite quiver with the set of vertices $Q_0=\{1,\,2,\, \cdots,\,n\}$ and with the set of arrows

\[ Q_1=\{\alpha^{(g)}_{ij}\colon j\rightarrow i\mid(i,j)\in \Omega, 1\leq g\leq {\rm gcd}(c_{ij},c_{ji})\}\sqcup\{\varepsilon_i\colon i\rightarrow i\mid 1\leq i\leq n\}. \]

Let $Q^{\circ }=Q^{\circ }(C,\,\Omega )$ be the quiver obtained from $Q$ by deleting all the loops $\varepsilon _i$. By the condition (O2), we infer that the finite quiver $Q^{\circ }$ is acyclic.

We recall the finite EI quiver $(Q^\circ,\, X)$. The assignment $X$ is given as follows: $X(i)=\langle \eta _i\; |\; \eta _i^{c_i}=1\rangle$ is a cyclic group of order $c_i$; for each $(i,\, j)\in \Omega$, we set $G_{ij}=\langle \eta _{ij}\; |\; \eta _{ij}^{{\rm gcd}(c_i,\, c_j)}=1\rangle$ to be a cyclic group of order ${\rm gcd}(c_i,\, c_j)$. There are injective group homomorphisms

\[ G_{ij}\hookrightarrow X(i), \; \eta_{ij}\mapsto \eta_i^{{c_i}/{{\rm gcd}(c_i, c_j)}} \]

and

\[ G_{ij}\hookrightarrow X(j), \; \eta_{ij}\mapsto \eta_j^{{c_j}/{{\rm gcd}(c_i, c_j)}}. \]

Then we have the $(X(i),\, X(j))$-biset $X(i)\times _{G_{ij}} X(j)$. We set

\[ X(\alpha_{ij}^{(g)})=X(i)\times_{G_{ij}} X(j) \]

for each $1\leq g\leq {\rm gcd}(c_{ij},\, c_{ji})$.

Let $\mathbb {K}$ be a field. The following algebras [Reference Geiss, Leclerc and Schröer11] play a fundamental role in categorifying the root lattices for non-symmetric Caran matrices. For more background, we refer to [Reference Geiss10].

Definition 6.2 [Reference Geiss, Leclerc and Schröer11, Section 1.4]

Let $(C,\, D,\, \Omega )$ be a Cartan triple with $Q=Q(C,\,\Omega )$. Consider the following $\mathbb {K}$-algebra

\[ H(C,D,\Omega)=\mathbb{K}Q/I, \]

where $\mathbb {K}Q$ is the path algebra of $Q$, and $I$ is the two-sided ideal of $\mathbb {K}Q$ generated by the following set

\begin{align*} & \{\varepsilon_k^{c_k}, \; \varepsilon_i^{{c_i}/{{\rm gcd}(c_i,c_j)}} \alpha^{(g)}_{ij}-\alpha^{(g)}_{ij}\varepsilon_j^{{c_j}/{{\rm gcd}(c_i,c_j)}}\mid k\in Q_0, \; (i,j)\in \Omega, \; 1\\ & \quad \leq g\leq {\rm gcd}(c_{ij},c_{ji})\}. \end{align*}

We will recall from [Reference Chen and Wang4, Subsection 4.2] the construction of a new Cartan triple $(C',\, D',\, \Omega ')$ from a given one $(C,\, D,\, \Omega )$, which depends on the characteristic of $\mathbb {K}$. Recall that $D={\rm diag}(c_1,\, \cdots,\, c_n)$.

Construction $(\dagger )$ for the case ${\rm char}(\mathbb {K})=p>0$. Assume that $c_i=p^{r_i}d_i$ satisfying $r_i\geq 0$ and ${\rm gcd}(p,\, d_i)=1$. For each $1\leq i,\, j\leq n$, we set

\[ \Sigma_{ij}^p=\{ (l_i,l_j)\; |\; 0\leq l_i< d_i, 0\leq l_j< d_j, l_ip^{r_i}\equiv l_jp^{r_j} ({\rm mod}\ {\rm gcd}(d_i, d_j))\}. \]

The rows and columns of the Cartan matrix $C'$ and its symmetrizer $D'$ are indexed by the following set

\[ M=\bigsqcup_{1\leq i\leq n} \{(i, l_i)\; |\; 0\leq l_i< d_i\}. \]

The diagonal entries of $C'$ are $2$, and the off-diagonal entries are given as follows:

\[ c'_{(i, l_i), (j, l_j)}=\begin{cases} -{\rm gcd}(c_{ij},c_{ji}) p^{r_j-{\rm min}(r_i, r_j)}, & \text{ if } (l_i, l_j)\in \Sigma_{ij}^p; \\ 0, & \text{otherwise.} \end{cases} \]

Let $D'$ be a diagonal matrix, whose $(i,\, l_i)$-th component is given by $p^{r_i}$. Set

\[ \Omega'=\{((i, l_i),(j, l_j))\; |\; (i, j)\in \Omega, (l_i, l_j)\in \Sigma_{ij}^p\}, \]

which is an orientation of $C'$.

Construction $(\dagger )$ for the case ${\rm char}(\mathbb {K})=0$. This is very similar to the above construction. We put $d_i=c_i$ and replace $\Sigma _{ij}^p$ by

(6.1)\begin{equation} \Sigma_{ij}=\{ (l_i,l_j)\; |\; 0\leq l_i< c_i, 0\leq l_j< c_j, l_i\equiv l_j ({\rm mod}\ {\rm gcd}(c_i, c_j))\}. \end{equation}

The off-diagonal entries of $C'$ is given by

\[ c'_{(i, l_i), (j, l_j)}=\begin{cases} -{\rm gcd}(c_{ij},c_{ji}) , & \text{ if } (l_i, l_j)\in \Sigma_{ij}; \\ 0, & \text{otherwise.} \end{cases} \]

We observe that $C'$ is symmetric and that $D'$ is the identity matrix.

We say that $\mathbb {K}$ has enough roots of unity for $D$, if for each $1\leq i\leq n$, the polynomial $t^{c_i}-1$ splits in $\mathbb {K}[t]$.

Theorem 6.3 Assume that $(C,\, D,\, \Omega )$ is a Cartan triple and that $\mathbb {K}$ has enough roots of unity for $D$. Keep the notation in Construction $(\dagger )$. Then there is an isomorphism of algebras

\[ \mathbb{K}\mathcal{C}(C, D, \Omega)\simeq H(C', D', \Omega'). \]

We are mainly interested in the following special case.

Proposition 6.4 Assume that ${\rm char}(\mathbb {K})=p>0$ and that $(C,\, D,\, \Omega )$ is a Cartan triple such that each $c_i$ is a $p$-power. Then there is an isomorphism of algebras

\[ \mathbb{K}\mathcal{C}(C, D, \Omega)\simeq H(C, D, \Omega), \]

which identifies ${\rm Span}_\mathbb {K}\{{\rm Id}_i,\, \eta _i,\, \cdots,\, \eta _i^{c_i-1}\}$ with ${\rm Span}_\mathbb {K}\{e_i,\, \varepsilon _{i},\, \cdots,\, \varepsilon _{i}^{c_i-1}\}$.

Here, ${\rm Span}_\mathbb {K}$ means the subspace spanned by the mentioned elements. Both ${\rm Id}_i$ and $\eta _i$ are viewed as automorphisms of $i$ in $\mathcal {C}(C,\, D,\, \Omega )$. Similarly, the trivial path $e_i$ and the loop $\varepsilon _i$ are viewed as elements in $H(C,\, D,\, \Omega )$.

6.2. From quotient to Cartan type

We study the situation of corollary 5.4. Let $G$ be a finite group and let $\Delta$ be a finite acyclic quiver with a $G$-action. We give conditions on when the quotient EI quiver $(\overline {\Delta },\, \overline {U}_{\rm tr})$ is of Cartan type.

The following natural conditions will be imposed on the quiver $\Delta$.

1. (1) For each $i$, the stabilizer $G_i=\langle \xi _i\mid \xi _i^{a_i}=1\rangle$ is cyclic with order $a_i$.

2. (2) For each arrow $\alpha \colon i\rightarrow j$, we have that $\xi _i^{{a_i}/{{\rm gcd}(a_i,\,a_j)}}=\xi _j^{{a_j}/{{\rm gcd}(a_i,\,a_j)}}$ and both belong to $G_\alpha$.

3. (3) For each $g\in G$, we have $\xi _{g(i)}=g\xi _ig^{-1}$.

The condition (${\dagger} 3$) means that the choice of the specific generators $\xi _i$ for $G_i$ is compatible with the $G$-action. It follows from (${\dagger} 2$) that the inclusion $G_\alpha \subseteq G_i\cap G_j$ is an equality.

Associated to the $G$-action on $\Delta$, we will define a Cartan triple $(C,\, D,\, \Omega )$; compare [Reference Lusztig23, Section 14.1]. The rows and columns of $C$ and $D$ are indexed by the orbit set $\overline {\Delta }_0=\Delta _0/G$. For each $G$-orbit $\boldsymbol {i}$ of vertices, the corresponding diagonal entry of $D$ is

\[ c_{\boldsymbol{i}}=\frac{|G|}{|\boldsymbol{i}|}, \]

where $|\boldsymbol {i}|$ denotes the cardinality of the $G$-orbit. For any vertices $\boldsymbol {i}$ and $\boldsymbol {j}$, we denote by $N_{\boldsymbol {i},\, \boldsymbol {j}}$ the number of arrows in $\Delta$ between the $G$-orbit $\boldsymbol {i}$ and $G$-orbit $\boldsymbol {j}$. The corresponding off-diagonal entry of $C$ is given by

\[ c_{\boldsymbol{i}, \boldsymbol{j}}=\frac{-N_{\boldsymbol{i}, \boldsymbol{j}}}{|\boldsymbol{j}|}. \]

The orientation $\Omega$ is consistent with that of $\overline {\Delta }$, that is, $(\boldsymbol {j},\, \boldsymbol {i})\in \Omega$ if and only if there is an arrow from $\boldsymbol {i}$ to $\boldsymbol {j}$ in $\overline {\Delta }$.

By the equality $c_{\boldsymbol {i}}c_{\boldsymbol {i},\, \boldsymbol {j}}=c_{\boldsymbol {j}}c_{\boldsymbol {j},\, \boldsymbol {i}}$, we infer the following useful identity

(6.2)\begin{equation} \frac{-c_{\boldsymbol{i}, \boldsymbol{j}}}{{\rm gcd}(c_{\boldsymbol{i}, \boldsymbol{j}}, c_{\boldsymbol{j}, \boldsymbol{i}})}=\frac{c_{\boldsymbol{j}}}{{\rm gcd}(c_{\boldsymbol{i}}, c_{\boldsymbol{j}})}. \end{equation}

Theorem 6.6 Assume that the $G$-action on $\Delta$ satisfies $({\dagger} 1)\mbox {-}({\dagger} 3)$ and that the associated Cartan triple is $(C,\, D,\, \Omega )$. Denote by $(Q^\circ,\, X)$ the corresponding EI quiver of Cartan type. Then there is an isomorphism of EI quivers

\[ (\overline{\Delta}, \overline{U}_{\rm tr}) \simeq (Q^\circ, X). \]

Moreover, we have the following immediate consequences.

1. (1) There is an equivalence of categories

   \[ \mathcal{P}_\Delta\rtimes G\simeq \mathcal{C}(C, D, \Omega). \]

2. (2) Assume that $\mathbb {K}$ has enough roots of unity for $D$ and that the Cartan triple $(C',\, D',\, \Omega ')$ is given in Construction $(\dagger )$. Then the algebras $\mathbb {K}\Delta \# G$ and $H(C',\, D',\, \Omega ')$ are Morita equivalent.

3. (3) Assume that ${\rm char}(\mathbb {K})=p>0$ and that $G$ is a $p$-group. Then the algebras $\mathbb {K}\Delta \# G$ and $H(C,\, D,\, \Omega )$ are Morita equivalent.