#  The Hilbert series of the preprojective algebras

Hiroyuki Minamoto  Department of Mathematics Osaka Metropolitan University,
Sakai City, Japan<EMAIL_ADDRESS>
###### Abstract.

The aim of this short note is to prove the formula of the Hilbert series of
the preprojective algebras in arbitrary characteristic by making effective use
of the formulas of the Hilbert series of differential graded (dg) algebras
with Adams grading. We also compute the Hilbert series of the quiver
Heisenberg algebras, a special class of central extensions of preprojective
algebras.

###### Key words and phrases:

Hilbert series. Preprojective algebras, Quiver Heisenberg algebras, Central
extension of preprojective algebras

###### 2020 Mathematics Subject Classification:

16G20, 16G70, 18G80

##  1\. Introduction

In this note, ${\mathbf{k}}$ denotes a field and a quiver $Q$ is assumed to be
connected, finite and acyclic unless otherwise is mentioned.

The aim of this short note is to prove the formula of the Hilbert series of
the preprojective algebras $\Pi(Q)$ (Theorem 3.1, Theorem 3.2) by making
effective use of the formulas of the Hilbert series of differential graded
(dg) algebras with Adams grading.

The formula were first obtained by Malkin, Ostrik and Vybornov [8, p.516,
Theorem] in the characteristic zero case. They used theory of module
categories over representations of quantum $\operatorname{SL}(2)$. (We note
that they dealt with the preprojective algebra $\Pi(Q)$ of $T$-type, which is
not dealt in this paper.)

Later Etingof and Eu [3, Theorem 3.4.2] dealt with a non-Dynkin quiver $Q$
which is _not necessarily acyclic_ and proved that the same formula holds true
in any characteristic. As a consequence they showed that the preprojective
algebra $\Pi(Q)$ is Koszul where $Q$ is non-Dynkin and not necessarily
acyclic. Their idea is to introduced the _partial preprojective algebras_ that
allows them to reduce the problem of $Q$ to that of a smaller quiver
$Q^{\prime}$. However, as one of the point to which the problem for a general
quiver is finally reduced, it is necessary for them to compute the Hilbert
series $h_{\Pi(Q)}$ for an extended-Dynkin quiver $Q$ before establishing
general formula. This was done in [3, Proposition 3.2.1] by a rather
complicated method.

In this paper, we see that the formula of the Hilbert series $h_{\Pi(Q)}$ for
a non-Dynkin quiver can be easily deduced from a well-know fact in
representation theory of a quiver, if once we establish basic properties of
the Hilbert series of differential graded algebras with Adams grading. By the
same method, we also compute the Hilbert series of the quiver Heisenberg
algebras (QHA), a special class of central extensions of preprojective
algebras [4, 7]. Finally, using a result of QHA and basic fact of
representation theory of quiver, we establish the formula for the Hilbert
series $h_{\Pi(Q)}$ for a Dynkin quiver $Q$ in arbitrary characteristic.

## Acknowledgment

The author thanks Martin Herschend for a suggestion that leads to Lemma A.1.

This note stemmed from discussion in the preparations of Winter School “Koszul
Algebra and Koszul Duality” at Osaka City University 2021, which the author
co-organized with Aaron Chan, Ryo Kanda and Hyohe Miyachi. The author thanks
them for helpful discussions. This work was partly supported by Osaka City
University Advanced Mathematical Institute : MEXT Joint Usage Research Center
on Mathematics and Theoretical Physics JPMXP 0619217849

The author was partially supported by JSPS KAKENHI Grant Number JP21K03210.

##  2\. Hilbert series of complexes and differential graded (dg) algebras with
Adams grading

When we deal with a differential graded (dg) algebra equipped with an extra
grading, to distinguish it from the cohomological grading, the extra grading
is often called _Adams grading_. In the paper we follow this terminology. Then
a usual graded algebra can be said as an algebra with Adams grading.

In this Section 2, $r\geq 1$ is a positive integer and $A={\mathbf{k}}^{\times
r}={\mathbf{k}}e_{1}\times\cdots\times{\mathbf{k}}e_{r}$ where $e_{i}$ is the
$i$-th canonical idempotent element:

$e_{i}=(\stackrel{{\scriptstyle 1}}{{0}},\cdots,\stackrel{{\scriptstyle
i-1}}{{0}},\stackrel{{\scriptstyle i}}{{1}},\stackrel{{\scriptstyle
i+1}}{{0}},\cdots,\stackrel{{\scriptstyle r}}{{0}}).$

We denote _the dimension matrix_ $(\dim e_{i}Me_{j})_{ij}$ of a finite
dimensional $A$-$A$-bimodule $M$ by $[M]$. Then, it is clear that

$[A]=1\ \textup{(the unit matrix)},\ [M^{*}]=[M]^{t}\ \textup{(transpose)},\
[M\otimes_{A}N]=[M][N]$

where $M^{*}=\operatorname{Hom}_{{\mathbf{k}}}(M,{\mathbf{k}})$.

We regard $A$ as an Adams graded algebra concentrated in degree $0$. Recall
that for an Adams graded $A$-$A$-bimodule $M=\bigoplus_{i}M_{i}$ which is
locally finite dimensional (i.e., $\dim M_{i}<\infty$ for all
$i\in{\mathbb{Z}}$) and bounded below (i.e., $M_{i}=0$ for $i\ll 0$), its
Hilbert series $h_{M}(t)$ is defined to be

$h_{M}(t):=\sum_{i}[M_{i}]t^{i}.$

To define the Hilbert series of complexes of $A$-$A$-bimodules with Adams
grading, we need to introduce a boundedness condition.

######  Definition 2.1.

We call a complex $M=(\bigoplus_{n}M^{n},d_{M})$ of graded $A$-bimodules
$M^{n}=\bigoplus_{i}M_{i}^{n}$ _suitably bounded_ if it satisfies the
following two properties:

  1. (1)

For each $i\in{\mathbb{Z}}$, we have
$\dim\oplus_{n\in{\mathbb{Z}}}M_{i}^{n}<\infty$.

In other words, for each $i$, the complex $M_{i}^{\bullet}$ is bounded and
each term $M_{i}^{n}$ is finite dimensional.

  2. (2)

There exists $i_{0}\in{\mathbb{Z}}$ such that $M_{i}^{\bullet}=0$ for all
$i<i_{0}$.

For a suitably bounded complex $M$, its _Hilbert series_ $h_{M}(t)$ is defined
to be

$h_{M}(t):=\sum_{n}(-1)^{n}h_{M^{n}}(t)=\sum_{i,n}(-1)^{n}[M_{i}^{n}]t^{i}.$

######  Remark 2.2.

It may be worth noting that $h_{M}(t)$ does not depend on the differential
$d_{M}$ but only depends on the underlying cohomological graded
$A$-$A$-bimodule $M^{\\#}$ with Adams grading.

Let $n\in{\mathbb{Z}}$. We denote by $(n)$ (resp. $[n]$) the shift operation
with respect to the Adams degree (resp. the cohomolgical degree), i.e.,
$(M[m](j))_{i}^{n}:=M_{i+j}^{n+m}$.

We collect basic properties which can be proved by standard arguments.

######  Lemma 2.3.

Let $M$ and $N$ be suitably bounded complexes of graded $A$-bimodules. The
following assertions hold.

  1. (1)

$h_{M[-n](-m)}(t)=(-1)^{n}t^{m}h_{M}(t)$.

  2. (2)

$h_{M}(t)=h_{{\textup{H}}(M)}(t)$.

  3. (3)

If $M$ and $N$ are quasi-isomorphic, then $h_{M}(t)=h_{N}(t)$.

  4. (4)

$h_{M\otimes_{A}N}(t)=h_{M}(t)h_{N}(t)$.

From the last assertion of the lemma, we deduce

######  Corollary 2.4.

Let $M$ be a suitably bounded complex of Adams graded $A$-$A$-bimodules and
$T$ denotes the tensor algebra of $M$

$T={\textup{T}}_{A}M=A\oplus M\oplus M\otimes_{A}M\oplus\cdots.$

Assume that $M_{i}^{\bullet}=0$ for all $i\leq 0$. Then $T$ is suitably
bounded and the Hilbert series $h_{T}(t)$ is

$h_{T}(t)=\sum_{n\geq 0}h_{M}(t)^{n}=\frac{1}{1-h_{M}(t)}.$

##  3\. The Hilbert series of the preprojective algebras

###  3.1. The preprojective algebras

Let $Q$ be a connected finite acyclic quiver with the vertexes
$Q_{0}=\\{1,2,\dots,r\\}$. We set
$A:={\mathbf{k}}Q_{0}={\mathbf{k}}e_{1}\times\cdots\times{\mathbf{k}}e_{r}$
and equip the arrow space $V:={\mathbf{k}}Q_{1}$ with a canonical
$A$-$A$-bimodule structure. If we regard the path algebra ${\mathbf{k}}Q$ as
an Adams graded algebra by path length, then it is the tensor algebra over $A$
of the bimodule $V(-1)$

${\mathbf{k}}Q={\textup{T}}_{A}(V(-1))$

and hence by Corollary 2.4 the Hilbert series $h_{{\mathbf{k}}Q}(t)$ is

$h_{{\mathbf{k}}Q}(t)=\sum_{n\geq 0}([V]t)^{n}=\frac{1}{1-[V]t}.$

We denote by $\overline{Q}$ the double of $Q$. Namely, $\overline{Q}$ is
obtained from $Q$ by formally adding an opposite arrow $\alpha^{*}:j\to i$ for
each arrow $\alpha:i\to j$ of the original quiver $Q$.

$\textstyle{\boxed{\ Q\
}}$$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{j}$
$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha}$$\textstyle{j\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\alpha^{*}}$$\textstyle{\boxed{\
\overline{Q}\ }}$

Then the arrow space ${\mathbf{k}}\overline{Q}_{1}$ can be identified with
$V\oplus V^{*}$ where
$V^{*}:=\operatorname{Hom}_{{\mathbf{k}}}(V,{\mathbf{k}})$. By the definition,
the adjacency matrix $C$ of $\overline{Q}$ is given by

$C=[V]+[V]^{t}=[V]+[V^{*}]=[{\mathbf{k}}\overline{Q}_{1}].$

We recall the definition of the preprojective algebra $\Pi(Q)$ of a quiver
$Q$. For a vertex $i\in Q_{0}$, the _mesh relation $\rho_{i}$ at $i$_ is the
element of ${\mathbf{k}}\overline{Q}$ that is given by

$\rho_{i}:=\sum_{\alpha\in Q_{1}:t(\alpha)=i}\alpha\alpha^{*}-\sum_{\alpha\in
Q_{1}:h(\alpha)=i}\alpha^{*}\alpha.$

The total sum $\rho:=\sum_{i\in Q_{0}}\rho_{i}$ is also called the _mesh
relation_. The _preprojective algebra_ is defined to be the path of
$\overline{Q}$ with mesh relations:

$\Pi=\Pi(Q)=\frac{\ {\mathbf{k}}\overline{Q}\ }{\ (\rho)\
}=\frac{{\mathbf{k}}\overline{Q}}{(\rho_{i}\mid i\in Q_{0})}.$

We equip $\overline{Q}$ with an Adams grading by path length:

$\textup{Ad}\deg\alpha:=1,\ \textup{Ad}\deg\alpha^{*}:=1\textup{ for
}\alpha\in Q_{1}.$

The mesh relations $\rho_{i}$ are homogeneous of Adams degree $2$ and
consequently $\Pi(Q)$ is a graded algebra. Since the degree $n$-part of
$\Pi(Q)_{n}$ of $\Pi(Q)$ is finite dimensional for all $n\geq 0$, we can take
the Hilbert series

$h_{\Pi(Q)}(t):=\sum_{n\geq 0}[\Pi(Q)_{n}]t^{n}.$

The formula for a non-Dynkin quiver is given in the following theorem.

######  Theorem 3.1 ([3, 8]).

Assume that $Q$ is a non-Dynkin quiver. Then the Hilbert series $h_{\Pi}(t)$
is given by the following formula:

$h_{\Pi}(t)=\frac{1}{1-Ct+t^{2}}.$

Proof is given in Section 3.3.

####  3.1.1. The Dynkin case

Let $Q$ be a Dynkin quiver. To give a formula of the Hilbert series of
$\Pi:=\Pi(Q)$, we need to introduce the Nakayama permutation matrix
$P:=P_{\nu}$.

First recall that a Nakayama automorphism $\nu:\Pi\to\Pi$ is an algebra
automorphism such that there is an isomorphism
${}_{\nu}\Pi\cong{\textup{D}}(\Pi)$ of $\Pi$-$\Pi$-bimodules. We use the
explicit form of a Nakayama automorphism $\nu$ of $\Pi$ is given by Brenner-
Butler-King [1, Definition 4.6]. Then $\nu$ induces a permutation of the
vertexes $Q_{0}$ which we denote by the same symbol $\nu$, i.e., for each
$i\in Q_{0}$, there exists a unique vertex $\nu(i)$ such that
$\Pi\nu(e_{i})\cong\Pi e_{\nu(i)}$ as $\Pi$-modules. Then the Nakayama
permutation matrix $P:=P_{\nu}$ is defined to be
$P:=(\delta_{\nu(i)j})_{i,j\in Q_{0}}$ so that we have $[{}_{\nu}M]=P[M]$ for
any $\Pi$-$\Pi$-bimodules $M$. We note that the matrices $P$ and $C$ are
commutative. Indeed, let $J$ denotes the Jacobson radical of $\Pi$. Then
$C=[{\mathbf{k}}\overline{Q}_{1}]=[J/J^{2}]$. It follows from Lemma A.1 that
$PC=CP$.

Now we can give the formula for a Dynkin quiver $Q$.

######  Theorem 3.2 ([8] in the characteristic zero case).

Assume that $Q$ is a Dynkin quiver. Then the Hilbert series
$h_{\widetilde{\Pi}}(t)$ is given by the following formula

$h_{\widetilde{\Pi}}(t)=\frac{1+Pt^{h}}{1-Ct+t^{2}}.$

Proof is given in Section 4.5.

###  3.2. The derived preprojective algebras of a finite acyclic quiver $Q$

We recall the definition of the derived preprojective algebra
$\widetilde{\Pi}:=\widetilde{\Pi}(Q)$ of a quiver $Q$. The underlying
cohomological graded algebra $\widetilde{\Pi}^{\\#}$ with Adams grading of
$\widetilde{\Pi}$ is given by the cohomological graded quiver $\widehat{Q}$
obtained from $\overline{Q}$ by formally adding a loop $s_{i}$ of
$\textup{Ad}\deg s_{i}=2,\textup{coh}\deg s_{i}=-1$ to each vertex $i\in
Q_{0}$.

$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{i}}$$\scriptstyle{\alpha}$$\textstyle{j\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{s_{j}}$$\scriptstyle{\alpha^{*}}$$\textstyle{\
\boxed{\ \widehat{Q}\ }}$

In other words, $\widetilde{\Pi}^{\\#}$ is the tensor algebra over
$A={\mathbf{k}}Q_{0}$ of the bimodule $V(-1)\oplus V^{*}(-1)\oplus A[1](-2)$

$\widetilde{\Pi}^{\\#}={\textup{T}}_{A}(V(-1)\oplus V^{*}(-1)\oplus A[1](-2))$

The differential $d$ of $\widetilde{\Pi}$ is given by
$d(\alpha):=0,d(\alpha^{*}):=0,d(s_{i}):=-\rho_{i}.$ The values of $d$ for
general homogeneous elements are determined from the Leibniz rule
$d(xy)=d(x)y+(-1)^{|x|}xd(y)$.

Since the algebra $\widetilde{\Pi}^{\\#}$ is a tensor algebra over $A$, we can
immediately obtain a formula for the Hilbert series $h_{\widetilde{\Pi}}$.

######  Proposition 3.3.

The Hilbert series $h_{\widetilde{\Pi}}(t)$ of the derived preprojective
algebra $\widetilde{\Pi}$ is given by the following formula

$h_{\widetilde{\Pi}}(t)=\frac{1}{1-Ct+t^{2}}.$

###### Proof.

First, we observe

$h_{V(-1)\oplus V^{*}(-1)\oplus A[1](-2)}(t)=[V]t+[V^{*}]t-[A]t^{2}=Ct-t^{2}.$

Applying Corollary 2.4, we complete the calculation

$\begin{split}h_{\widetilde{\Pi}}(t)=h_{\widetilde{\Pi}^{\\#}}(t)=\frac{1}{1-(Ct-t^{2})}=\frac{1}{1-Ct+t^{2}}.\end{split}$

∎

#### 3.2.1.

We recall basic properties of the derived preprojective algebra. There is a
canonical homomorphism $\widetilde{\Pi}(Q)\to\Pi(Q)$ of dg-algebras with Adams
grading, which induces an isomorphism
${\textup{H}}^{0}(\widetilde{\Pi}(Q))\xrightarrow{\cong}{\textup{H}}^{0}(\Pi(Q))=\Pi(Q)$
of Adams graded algebras. The dg-algebra $\widetilde{\Pi}(Q)$ is a model of
Keller’s $2$-Calabi-Yau completion of ${\mathbf{k}}Q$ [5], i.e., it is quasi-
isomorphic to the derived tensor algebra of
$\operatorname{\mathbb{R}Hom}_{{\mathbf{k}}Q^{\mathrm{e}}}({\mathbf{k}}Q,{\mathbf{k}}Q^{\mathrm{e}})[1]\cong\operatorname{\mathbb{R}Hom}_{{\mathbf{k}}Q}(({\mathbf{k}}Q)^{*},{\mathbf{k}}Q)[1]$
over ${\mathbf{k}}Q$ (see for example [7, Section 6.1]). We note that the
endofunctor
$\operatorname{\mathbb{R}Hom}_{{\mathbf{k}}Q}(({\mathbf{k}}Q)^{*},{\mathbf{k}}Q)[1]\otimes^{\mathbb{L}}_{{\mathbf{k}}Q}-$
of the derived category
${\mathsf{D}}^{\mathrm{b}}({\mathbf{k}}Q\operatorname{mod})$ is the inverse
$\nu^{-1}_{1}$ of $(-1)$-shifted Nakayama
$\nu_{1}:=({\mathbf{k}}Q)^{*}[-1]\otimes^{\mathbb{L}}_{{\mathbf{k}}Q}-$.

$\widetilde{\Pi}(Q)\simeq\bigoplus_{n\geq
0}(\operatorname{\mathbb{R}Hom}_{{\mathbf{k}}Q}(({\mathbf{k}}Q)^{*},{\mathbf{k}}Q)[1])^{\otimes^{\mathbb{L}}_{{\mathbf{k}}Q}n}\simeq\bigoplus_{n\geq
0}\nu_{1}^{-n}({\mathbf{k}}Q)$

###  3.3. Proof of Theorem 3.1

In the case where $Q$ is non-Dynkin, it is well-known that the complex
$\nu_{1}^{-n}({\mathbf{k}}Q)$ is concentrated in $0$-th cohomological degree
for all $n\geq 0$ and hence $\widetilde{\Pi}(Q)$ is quasi-isomorphic to
$\Pi(Q)$ (see [6, Example 2.8]). Therefore, we conclude

$h_{\Pi}(t)=h_{\widetilde{\Pi}}(t)=\frac{1}{1-Ct+t^{2}}.$

∎

##  4\. The Hilbert series of the quiver Heisenberg algebras

###  4.1. The quiver Heisenberg algebras

Let $Q$ be a quiver. For an element $v=(v_{i})_{i\in
Q_{0}}\in{\mathbf{k}}Q_{0}$, we define the central extension
${}^{v}\\!\Lambda:={}^{v}\\!\Lambda(Q)$ of the preprojective algebra to be

${}^{v}\\!\Lambda(Q):=\frac{{\mathbf{k}}[z]\overline{Q}}{(\rho_{i}-v_{i}ze_{i}\mid
i\in Q_{0})}.$

This is a restricted version of the central extensions of preprojective
algebras studied in [4]. Looking from another viewpoint ${}^{v}\\!\Lambda(Q)$
can be regarded as a deformation family of a preprojective algebra. More
general deformation families are studied in [2, 9]. We set $\textup{Ad}\deg
z:=2$ so that ${}^{v}\Lambda(Q)$ acquires Adams grading.

In [7], the algebra ${}^{v}\\!\Lambda$ is studied under the name _quiver
Heisenberg algebras_ mainly in the case where $v\in{\mathbf{k}}Q_{0}$ is
sincere (i.e., $v_{i}\neq 0$ for all $i\in Q_{0}$). We set
${}^{v}\varrho_{i}:=v_{i}^{-1}\rho_{i},\ {}^{v}\\!\varrho:=\sum_{i\in
Q_{0}}{}^{v}\\!\varrho_{i}$ for $i\in Q_{0}$ and
${}^{v}\\!\eta_{a}:=a{}^{v}\\!\varrho-{}^{v}\\!\varrho a$ for
$a\in\overline{Q}_{0}$. It is shown in [7, Lemma 1.6] that if
$v\in{\mathbf{k}}Q_{0}$ is sincere, then the algebra ${}^{v}\\!\Lambda$ has an
isomorphisms

${}^{v}\Lambda\xrightarrow{\cong}\frac{{\mathbf{k}}\overline{Q}}{\
({}^{v}\\!\eta_{a}|a\in\overline{Q}_{1})\ }$

that preserves $\overline{Q}$ and sends $z$ to ${}^{v}\\!\varrho$. It is clear
that this isomorphism preserves Adams grading.

To establish a formula of the Hilbert series of ${}^{v}\\!\Lambda(Q)$, we
introduce the derived quiver Heisenberg algebra
${}^{v}\widetilde{\Lambda}(Q)$.

###  4.2. The derived quiver Heisenberg algebras

Assume that $v\in{\mathbf{k}}Q_{0}$ is sincere. We define the _derived quiver
Heisenberg algebra_
${}^{v}\\!\widetilde{\Lambda}:={}^{v}\\!\widetilde{\Lambda}(Q)$ in the
following way. The underlying cohomological graded algebra
${}^{v}\\!\widetilde{\Lambda}^{\\#}$ is given by

(4-1) ${\mathsf{T}}_{A}(V(-1)\oplus V^{*}(-1)\oplus V^{*}[1](-3)\oplus
V[1](-3)\oplus A[1](-4))$

where $A={\mathbf{k}}Q_{0}$ and $V={\mathbf{k}}Q_{1}$. A part of corresponding
cohomorogical graded quiver with Adams grading is given as below where we
denote by $\alpha^{\circledast}$ the element of $V[1](-3)$ corresponding to
$\alpha\in V$ and by $\alpha^{\circ}$ the element of $V^{*}[1](-3)$
corresponding to $\alpha^{*}\in V^{*}$.

$\textstyle{i\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t_{i}}$$\scriptstyle{\alpha^{\circledast}}$$\scriptstyle{\alpha}$$\textstyle{j\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{t_{j}}$$\scriptstyle{\alpha^{*}}$$\scriptstyle{\alpha^{\circ}}$

The differential $d$ of ${}^{v}\\!\widetilde{\Lambda}$ is defined in the
following way. Then the underlying cohomological graded algebra (4-1) is
freely generated by
$\alpha,\alpha^{*},\alpha^{\circ},\alpha^{\circledast},t_{i}$ for $\alpha\in
Q_{1}$ and $i\in Q_{0}$. The values of $d$ for these generators are defined by
the formulas:

$\begin{split}&d(\alpha):=0,d(\alpha^{*}):=0,d(\alpha^{\circ}):=-{}^{v}\\!\eta_{\alpha^{*}},d(\alpha^{\circledast}):={}^{v}\\!\eta_{\alpha},\\\
&d(t_{i})=\sum_{\alpha:t(\alpha)=i}\alpha\alpha^{\circ}-\sum_{\alpha:h(\alpha)=i}\alpha^{\circ}\alpha+\sum_{\alpha:h(\alpha)=i}\alpha^{*}\alpha^{\circledast}-\sum_{\alpha:t(\alpha)=i}\alpha^{\circledast}\alpha^{*}.\end{split}$

The values of $d$ for general homogeneous elements are determined from the
Leibniz rule $d(xy)=d(x)y+(-1)^{|x|}xd(y)$.

In the same way of Proposition 3.3 we can compute the Hilbert series of the
derived quiver Heisenberg algebra ${}^{v}\\!\widetilde{\Lambda}$.

######  Proposition 4.1.

In the above setting, we have

$h_{{}^{v}\\!\widetilde{\Lambda}}(t)=\frac{1}{1-Ct+Ct^{3}-t^{4}}=\frac{1}{(1-Ct+t^{2})(1-t^{2})}.$

###  4.3. The non-Dynkin case

######  Theorem 4.2.

Let $Q$ be a non-Dynkin quiver and $v\in{\mathbf{k}}Q_{0}$. Then we have

$h_{{}^{v}\\!\Lambda}(t)=\frac{1}{(1-Ct+t^{2})(1-t^{2})}.$

###### Proof.

First assume that $v\in{\mathbf{k}}Q_{0}$ is sincere. Then if $Q$ is non-
Dynkin, then the derived quiver Heisenberg algebra
${}^{v}\\!\widetilde{\Lambda}$ is quasi-isomorphic to the quiver Heisenberg
algebra ${}^{v}\\!\Lambda$ by [7, Proposition 7.1]. Thus we deduce the desired
conclusion from Proposition 4.1.

We deal with the general case. Recall that we label the vertexes as
$Q_{0}=\\{1,2,\dots,r\\}$. We set $R:={\mathbf{k}}[x_{1},\ldots,x_{r}]$ with
$\textup{Ad}\deg x_{1}:=2,\dots,\textup{Ad}\deg x_{r}:=2$. It is well-known
that the deformation family

$\Pi_{\bullet}:=\Pi(Q)_{\bullet}:=\frac{R\overline{Q}}{(\rho_{i}-x_{i}e_{i}\mid
i\in Q_{0})}$

introduced by Crawley-Boevey-Holland [2] is flat over $R$ (see e.g. [7,
Theorem 7.3]). Let $S:={\mathbf{k}}[y_{1},\dots,y_{r},z]$ with
$\textup{Ad}\deg y_{1}:=0,\dots,\textup{Ad}\deg y_{r}:=0,\textup{Ad}\deg z:=2$
and $\phi:R\to S$ the ${\mathbf{k}}$-algebra homomorphism such that
$\phi(x_{i})=y_{i}z$ for all $i\in Q_{0}$. We set
${}^{\bullet}\\!\Lambda:=S\otimes_{R}\Pi_{\bullet}$. Let
$T:={\mathbf{k}}[y_{1},\dots,y_{r}]$ and $\psi:T\hookrightarrow S$ the
canonical inclusion. Since $S$ is flat over $T$, the algebra
${}^{\bullet}\\!\Lambda$ is flat over $T$. Observe that for all $n\geq 0$, the
Adams degree $n$-part ${}^{\bullet}\\!\Lambda_{n}$ is a finite $T$-module and
hence it is a finite flat module over $T$.

For $v\in{\mathbf{k}}Q_{0}$, we set
$\kappa(v):=T/(y_{1}-v_{1},\dots,y_{r}-v_{r})\cong{\mathbf{k}}$, the
corresponding residue field. Then the Adams graded algebra
$\kappa(v)\otimes_{T}{}^{\bullet}\\!\Lambda$ is isomorphic to
${}^{v}\\!\Lambda$. Hence we have an isomorphism
$\kappa(v)\otimes_{T}{}^{\bullet}\\!\Lambda_{n}\cong{}^{v}\\!\Lambda_{n}$ of
${\mathbf{k}}$-vector spaces for each $n\geq 0$. It follows from flatness of
${}^{\bullet}\\!\Lambda_{n}$ over $T$ that the ${\mathbf{k}}$-dimension of
${}^{v}\\!\Lambda_{n}$ is independent from $v\in{\mathbf{k}}Q$. Thus we
conclude that the desired formula holds true eve when $v$ is not sincere. ∎

###  4.4. The Dynkin case

In this section $Q$ is a Dynkin quiver. To deal with the Dynkin case, we need
to introduce the notion of regularity. An element $v\in{\mathbf{k}}Q_{0}$ is
called _regular_ if $\sum_{i\in Q_{0}}v_{i}\dim(e_{i}N)\neq 0$ for any
indecomposable ${\mathbf{k}}Q$-module $N$. We note that a regular element is
sincere. It is shown in [7, Theorem 9.1] that ${}^{v}\\!\Lambda(Q)$ is finite
dimensional if and only if $v$ is regular.

######  Theorem 4.3 ([4] in the characteristic zero case).

Let $Q$ be a Dynkin quiver. Assume that $v\in{\mathbf{k}}Q_{0}$ is regular.
Then we have

$h_{{}^{v}\\!\Lambda}(t)=\frac{1-t^{2h}}{(1-Ct+t^{2})(1-t^{2})}.$

###### Proof.

By [7, Theorem 9.9], we have an isomorphism
${\textup{H}}({}^{v}\\!\widetilde{\Lambda})\cong{}^{v}\\!\Lambda[u]$ of Adams
graded algebras where the right hand side is the polynomial algebra in a
single variable $u$ with cohomological degree $-2$ and Adams degree $2h$. Thus
we have

$h_{{}^{v}\\!\widetilde{\Lambda}}=\sum_{n\geq
0}t^{2nh}h_{{}^{v}\\!\Lambda}=(1-t^{2h})^{-1}h_{{}^{v}\\!\Lambda}.$

∎

###  4.5. Proof of Proposition 3.2

First note that by [1, Definition 4.6], we have
$\nu^{2}=\operatorname{id}_{\Pi}$. It follows that $P^{2}=1$ and
$(1-Pt^{h})(1+Pt^{h})=1-t^{2h}$.

Taking a field extension of ${\mathbf{k}}$ (if it is necessary), we may assume
that there is a regular element $v\in{\mathbf{k}}Q$ (see [7, Example
5.11(2)]). By [7, Corollary 9.12, Remark 9.13], there exists an exact sequence
of the following form

$0\to{\textup{D}}(\Pi)(-2h+2)\to{}^{v}\\!\Lambda(-2)\xrightarrow{{\mathsf{r}}_{\varrho}}{}^{v}\\!\Lambda\to\Pi\to
0$

where ${\mathsf{r}}_{\varrho}$ denotes the multiplication map by $\varrho$. On
the other hand, we have ${\textup{D}}(\Pi)\cong{}_{\nu}\\!\Pi(h-2)$ by [1].
Thus, we have the equality
$h_{\Pi}(t)-h_{{}^{v}\\!\Lambda}(t)+t^{2}h_{{}^{v}\\!\Lambda}(t)-t^{h}Ph_{\Pi}(t)=0$.
Thus we obtain

$h_{\Pi}(t)=\frac{1-t^{2}}{1-Pt^{h}}h_{{}^{v}\\!\Lambda}(t)=\frac{1-t^{2h}}{(1-Pt^{h})(1-Ct+t^{2})}=\frac{1+Pt^{h}}{1-Ct+t^{2}}.$

∎

## Appendix A

Let $B$ be a finite dimensional basic algebra and $\\{e_{i}\\}_{i=1}^{r}$ be a
complete set of primitive orthogonal idempotent elements. We may regard $B$ as
an algebra over $A:=\prod_{i=1}^{r}{\mathbf{k}}e_{i}$. Let $\phi:B\to B$ be an
algebra automorphism. We denote by $\overline{\phi}$ the induced permutation
of order $n$, i.e., for $i=1,2,\dots,r$, $\overline{\phi}(i)$ denotes a unique
number such that $B\phi(e_{i})\cong Be_{\overline{\phi}(i)}$. We set
$P:=(\delta_{\overline{\phi}(i)j})_{i,j=1}^{r}$. Finally let $J$ be the
Jacobson radical of $B$ and $C:=[J/J^{2}]$ the dimension matrix of
$B$-$B$-bimodule $J/J^{2}$.

######  Lemma A.1.

We have

$PC=CP.$

###### Proof.

For a $B$-$B$-bimodule $M$, it is straightforward to check that
$[{}_{\phi}M]=P_{\overline{\phi}}[M]$ and
$[M_{\phi^{-1}}]=[M]P_{\overline{\phi}}$.

It is clear that the automorphism $\phi^{-1}:B\to B$ preserves $J$ and $J^{2}$
and hence induces an isomorphisms $J/J^{2}\to J/J^{2},\ j\
\operatorname{mod}J^{2}\mapsto\phi^{-1}(j)\ \operatorname{mod}J^{2}$ of
${\mathbf{k}}$-vector spaces. This map gives an isomorphism
${}_{\phi}(J/J^{2})\xrightarrow{\cong}(J/J^{2})_{\phi^{-1}}$ of
$B$-$B$-bimodules. Combining the above observation, we obtain the desired
conclusion. ∎

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