(\Pa\Pa)^{\vee} \cong \Pa\Pa, \ (\Pa V \Pa)^{\vee} \cong \Pa V^{*} \Pa.
\end{equation}

Moreover, as $(\Pa^{\mre})^{\vee} \cong \Pa^{\mre}$, we may identify the $\Pa^{\mre}$-dual $m^{\vee}$ of the canonical map $m: \Pa^{\mre} \to \Pa\Pa$
with the embedding

\begin{equation}\label{202012091717}
\Pa\Pa = \bigoplus_{ i \in Q_{0}} \Pa e_{i} \otimes_{\kk} e_{i} \Pa \to \bigoplus_{ i,j \in Q_{0} } \Pa e_{i} \otimes_{\kk} e_{j} \Pa = A^{\mre}.
\end{equation}

The following lemma gives a description of the complex $\PPa^{\vee}$.

The identifications (<ref>)
provides the following isomorphism of complexes

\[
\PPa^{\vee} \cong
\left( \Pa V^{*} \Pa[-1] \oplus \Pa\Pa, \begin{pmatrix} 0 & \uparrow  \hat{\rho}\\  0 & 0 \end{pmatrix} \right).
\]

It is easy to show that
for $f \in \Hom_{\Pa^{\mre}}(\Pa\Pa, \Pa^{\mre})$,
the composition $f \circ \hat{\mu}: \Pa V\Pa \to \Pa^{\mre}$ corresponds to $- \hat{\rho}(f)$
via the isomorphism  $F_{V}$  given in Lemma <ref>.

Since the $0$-th differential $d_{\Hom}^{0}$ of the complex $\cpxHom_{\Pa^{\mre}}(\PPa, \Pa^{\mre})$
is $d_{\Hom}^{0}(f) = d_{\Pa^{\mre}}^{0} \circ f - f\circ d_{\PPa}^{-1} =- f\circ \hat{\mu}$, the assertion. follows.

Since $\PPa$ is a projective resolution of $\Pa$ over $\Pa^{\mre}$, the complex $\PPa^{\vee}$ over $\Pa^{\mre}$ is a representative of
$\Pa^{\vvee} = \RHom_{\Pa}(\Pa, \Pa^{\mre})$.
We set $\PPi_{1} := \PPa^{\vee}[1]$, since  it is the $*$-degree $1$ part of the derived preprojective algebra $\PPi$ (see Section <ref>).
Note that the functor $\PPi_{1} \lotimes_{\Pa}- : \Dbmod{\Pa} \to \Dbmod{\Pa}$ is the inverse $\nu_{1}^{-1}$ of the $-1$-shifted
Nakayama functor $\nu_{1} ( -) := \tuD(\Pa)[-1] \lotimes_{\Pa} -$.

§.§ Weighted trace and weighted Euler characteristic

§.§.§ Trace of an endomorphism of a   complex over $\kk$

Let $V \in \sfC^{\mrb}(\kk)$ and $f: V \to V$, we define the trace $\Tr(f)$ to be

\[
\Tr(f) := \sum_{i \in \ZZ} ( -1)^{i}\Tr(f^{i})
\]

where $\Tr(f^{i})$ in the right hand side is the trace of the linear map $f^{i}: V^{i} \to V^{i}$.
We note that the Euler characteristic  $\Euch(V) := \sum_{i \in \ZZ} ( -1)^{i}\dim V^{i}$ of $V$ is obtained as
$\Euch (V) = \Tr(\id_{V} )$.

§.§.§ Weighted trace and  weighted Euler characteristic

Let $A = \kk Q$ and  $ v= (v_{i}) \in \kk Q_{0}$.
Let $M \in \sfC^{\mrb}(A)$ and $f: M \to M$.
Observe that a  vertex $i \in Q_{0}$ induces a morphism  $e_{i}f: e_{i} M \to e_{i} M$ in $\sfC^{\mrb}(\kk)$.
We define the weighted trace ${}^{v}\!\Tr(f)$ of $f$ to be the weighted sum of traces of $e_{i} f$.

\[
{}^{v}\!\Tr(f) := \sum_{i \in Q_{0}} v_{i} \Tr(e_{i} f).
\]

We define the weighted Euler characteristic ${}^{v}\!\Euch(M)$ to be $ {}^{v}\!\Euch(M) := \sum_{i \in Q_{0} } v_{i} \Euch(e_{i}M)$
so that we have
${}^{v}\!\Tr(\id_{M} ) = {}^{v}\!\Euch(M)$.

We denote the dimension vector of $M$ by $\Euvect(M)$.

\[
\Euvect(M) := (\Euch(e_{1} M), \Euch(e_{2}M), \ldots, \Euch(e_{r}M) )^{t}  \in \ZZ Q_{0}.
\]

Then the weighted Euler characteristic ${}^{v}\!\Euch(M)$ is obtained as

\begin{equation}\label{202102051939}
{}^{v}\!\Euch(M)  = v^{t}\Euvect(M),
\end{equation}

i.e., the product of the row vector $v^{t}$ and the column vector $\Euvect(M)$.

It is straightforward to check that these notions descend to objects $M \in \Dbmod{A}$ and endomorphisms $f: M \to M$.
We leave the verification of the following lemma to the reader.

* ${}^{v}\!\Tr(f) = \sum_{ i \in \ZZ}(-1)^{i}\ {}^{v}\!\Tr(\tuH^{i}(f))$.

* ${}^{v}\!\Tr(\id_{M}) = \sum_{i \in \ZZ} (-1)^{i} \ {}^{v}\!\Euch(\tuH^{i}(M))$.

* If $f$ is nilpotent in $\sfD(\Pa)$, then ${}^{v}\!\Tr(f) = 0$.

Recall that we may identify the Grothendieck group $K_{0}(A)$ with $\ZZ Q_{0}$ via the map $[M] \mapsto \Euvect(M)$.
Let  $F$ be an autoequivalence  of $\Dbmod{A}$.
We denote by $\underline{F}: \ZZ Q_{0} \to \ZZ Q_{0}$ the automorphism  induced from $F$.
In other words, $\underline{F}$ is the square matrix that satisfies

\[
\underline{F}\Euvect(M) = \Euvect(F(M))
\]

for all $M \in \Dbmod{A}$.
We note the following equation that
follows from (<ref>).

\begin{equation}\label{202102061837}
{}^{v}\!\Euch(F(M)) = {}^{\underline{F}^{t}(v)}\Euch(M).
\end{equation}

§.§.§ The Cartan matrix and the Coxeter matrix

We recall that  the Cartan matrix $C$  is a matrix defined by

\[
C := (\Euch(e_{i} \Pa e_{j}) )_{i,j}.
\]

Then the Coxeter matrix $\Phi$ (for left modules) is defined by

\[
\Phi := - C^{t} C^{-1}.
\]

Note that in the notation of  Section <ref>, we have
$\Phi = \underline{\nu_{1}}$.
In other words, we have

\[
\Euvect(\nu_{1}(M)) = \Phi \Euvect(M)
\]

for $M \in \Dbmod{\Pa}$.
Therefore setting $\Psi := \Phi^{-t}$, we have

\begin{equation}\label{202111101530}
{}^{v}\!\Euch(\PPi_{1} \lotimes_{\Pa} M ) =\Euvect(\nu_{1}^{-1}(M)) = {}^{\Psi(v)} \Euch(M).
\end{equation}

§.§ Trace formula

Let $i \in Q_{0}$.
We define a morphism $\hat{e}_{i}: AA \to AA$ of $A^{\mre}$-modules by the formula

\[
\hat{e}_{i}(e_{j} ) := \delta_{i,j} e_{i}
\]

for $j \in Q_{0}$.
We define  a morphism
$\tilde{e}_{i} :  \PPa^{\vee}\to   \PPa$ of complexes of $\Pa$-$\Pa$-bimodules to be

\begin{equation}\label{2020081418071}
\tilde{e}_{i}: \small
\PPa^{\vee}  = \left( \Pa V^{*} \Pa[-1] \oplus \Pa \Pa,  \begin{pmatrix} 0 & \uparrow \hat{\rho} \\ 0 & 0\end{pmatrix} \right)
\xrightarrow{ \tiny \begin{pmatrix} 0 & \hat{e}_{i} \\ 0 & 0 \end{pmatrix}}
\left( \Pa \Pa \oplus \Pa V\Pa [1], \begin{pmatrix} 0 & \hat{\mu} \uparrow \\ 0 & 0\end{pmatrix} \right) =
\PPa.
\end{equation}

Recall that  $\PPa^{\vee} \cong  \Pa^{\vvee}$  in $\sfD(\Pa^{\mre})$ and
there are canonical isomorphisms

\[
\Hom_{\Pa^{\mre}}(\Pa^{\vvee}, \Pa) \cong \tuHH_{0}(\Pa) \cong \Pa/[\Pa, \Pa] \cong \kk Q_{0}.
\]

Note that $e_i \in \kk Q_{0}$ in the right hand side  corresponds to $\tilde{e}_{i}$ in the left hand side.
Thus the set $\{ \tilde{e}_{i}\}_{i \in Q_{0}}$ forms a basis of $\Hom_{\Pa^{\mre}}(\Pa^{\vvee}, \Pa)$.

For $v \in \kk Q_{0}$, we denote by ${}^{v}\!\ttheta$ the corresponding  element of $\Hom_{\Pa^{\mre}}(\Pa^{\vvee}, \Pa)$.
In other words,
for $v \in \kk Q_{0}$, we set

\[
{}^{v}\!\ttheta := \sum_{ i \in Q_{0}} v_{i} \tilde{e}_{i}.
\]

Since $(A^{\vvee})^{\vvee} \cong A$ in $\sfD(\Pa^{\mre})$,
the $A^{\mre}$-duality $(-)^{\vvee}$ induces an endomorphism
$(-)^{\vvee}: \Hom_{\Pa^{\mre}}(A^{\vvee}, A) \to \Hom_{\Pa^{\mre}}(\Pa^{\vvee}, \Pa)$.
By a general result  due to Van den Bergh <cit.>, this is the identity map (which in our case can be proved by an easy  computation).
Thus  we have

\begin{equation}\label{202105111259}
({}^{v}\!\ttheta)^{\vvee} = {}^{v}\!\ttheta.
\end{equation}

Recall from [34] that the functor $\Pa^{\vvee} \lotimes -: \Dbmod{\Pa} \to \Dbmod{\Pa}$ is the inverse of a Serre functor.
$\isagl{-,+}: \Hom_{\Pa}(M, N)  \otimes\Hom_{\Pa}(\Pa^{\vvee} \lotimes_{\Pa} N, M) \to \kk$ be the pairing of Serre duality.

Let $M \in \Dbmod{\Pa}$. We recall our convention that

\[{}^{v}\!\ttheta_{M} = {}^{v}\!\ttheta \lotimes_{\Pa} M: \PPa^{\vvee} \lotimes_{\Pa} M \to M.\]

The following theorem gives a formula that computes the pairing $\isagl{f, {}^{v}\!\ttheta_{M}}$ for $f \in \End_{\Pa}(M)$.

Let $M \in \Dbmod{\Pa}$ and $f \in \End_{\Pa}(M)$.
Then the following equality holds

\[
\isagl{f, {}^{v}\!\ttheta_{M}}  = {}^{v}\!\Tr(f).
\]

In subsequent work [38],
we will prove Theorem <ref> for homologically smooth proper dg-algebras.

By linearity, it is enough to show that
$\isagl{f,  \tilde{e}_{i,M}} = \Tr(e_{i} f)$ for each $i \in Q_{0}$.

We recall the isomorphisms in $\sfD(\kk)$ which proves Serre duality from [34] (see also Section <ref>).

\[
\RHom_{\Pa}(M,N)
\xrightarrow{ \cong \  F }\tuD(M) \lotimes_{\Pa} \Pa^{\vvee} \lotimes_{\Pa} N
\xrightarrow{ \cong \ G} \tuD \RHom_{\Pa}( \Pa^{\vvee} \lotimes_{\Pa}N, M).
\]

we explain the construction of $G$.
Let $L,M \in \sfC(\Pa)$.
We define a cochain map
$\tilde{\Phi}_{L,M}: \tuD(M) \otimes_{\Pa} L \to \tuD\cpxHom_{\Pa}(L,M)$ by the formula

\[
\tilde{\Phi}_{L,M}(\phi \otimes l) (f) := (-1)^{|l||f|}\phi(f(l))
\]

for homogeneous elements $l, \phi, f$ of $L, \tuD(M)$ and $\cpxHom_{\Pa}(L,M)$.
This map is natural in $L,M$ and descents to a morphism $\Phi_{L,M}:  \tuD(M) \lotimes_{\Pa} L \to \tuD \RHom_{\Pa}(L,M)$
in $\sfD(\kk)$.
The morphism $G$ is set to be $G := \Phi_{\Pa^{\vvee} \lotimes N, M}$.

we explain the construction of $F$ in the case $N = M$.
Let $P \in \sfC^{\mrb}( \Pa \proj)$ be a complex of projective $\Pa$-module quasi-isomorphic to $M$.
Then $F$ is given by the following quasi-isomorphism in $\sfC(\kk)$ which is denoted by the same symbol

\[
\begin{split}
F: \cpxHom_{\Pa}(P,P)
& \xrightarrow{ \cong } \cpxHom_{\Pa^{\mre}}(\Pa, \cpxHom_{\kk}(P,P) ) \\
&\xrightarrow{\sim} \cpxHom_{\Pa^{\mre}}( \PPa, \cpxHom_{\kk}(P,P) ) \cong
\tuD(P) \otimes_{\Pa} \PPa^{\vee} \otimes_{\Pa} P
\end{split}
\]

where the second morphism is induced from $\mu: \PPa \to \Pa$.

To provide an explicit formula of $F$, first we fix a way to exhibit elements of $\Hom^{0}_{\Pa}(P,P)$.
For this we recall that $\Hom_{\Pa}^{0}(P, P)$ is a subspace $\Hom_{\kk}^{0}(P, P)$ of $\kk$-linear morphisms of degree $0$.
The underlying cohomologically graded module of $P$ is
$\bigoplus_{n \in \ZZ} P^{n}[-n]$ and
hence $\Hom_{\kk}^{0}(P, P) =\prod_{n \in \ZZ} ( P^{n}[-n]) \otimes_{\kk} (\tuD(P^{n})[n])$.
For $j \in Q_{0} $ and $ n \in \ZZ$,
we fix a $\kk$-basis  $\{\phi_{s}^{(j,n)}\}_{ s= 1}^{d_{j,n}}$ of $e_{j} P^{n}$ and denote by $\{\phi_{s}^{*(j,n)}\}_{s=1}^{d_{j,n}}$
the dual $\kk$-basis.
For simplicity we set $\tilde{\phi}_{s}^{(j,n)} :=
\phi_{s}^{(j, n)}[-n]$ and $\tilde{\phi}_{t}^{*(j, n)} := \phi_{t}^{*(j,n)}[n]$.
regarding $\Hom_{\Pa}^{0}(P, P)$ as a subspace of $\Hom_{\kk}^{0}(P, P)$,
we may write  an element  $f$ of $\Hom_{\Pa}^{0}(P,P)$ as

\begin{equation}\label{202012091744}
f = \sum_{s,t,n,j,k} f_{st}^{(j,k, n)} \tilde{\phi}_{s}^{(j, n)}\otimes \tilde{\phi}_{t}^{*(k, n)}
\end{equation}

for some $f_{s,t}^{(j,k, n)} \in \kk$.

Recall that as a cohomologically graded modules
$\PPa = \Pa\Pa \oplus \Pa V\Pa [1]$ and the morphism $\mu: \PPa \to \Pa$ factors through the morphism
$\Pa\Pa \to \Pa$.
It follows that the image of the induced morphism $\cpxHom_{\Pa^{\mre}}(\Pa, \cpxHom_{\kk}(P, P)) \to
\cpxHom_{\Pa^{\mre}}(\PPa, \cpxHom_{\kk}(P, P))$
is contained in the direct summand $\cpxHom_{\Pa^{\mre}}(\Pa\Pa, \cpxHom_{\kk}(P, P))$.
Let $\tilde{m}: \Pa^{\mre} \to \PPa$ be the morphism in $\sfC(A^{\mre})$ induced from the canonical map $m: \Pa^{\mre} \to \Pa\Pa$.
Then $\tilde{m}$ and $\mu$ induce the following commutative diagram,
the left column of which is $F$.

\[
\begin{xymatrix}{
\cpxHom_{\Pa}(P,P) \ \  \ar@{=}[r] \ar[dd]_{\simeq}  &
\cpxHom_{\Pa}(P,P) \ \  \ar@{^(->}[r] \ar[d]^{\cong} & \cpxHom_{\kk}(P, P) \ar[dd]^{\cong}  \\
& \cpxHom_{\Pa^{\mre}}(\Pa, \cpxHom_{\kk}(P,P)) \ar@{^(->}[d] & \\
\cpxHom_{\Pa^{\mre}}(\PPa, \cpxHom_{\kk}(P,P))  \ar[r]  \ar[d]_{\cong} &
\cpxHom_{\Pa^{\mre}}(\Pa\Pa, \cpxHom_{\kk}(P,P)) \ar@{^(->}[r] \ar[d]_{\cong} &
\cpxHom_{\Pa^{\mre}}(\Pa^{\mre}, \cpxHom_{\kk}(P,P)) \ar[d]^{\cong}  \\
\tuD(P) \otimes_{\Pa} \PPa^{\vee} \otimes_{\Pa} P  \ar@{->>}[r]  &
\tuD(P) \otimes_{\Pa} (\Pa\Pa)^{\vee} \otimes_{\Pa} P \ar@{^(->}[r] &
\tuD(P) \otimes_{\Pa} (\Pa^{\mre})^{\vee} \otimes_{\Pa} P
\]

Using this  diagram and (<ref>), we see that if $f$ is of the form of (<ref>), then $F(f)$ is
an element of $\tuD(P) \otimes_{\Pa} (\Pa\Pa)^{\vee} \otimes_{\Pa} P \subset \tuD(P) \otimes_{\Pa} \PPa^{\vee} \otimes_{\Pa} P$
given as

\begin{equation}\label{202006151617}
F(f) =\sum_{s,t,n,j} ( -1)^{n} f_{st}^{(j,j,n)} \tilde{\phi}_{t}^{*(j,n)}  \otimes e_{j} \otimes \tilde{\phi}_{s}^{(j, n)}
\end{equation}

we use the identification

\[
\tuD(P) \otimes_{\Pa} (\Pa\Pa)^{\vee} \otimes_{\Pa} P \cong
\tuD(P) \otimes_{\Pa} \Pa\Pa \otimes_{\Pa} P \cong
\tuD(P) \otimes_{\Pa_{0}}\Pa_{0} \otimes_{\Pa_{0}} P.
\]

Let $\mu_{P}:  \PPa  \otimes_{\Pa}  P  \to P$ be the canonical quasi-isomorphism, which induces the identification
$A \lotimes_{A} P \cong P$.
Now we can check that the desired result holds as follows

\[
\begin{split}
\isagl{f,   \hat{e}_{i, M}}
& = G ( F(f) ) (\mu_{P}\tilde{e}_{i, M})\\
& =  \sum_{s,t, n,j,k} (-1)^{n} f_{s,t}^{(j,j, n)}
\tilde{\phi}_{t}^{*(j,n)}  \left(  (\mu_{P} \hat{e}_{i ,M} )( e_{j} \otimes \tilde{\phi}_{s}^{(j, n)}) \right) \\
& = \sum_{s,t,n,j }  (-1)^{n} f_{s,t}^{(i,i,n)}\tilde{\phi}_{t}^{*(i,n)} \left( \tilde{\phi}_{s}^{(i, n)}\right)\\
&= \sum_{s,n}  (-1)^{n} f_{s,s}^{(i,i,n)}  \\
& = \Tr(e_{i} f).
\end{split}
\]

Let $i\in Q_{0}$ be a vertex.
Applying Theorem <ref> to the simple module $S_{i}$
and the identity map $\id_{S_{i}}$, we obtain

\[
\isagl{ \id_{S_{i}}, {}^{v}\!\ttheta_{S_{i}} } = {}^{v}\!\Euch(S_{i}) = v_{i}.
\]

We point out the following consequence.

Let $\phi \in \Hom_{\Pa^{\mre}}(\Pa^{\vvee}, \Pa)$.
Then we have

\[
\phi = \sum_{i \in Q_{0}} \isagl{ \id_{S_{i}}, \phi_{S_{i}} } \tilde{e}_{i}.
\]

§.§ The universal Auslander-Reiten triangle

Let $v \in \kk Q_{0}$.
We set  ${}^{v}\!\YYM_{1} := \cone({}^{v}\!\ttheta)$,
since, as is explained in Section <ref>,
it is the $*$-degree $1$ part of the derived quiver Heisenberg  algebra ${}^{v}\!\YYM$
provided that $v_{i} \neq 0$ for all $i \in Q_{0}$. .

We set  notation  for the exact triangle ${}^{v}\!\sfAR$ obtained from the morphism ${}^{v}\!\ttheta: A^{\vvee} \to A$
as below.

\[
{}^{v}\!\sfAR: \Pa \xrightarrow{ \ {}^{v}\!\rrho \ }  {}^{v}\!\YYM_{1} \xrightarrow{{}^{v}\!\ppi_{1}} \PPi_{1}
\xrightarrow{-{}^{v}\!\ttheta[1]} \Pa[1].
\]

A morphism is called AR-coconnecting if it is the co-connecting morphism for an AR-triangle.
Combining Theorem <ref> and the Happel criterion for AR-coconnecting morphisms (reviewed in Section <ref>),
we can immediately
prove that if ${}^{v}\!\Euch(M) \neq 0$, then  ${}^{v}\!\theta_{M}$ is AR-coconnecting and the triangle ${}^{v}\!\sfAR_{M} := {}^{v}\!\sfAR\lotimes M$ is an AR-triangle.
The precise statement is the following.

Let $v \in \kk Q_{0}$ and $M \in \sfD^{\mrb}( \Pa \mod )$ an indecomposable object.
We denote by ${}^{v}\!\sfAR_{M}$ the following exact triangle in $\sfD(\Pa)$.

\[
{}^{v}\!\sfAR_{M} :
M  \xrightarrow{ \ {}^{v}\!\rrho_{M} \ } {}^{v}\!\YYM_{1} \lotimes_{\Pa} M \xrightarrow{ \ {}^{v}\!\ppi_{1, M} \ } \PPi_{1} \lotimes_{\Pa} M \xrightarrow{ -{}^{v}\!\ttheta_{M}[1]} M[1].
\]

Then the following statements hold.

* If ${}^{v}\!\Euch(M) \neq 0$ in $\kk$, then
the morphism ${}^{v}\!\ttheta_{M}$ is AR-coconnecting to $M$ and
the exact triangle ${}^{v}\!\sfAR_{M}$ is an Auslander-Reiten triangle starting from $M$.

* Assume moreover that $\dim \ResEnd_{\Pa}(M) = 1$.
the exact triangle ${}^{v}\!\sfAR_{M}$ is an Auslander-Reiten triangle starting from $M$
if and only if ${}^{v}\!\Euch (M) \neq 0$ in $\kk$.
In the case ${}^{v}\!\Euch(M) =0$, the exact triangle ${}^{v}\!\sfAR_{M}$ splits.

Let $M \in \sfD^{\mrb}( \Pa \mod )$ be an indecomposable object.
From  Lemma <ref> and  Theorem <ref>, we have

\[
\isagl{\id_{M}, \ttheta_{M}}  ={}^{v}\!\Euch(M), \
\isagl{f, \ttheta_{M}} = 0 \textup{ for } f \in \rad\End_{\Pa}(M).
\]

(1) Assume that ${}^{v}\!\Euch(M) \neq 0$ in $\kk$.
It follows from Happel's criterion (Theorem <ref>)
that  ${}^{v}\!\sfAR_{M}$ is an Auslander-Reiten triangle.

(2) follows from Lemma <ref>.

§.§.§ Regularity of weights

We recall from the introduction  the definitions of a sincere weight and  a regular weight
we add one more notion for weights.

* A weight $v \in \kk Q_{0}$ is called sincere if $v_{i} \neq 0$ for all $i \in Q_{0}$.

* A weight  $v \in \kk Q_{0}$  is called regular
if we have ${}^{v}\!\Euch(M) \neq 0$ in $\kk$ for all
$M \in \ind \kk Q$.

* A weight  $v \in \kk Q_{0}$  is called semi-regular
if we have ${}^{v}\!\Euch(M) \neq 0$ in $\kk$ for all indecomposable
prepeojective modules $M$ and indecomposable preinjective modules $M$.

Assume that the base field $\kk$ is algebraically closed and of characteristic $0$.
In the case $Q$ is  Dynkin,
the vector space $\kk Q_{0}$ may be identified with the Cartan subalgebra $\mathfrak{h}$ of the semi-simple Lie algebra $\mathfrak{g}$ corresponding to $Q$.
By Gabriel's theorem [19]
that says that the dimension vectors of indecomposable $\Pa$-modules are precisely the positive roots of $\mathfrak{g}$,
the regularity defined above   coincides with usual notion used about an element  of the Cartan subalgebra $\mathfrak{h}$.

Let $\cU_{\Pa} := \add \{ \nu_{1}^{-p}P_{i} \mid p \in \ZZ, i \in Q_{0}\}$.
Then a  weight  $v \in \kk Q_{0}$ is semi-regular
if and only if ${}^{v}\!\Euch(M) \neq 0$ for all $M \in \ind\cU_{\Pa}$.
We note that a regular weight is sincere.

We give examples of regular weight  $v \in \kk Q_{0}$.

* Assume that $\QQ \subset \kk$. If $v \in \kk^{\times} Q_{0}$ is strictly positive,
i.e., $v_{i}\in \QQ_{>0}$ for all $i \in Q_{0}$, then it is regular.

* Let $\PP$ be the prime field contained in $\kk$.
Assume that $\dim_{\PP} \kk \geq r $.
Taking linearly independent element $v_{1}, v_{2}, \ldots, v_{r} \in \kk$ over $\PP$,
we obtain a regular weight $v \in \kk Q_{0}$.

Recall that for an indecomposable object  $M \in \ind \Dbmod{A}$
there exist $M' \in \ind \kk Q$ and  a unique integer $p \in \ZZ $ such that  $M \cong M'[p]$
\[
{}^{v}\!\Euch(M)= ( -1)^{p}{}^{v}\!\Euch(M').
\]

Thus if $v \in \kk Q_{0}$ is regular, then  ${}^{v}\!\Euch(M) \neq 0$ for all $M \in \ind \Dbmod{A}$.

§.§.§ Universal Auslander-Reiten triangle

From Theorem <ref>, we conclude the following result.

Assume that $v \in \kk Q_{0}$ is regular  (resp. semi-regular).
Let $M $ be an object of $\sfD^{\mrb}( \Pa \mod )$ (resp. $\cU_{\Pa}[\ZZ]$).
Then the exact triangle ${}^{v}\!\sfAR_{M}$ is a direct sum of  Auslander-Reiten triangles.

\[
{}^{v}\!\sfAR_{M} :
M  \xrightarrow{ \ {}^{v}\!\rrho_{M} \ } {}^{v}\!\YYM_{1} \lotimes_{\Pa} M \xrightarrow{ \ {}^{v}\!\ppi_{1, M} \ } \PPi_{1} \lotimes_{\Pa} M \xrightarrow{ -{}^{v}\!\ttheta_{M}[1]} M[1].
\]

In other words,
the morphism ${}^{v}\!\rrho_{M}: M \to {}^{v}\!\YYM_{1} \lotimes_{A} M$ is a minimal left $\rad$-approximation  of $M$
the morphism ${}^{v}\!\ppi_{1, M}: {}^{v}\!\YYM_{1} \lotimes_{A} M \to \PPi_{1} \lotimes_{A} M$
is a minimal right $\rad$-approximation of $\PPi_{1} \lotimes_{A} M$.

We point out the following corollary.

Assume that $v \in \kk Q_{0}$ is semi-regular.
Then the object ${}^{v}\!\YYM_{1} \in \sfD(\Pa^{\mre})$ is concentrated in the $0$-th cohomological degree.

It is enough to show that the object ${}^{v}\!\YYM_{1} e_{i} \cong {}^{v}\!\YYM_{1} \lotimes_{\Pa} P_{i}$ of $\sfD(\Pa)$ is concentrated in the $0$-th cohomological degree for $i \in Q_{0}$.

Let $M \in \ind \Dbmod{\Pa}$. Recall that  $M$ is concentrated in a single cohomological degree and $\Hom_{\Pa}(P_{i}, M) = e_{i} M$.
It follows that if $\Hom_{\Pa}(P_{i}, M) \neq 0$, then $M$ is concentrated in the $0$-the cohomological degree.
Since the morphism  ${}^{v}\!\rrho_{P_{i}} : P_{i} \to {}^{v}\!\YYM_{1}e_{i}$ is minimal by Theorem <ref>,
we conclude that ${}^{v}\!\YYM_{1}e_{i}$ is concentrated in the $0$-th cohomological degree.

§.§ The action of an exact autoequivalence $F$ on ${}^{v}\!\ttheta$

Let  $\phi: A^{\vvee} \to A$ be a morphism in $\Dbmod{A^{\mre}}$.
We denote  the induced natural transformation  by the same symbol

\[
\phi: \sfS^{-1}= \Pa^{\vvee} \lotimes _{\Pa} - \xrightarrow{ \ \phi \lotimes -\ }  \Pa \lotimes_{\Pa} - = \id_{\sfD}
\]

where $\id_{\sfD}$ is the identity functor of $\Dbmod{\Pa}$.

Let $T \in \Dbmod{\Pa}$ be a two-sided tilting complex over $\Pa$ and
$F:= T \lotimes_{\Pa}-$  the associated  exact autoequivalence of $\Dbmod{\Pa}$.
We note that there is a natural isomorphism $\gamma_{F}: \sfS^{-1} F \to F\sfS^{-1}$ induced from an
isomorphism $\gamma_{T}: \Pa^{\vvee} \lotimes_{\Pa} T \to T \lotimes_{\Pa} \Pa^{\vvee}$ of $\Dbmod{\Pa^{\mre}}$
(see Lemma <ref>).

We define  a natural transformation $F \cdot \phi: \sfS^{-1} \to \id$ to be the following composition

\[
F\cdot \phi : \sfS^{-1}\xrightarrow{\cong} FF^{-1} \sfS^{-1}  \xrightarrow{ F((\gamma_{F^{-1}})^{-1}) } F\sfS^{-1} F^{-1} \xrightarrow{ \ F \phi F^{-1} \ } F F^{-1}\xrightarrow{\cong} \id_{\sfD}.
\]

It follows from the above remark
that the natural transformation $F\cdot \phi$ is induced from
a morphism $F\cdot \phi: \Pa^{\vvee} \to \Pa$ of $\Dbmod{\Pa^{\mre}}$, which we denote by the same symbol.

We obtain the following lemma immediately from the definition of $F\cdot \phi$.

Let $T \in \Dbmod{\Pa}$ be a two-sided tilting complex over $\Pa$ and
$F:= T \lotimes_{\Pa}-$  the associated  exact autoequivalence of $\Dbmod{\Pa}$.
Then we have

\[
F( ( F^{-1}\cdot\phi)_{M}) = \phi_{F(M)} \gamma_{F, M}.
\]

In other words, the following diagram is commutative

\[
\begin{xymatrix}@C=60pt{
\sfS^{-1}F(M) \ar[r]^{\phi_{F(M)}}  \ar[d]_{\gamma_{F,M}}
& F(M) \\
F\sfS^{-1}(M) \ar[ur]_{ \  F( (F^{-1} \cdot \phi)_{M}) } &.
\]

The next theorem provides a way to compute $F\cdot ({}^{v}\!\ttheta)$.

Let $T \in \Dbmod{\Pa}$ be a two-sided tilting complex over $\Pa$ and
$F:= T \lotimes_{\Pa}-$  the associated  exact autoequivalence of $\Dbmod{\Pa}$.
Then, for $v \in \kk Q_{0}$,
we have the following equality in $\Hom_{A^{\mre}}(A^{\vvee}, A)$

\[
F^{-1}\cdot ( {}^{v}\!\ttheta) = {}^{\underline{F}^{t}(v)} \ttheta.
\]

For $M \in \Dbmod{\Pa}$ and $f\in \End_{\Pa}(M)$,
we have

\[
\isagl{f, (F^{-1}\cdot {}^{v}\!\ttheta )_{M}}  = \isagl{F(f), F(F^{-1}\cdot ({}^{v}\!\ttheta ))\gamma_{F} }
= \isagl{F(f), ({}^{v}\!\ttheta)_{F(M)}}.
\]

Therefore, for $i \in Q_{0}$, we have

\[
\begin{split}
\isagl{ \id_{S_{i}}, (F^{-1}\cdot {}^{v}\!\ttheta)_{ S_{i} } }
= \isagl{ \id_{F(S_{i}) } , {}^{v}\!\ttheta_{F(S_{i})}}  = {}^{v}\!\Euch(F(S_{i}))
=   {}^{\underline{F}^{t}(v)} \Euch(S_{i}) = (\underline{F}^{t} (v))_{i}
\end{split}
\]

where for the second equality we use Theorem <ref>
and for the third the equation (<ref>).
Thus we deduce the desired equality from Corollary <ref>.

Let $v \in \kk Q_{0}$ and $T \in \Dbmod{\Pa}$ be a two-sided tilting complex over $\Pa$.
We denote by
$F:= T \lotimes_{\Pa}-$  the associated  exact autoequivalence of $\Dbmod{\Pa}$.
Then for $M \in \ind \Dbmod{\Pa}$ such that $\dim \ResEnd_{\Pa}(M) =1$ and ${}^{v}\!\Euch(M) \neq 0$, the following holds.

* We have the following equality in $\Hom_{\Pa}(\sfS^{-1}(M), M)$

\[
(F^{-1}\cdot{}^{v}\!\ttheta)_{M} = \frac{{}^{v}\!\Euch (F(M))}{{}^{v}\!\Euch (M)} {}^{v}\!\ttheta_{M}.
\]

* We have the following equality in $\Hom_{A}(F\sfS^{-1}(M), F(M))$

\[
\frac{{}^{v}\!\Euch (F(M))}{{}^{v}\!\Euch (M)} F( {}^{v}\!\ttheta_{M}) \gamma_{F} = {}^{v}\!\ttheta_{F(M)}.
\]

In other words the following diagram is commutative:

\[
\begin{xymatrix}@C=60pt{
\sfS^{-1}F(M) \ar[r]^{{}^{v}\!\ttheta_{F(M)}}  \ar[d]_{\gamma_{F,M}} &
F(M) \\
F\sfS^{-1}(M) \ar[ur]_{x F({}^{v}\!\ttheta_{M}) }
\]

\[x: =\frac{{}^{v}\!\Euch (F(M))}{{}^{v}\!\Euch (M)}.
\]

We have the following equation

\[
\begin{split}
\isagl{ \id_{M},  ( F^{-1}\cdot {}^{v}\!\ttheta)_{M} } = \isagl{\id_{M}, {}^{\underline{F}^{t}(v)}\ttheta_{M}} = {}^{\underline{F}^{t}(v)} \Euch(M) = {}^{v}\!\Euch(F(M)).
\end{split}
\]

where for the first equality we use Theorem <ref>
and for the third the equation (<ref>).
We deduce the desired equation by using Lemma <ref>.

(2) follows from (1) together with Lemma <ref>.

From uniqueness of AR-triangles, we obtain the following corollary.

Let $v \in \kk Q_{0}$ and $F$ be an exact autoequivalence of $\Dbmod{A}$.
Then for $M \in \ind \Dbmod{\Pa}$ such that $\dim\ResEnd_{\Pa}(M) =1$ and  ${}^{v}\!\Euch(M) \neq 0,  {}^{v}\!\Euch(F(M)) \neq 0$,
there exists an isomorphism ${}^{v}\!\sfa_{F, M}: {}^{v}\!\YYM_{1} \lotimes_{A} F(M) \to F({}^{v}\!\YYM_{1} \lotimes_{A} M)$
that gives the following isomorphism of exact triangles:

\[
\begin{xymatrix}@C=60pt{
\sfS^{-1}F(M) \ar[r]^{{}^{v}\!\ttheta_{F(M)}}  \ar[d]_{\gamma_{F}} &
F(M) \ar[r]^{{}^{v}\!\rrho_{F(M)}} \ar@{=}[d]  &
{}^{v}\!\YYM_{1} \lotimes_{A}F(M) \ar[d]^{{}^{v}\!\sfa_{F, M} } \ar[r]^{{}^{v}\!\ppi_{1, F(M)} } &
\sfS^{-1}F(M) [1] \ar[d]^{\gamma_{F}[1]}    \\
F\sfS^{-1}(M) \ar[r]_{x F({}^{v}\!\ttheta_{M}) } &
F(M) \ar[r]_{F({}^{v}\!\rrho_{M})}  &
F({}^{v}\!\YYM_{1} \lotimes_{A} M) \ar[r]_{x^{-1}F({}^{v}\!\ppi_{1, M}) } &
F\sfS^{-1}(M) [1]   \\
\]

\[
x: =\frac{{}^{v}\!\Euch (F(M))}{{}^{v}\!\Euch (M)}.
\]

We remark that the morphism ${}^{v}\!\sfa_{F, M}$ is not uniquely determined, but it is determined modulo $\rad$.

§.§ Right modules and dualities

Applying the same argument for $A^{\op}$ we can obtain  right versions of above results.
In this subsection, we discuss relationships between the previous results and their right versions
through dualities.
For this purpose first we fix notations and terminologies for right modules.

§.§.§ The weighted Euler characteristic for right modules

Let $N \in \Dbmod{\Pa^{\op}}$.
We set
\[
\Euvect(N) : = (\Euch(N e_{1} ), \ldots, \Euch(N e_{r}))\]

and regard it as a row vector.
The weighted Euler characteristic for   $v \in \kk Q_{0}$ is defined by the formula

\[
{}^{v}\!\Euch(N) := \Euvect(N) v = \sum_{i \in Q_{0} }v_{i} \Euch(N e_{i}).
\]

§.§.§ The properties (I) and  (I') for right modules

The $\kk$-duality $\tuD(-)$ descents to the transpose of dimension vectors i.e, $\Euvect(\tuD(M)) = \Euvect(M)^{t}$.
It follows that  ${}^{v}\!\Euch(\tuD(M)) = {}^{v}\!\Euch(M)$. Since the functor $\tuD(-): \Pa\mod \to \Pa^{\op}\mod$ gives a contravariant equivalence of categories
which sends the preprojective modules to the preinjective modules and the preinjective modules to the preprojective modules, we obtain the following lemma.

* An element  $v \in \kk Q_{0}$  is regular (for left modules)
if and only if it satisfies regularity for right modules,
we have ${}^{v}\!\Euch(N) \neq 0$ in $\kk$ for all
$N \in \ind \kk Q^{\op}$.

* An element  $v \in \kk Q_{0}$ is semi-regular (for left modules)
if and only if it satisfies semi-regularity  for right modules,
if we have ${}^{v}\!\Euch(N) \neq 0$ in $\kk$ for all
prepeojective  $\Pa^{\op}$-modules $N$ and preinjective $\Pa^{\op}$ modules $N$.

§.§.§ The Coxeter matrix $\Phi_{\textup{right}}$ for right modules

We remark that the Coxeter matrix $\Phi_{\textup{right}}$ for right modules is different from that for left modules which we wrote $\Phi$.
Later we deal with left modules and right modules at the same time, for this we need to know  relationship between $\Phi$ and $\Phi_{\textup{right}}$.

Let $C$ be the Cartan matrix of $\Pa = \kk Q$.
Then the Coxeter matrix for right modules is given as
$\Phi_{\textup{right}}: = - C^{-1} C^{t}$. Then, we have the equality below for all $N \in \Dbmod{\Pa^{\op}}$.

\[
\Euvect(\nu_{1}(N)) = \Euvect(N) \Phi_{\textup{right}}.
\]

Observe that we have $\Psi= \Phi^{-t} = - C^{-1}C^{t} = \Phi_{\textup{right}}$.
Consequently we have

\[
\begin{split}
&\Euvect(N \lotimes_{\Pa} \PPi_{1} ) = \Euvect(N) \Phi_{\textup{right}}^{-1} =\Euvect(N)\Psi^{-1}, \\  {}^{v}& \Euvect(N \lotimes_{\Pa} \PPi_{1}) = {}^{\Psi^{-1}(v)}\Euvect(N).
\end{split}
\]

In particular we deduce the following assertion.

Assume that $v \in \kk Q_{0}$ is an eigenvector of $\Psi$ with the eigenvalue $\lambda$.
Then for $N \in \Dbmod{\Pa^{\op}}$, we have

\[
{}^{v}\!\Euch(N \lotimes_{\Pa} \PPi_{1}) = \frac{1}{\lambda} {}^{v}\!\Euch(N).
\]

We give  relationships of the weighted Euler characteristic ${}^{v}\!\Euch(M)$ for $M \in \Dbmod{\Pa}$ and that of $\Pa$-dual
$M^{\lvvee} := \RHom_{\Pa}(M, \Pa) $ of $M$, which is an object of $\Dbmod{\Pa^{\op}}$.

We have an isomorphism $M^{\lvvee} \cong \tuD(\nu_{1}(M))[-1]$.
Consequently we have

\begin{equation}\label{202104021807}
\begin{split}
&\Euvect(M^{\lvvee} ) = (-\Phi \Euvect(M))^{t}, \ \
{}^{v}\!\Euch(M^{\lvvee}) =- {}^{v}\!\Euch(\nu_{1}(M)) = - {}^{\Psi^{-1}(v)}\Euch(M).
\end{split}
\end{equation}

§.§.§ Duality

Let $N \in \ind \Dbmod{\Pa^{\op}}$.
Assume that ${}^{v}\!\Euch(N^{\rvvee} )\neq 0, {}^{v}\!\Euch( \tuD(\Pa) [-1] \lotimes_{\Pa} N^{\rvvee} ) \neq 0$.
there exists an isomorphism

\[
{}^{v}\!\sfb_{N}: \RHom_{A^{\op}}(N \lotimes_{A} {}^{v}\!\YYM_{1}, \PPi_{1}) \to {}^{v}\!\YYM_{1} \lotimes_{A} N^{\rvvee}
\]

that completes the following commutative diagram

\begin{equation}\label{202103051621}
\begin{xymatrix}{
(N[1], \PPi_{1}) \ar[r] \ar[d]_{\cong}  &
(N\lotimes_{A} \PPi_{1}, \PPi_{1}) \ar[d]_{\cong} \ar[r] &
(N\lotimes_{A} {}^{v}\!\YYM_{1}, \PPi_{1}) \ar[d]_{\cong}^{{}^{v}\!\sfb_{N}} \ar[r] &
(N[1], \PPi_{1})[1] \ar[d]_{\cong} \\
A^{\vvee} \lotimes_{A} N^{\rvvee}
\ar[r]_{  -y({}^{v}\!\ttheta_{N^{\rvvee} })}
&  N^{\rvvee} \ar[r]_{{}^{v}\!\varrho_{N^{\rvvee}}}
& {}^{v}\!\YYM_{1} \lotimes_{A} N^{\rvvee} \ar[r]_{-y^{-1}{}^{v}\!\ppi_{1, N^{\rvvee}}}
& \PPi_{1} \lotimes_{A} N^{\rvvee}
\end{equation}

where we use the  abbreviation  $(-,+) =\RHom_{\Pa^{\op}}(-, +)$,
the top row is the exact triangle $\RHom( {}_{N}{}^{v}\!\sfAR, \PPi_{1})$
and we set

\[
y := \frac{{}^{v}\!\Euch(\tuD(A)[-1] \lotimes_{A} N^{\rvvee})}{{}^{v}\!\Euch(N^{\rvvee})}.
\]

First we claim that we have the following commutative diagram

\begin{equation}\label{202105231814}
\begin{xymatrix}@C=70pt{
(N[1])^{\rvvee} \ar[d]_{\cong}\ar[r]^{(-{}_{N}\!\! {}^{v}\! \ttheta [1])^{\rvvee}}
&  (N \lotimes \PPi_{1})^{\rvvee}\ar[d]_{\cong}\\
\tuD(A)[-1] \lotimes_{A}  A^{\vvee} \lotimes_{A} N^{\rvvee}
\ar[r]^{  -y({}_{\tuD(A)[-1]}\!\!{}^{v}\!\ttheta_{N^{\rvvee} })}
& \tuD(A) [-1]  \lotimes_{A} N^{\rvvee}
\end{equation}

Indeed we obtain this diagram in the following way:

\[
\begin{xymatrix}@C=70pt{
(N[1])^{\rvvee} \ar[d]_{\cong}\ar[r]^{(-{}_{N}\!\!{}^{v}\! \ttheta [1])^{\rvvee}}
&  (N \lotimes \PPi_{1})^{\rvvee}\ar[d]_{\cong}\\
(A[1])^{\vvee} \lotimes_{A} \tuD(A) \lotimes_{A} N^{\rvvee}
\ar[d]_{\cong}\ar[r]^{ (-{}^{v}\!\ttheta [1])^{\vvee}_{\tuD(A) \lotimes N^{\rvvee} }}
&  (A^{\vvee} [1])^{\vvee} \lotimes_{A} \tuD(A) \lotimes_{A} N^{\rvvee} \ar[d]_{\cong}\\
A^{\vvee} \lotimes_{A} \tuD(A)[-1] \lotimes_{A} N^{\rvvee}
\ar[d]_{\cong}\ar[r]^{ (-{}^{v}\!\ttheta)^{\vvee}_{\tuD(A)[-1]  \lotimes N^{\rvvee} }}
&  (A^{\vvee} )^{\vvee} \lotimes_{A} \tuD(A) [-1]  \lotimes_{A} N^{\rvvee} \ar[d]_{\cong}\\
A^{\vvee} \lotimes_{A} \tuD(A)[-1] \lotimes_{A} N^{\rvvee}
\ar[d]_{\cong}^{\gamma_{\tuD(A)[-1]} } \ar[r]^{ -{}^{v}\!\ttheta_{\tuD(A)[-1]  \lotimes N^{\rvvee} }}
& \tuD(A) [-1]  \lotimes_{A} N^{\rvvee} \ar@{=}[d]\\
\tuD(A)[-1] \lotimes_{A}  A^{\vvee} \lotimes_{A} N^{\rvvee}
\ar[r]^{  -y({}_{\tuD(A)[-1]}\!\!{}^{v}\!\ttheta_{N^{\rvvee} })}
& \tuD(A) [-1]  \lotimes_{A} N^{\rvvee}
\]

where for the commutativity of the first square
we use a  canonical isomorphism

\[
(L \lotimes_{A} X)^{\rvvee}
\cong X^{\rvvee} \lotimes_{A} L^{\rvvee} \cong X^{\vvee} \lotimes_{A} \tuD(A) \lotimes L^{\rvvee}.
\]

which is natural in $L \in \sfD(A^{\op})$ and $X \in \sfD(A^{\mre})$
(see Section <ref>).
For the third square we use (<ref>).
Finally the commutativity of the fourth  square follows from Proposition <ref>.

We apply $\PPi_{1}\lotimes_{\Pa} - $ to (<ref>) and use
canonical isomorphisms

\[
\RHom_{A^{\op}}(- ,\PPi_{1} ) \cong \PPi_{1} \lotimes_{A} (-)^{\rvvee}, \
\PPi_{1} \lotimes_{\Pa} \tuD(A) [-1] \cong \Pa,
\]

then we obtain a commutative square that appeared in the  diagram (<ref>).

We remark that $\sfb_{N}$ is not uniquely determined, but it is unique modulo $\rad$.
We point out that $\sfb_{N}$ has functoriality modulo $\rad$.

§.§ Bimodules version

In this subsection, we establish isomorphisms over $\Pa^{\mre}$ that involve ${}^{v}\!\YYM_{1}$.
The results of this sections are  used in Section <ref> and the reader can postpone them until then.

Combining Lemma <ref> and Theorem <ref> we deduce the following result.

Let $v \in \kk Q_{0}$ and $T$ be a two-sided tilting complex over $\Pa$.
there exists an isomorphism

\[
{}^{v}\!\sfc_{ T}:  {}^{v}\!\YYM_{1} \lotimes_{\Pa} T \to T  \lotimes_{\Pa} {}^{\underline{T}^{t}(v)}\!\YYM_{1}
\]

in $\sfD(\Pa^{\mre})$ that gives the following isomorphism of exact triangles:

\[
\begin{xymatrix}@C=60pt{
\Pa^{\vvee} \lotimes_{\Pa} T \ar[r]^{{}^{v}\!\ttheta_{T} }  \ar[d]_{\gamma_{T}}
T   \ar[r]^{{}^{v}\!\rrho_{T}} \ar@{=}[d]  &
{}^{v}\!\YYM_{1} \lotimes_{A}T \ar[d]^{{}^{v}\!\sfc_{T} } \ar[r]^{{}^{v}\!\ppi_{1, T} } &
\PPi_{1} \lotimes_{\Pa} T  \ar[d]  \\
T \lotimes_{\Pa} \Pa^{\vvee} \ar[r]_{ {}_{T}  {}^{\underline{T}^{t}(v)} \ttheta } &
T  \ar[r]_{  {}_{T} ({}^{\underline{T}^{t}(v)} \rrho) }  &
T \lotimes_{\Pa}  {}^{\underline{T}^{t}(v)} \!\YYM_{1} \ar[r]_{ {}_{T} ({}^{\underline{T}^{t}(v)}\ppi_{1}) } &
T \lotimes_{\Pa} \PPi_{1}.   \\
\]

In the case where the weight $v \in \kk Q_{0}$ is an eigenvector of $\underline{T}^{t}$ we obtain the following corollary
by using the isomorphism (<ref>).

Let $v \in \kk Q_{0}$ and $T$ be a two-sided tilting complex over $\Pa$.
Assume that $v$ is an eigenvector of  $\underline{T}^{t}$ with the eigenvalue $\lambda$.
there exists an isomorphism

\[{}^{v}\!\sfc'_{T}: {}^{v}\!\YYM_{1} \lotimes_{\Pa} T \to T \lotimes_{\Pa} {}^{v}\!\YYM_{1}
\]
in $\sfD(\Pa^{\mre})$
that gives the following isomorphism of exact triangles:

\[
\begin{xymatrix}@C=60pt{
\Pa^{\vvee} \lotimes_{\Pa} T \ar[r]^{{}^{v}\!\ttheta_{T} }  \ar[d]_{\gamma_{T}}
T   \ar[r]^{{}^{v}\!\rrho_{T}} \ar@{=}[d]  &
{}^{v}\!\YYM_{1} \lotimes_{A}T \ar[d]^{{}^{v}\!\sfc'_{T} } \ar[r]^{{}^{v}\!\ppi_{1, T} } &
\PPi_{1} \lotimes_{\Pa} T  \ar[d]   \\
T \lotimes_{\Pa} \Pa^{\vvee} \ar[r]_{  \lambda {}_{T}  {}^{v}\!\ttheta } &
T  \ar[r]_{  {}_{T} {}^{v}\!\rrho }  &
T \lotimes_{\Pa}  {}^{v}\!\YYM_{1} \ar[r]_{ \lambda^{-1} {}_{T}{}^{v}\!\ppi_{1}} &
T \lotimes_{\Pa} \PPi_{1}   \\
\]

We collect the case $T = \PPi_{1}$ of Corollary <ref>, since it plays a key role in Section <ref>.

Let $\Phi $ be the Coxeter matrix of $\kk Q$ for left modules.
For simplicity we set $\Psi := \Phi^{-t}= \underline{\PPi_{1}}^{t}$.

Assume that $v \in \kk Q_{0}$ is an eigenvector of  $\Psi$ with the eigenvalue $\lambda$.
there exists an isomorphism

\[{}^{v}\!\sfc'_{\PPi_{1}}: {}^{v}\!\YYM_{1} \lotimes_{\Pa} \PPi_{1} \to \PPi_{1} \lotimes_{\Pa} {}^{v}\!\YYM_{1}
\]
in $\sfD(\Pa^{\mre})$
that gives the following isomorphism of exact triangles:

\begin{equation}\label{202306211412}
\begin{xymatrix}@C=60pt{
\Pa^{\vvee} \lotimes_{\Pa} \PPi_{1} \ar[r]^{{}^{v}\!\ttheta_{\PPi_{1}} }  \ar[d]_{\gamma_{\PPi_{1}}}
\PPi_{1}   \ar[r]^{{}^{v}\!\rrho_{\PPi_{1}}} \ar@{=}[d]  &
{}^{v}\!\YYM_{1} \lotimes_{A}\PPi_{1} \ar[d]^{{}^{v}\!\sfc'_{\PPi_{1}} } \ar[r]^{{}^{v}\!\ppi_{1, \PPi_{1}} } &
\PPi_{1} \lotimes_{\Pa} \PPi_{1}  \ar@{=}[d]   \\
\PPi_{1} \lotimes_{\Pa} \Pa^{\vvee} \ar[r]_{  \lambda {}_{\PPi_{1}}  {}^{v}\!\ttheta } &
\PPi_{1}  \ar[r]_{  {}_{\PPi_{1}} {}^{v}\!\rrho }  &
\PPi_{1} \lotimes_{\Pa}  {}^{v}\!\YYM_{1} \ar[r]_{ \lambda^{-1} {}_{\PPi_{1}}{}^{v}\!\ppi_{1}} &
\PPi_{1} \lotimes_{\Pa} \PPi_{1}   \\
\end{equation}

It only remains to prove that the right most vertical arrow of the commutative diagram (<ref>) is the identity morphism.
This is proved as the case $n =2$ of Lemma <ref>(1).

§.§.§ The right duality

Recall that
we set $\Psi := \Phi^{-t}= \underline{\nu_{1}}^{-t}$,
that  $\nu_{1} =\tuD(A)[-1]\lotimes_{\Pa} -$.

There exists an isomorphism

\[{}^{v}\!\sfd: ({}^{v}\!\YYM_{1})^{\rvvee} \to \tuD(A)[-1]\lotimes_{\Pa} ({}^{\Psi^{-1}(v)} \!\YYM_{1}) \]

in $\sfD(\Pa^{\mre})$
that completes the  following commutative diagram

\begin{equation}\label{202105232003}
\begin{xymatrix}@C=40pt{
(A[1])^{\rvvee} \ar[d]_{\cong}\ar[r]^{(-{}^{v}\!\ttheta [1])^{\rvvee}} &
\PPi_{1}^{\rvvee}\ar[d]_{\cong}  \ar[r]^{ ({}^{v}\!\ppi_{1})^{\rvvee} } &
({}^{v}\!\YYM_{1})^{\rvvee} \ar[d]^{{}^{v}\!\sfd} \ar[r]^{ ({}^{v}\!\rrho)^{\rvvee}} &
A^{\rvvee} \ar[d]^{\cong} \\
\tuD(A)[-1] \lotimes_{A}  A^{\vvee}
\ar[r]_-{  -{}_{\tuD(A)[-1] }{}^{\Psi^{-1}(v)}\ttheta }
& \tuD(A) [-1]   \ar[r]_-{{}_{\tuD(A)[-1]} {}^{\Psi^{-1}(v)} \rrho} &
\tuD(A)[-1] \lotimes_{\Pa} ({}^{\Psi^{-1}(v)} \!\YYM_{1} )  \ar[r]_-{- {}_{\tuD(A)[-1] } {}^{\Psi^{-1}(v)}\ppi_{1} }
\tuD(A) [-1] \lotimes_{\Pa} \PPi_{1}.
\end{equation}

We  can verify commutativity of  the left most  square of (<ref>) in the following way

\[
\begin{xymatrix}@C=50pt{
(A[1])^{\rvvee} \ar[d]_{\cong}\ar[r]^{(-{}^{v}\!\ttheta [1])^{\rvvee}}
&  \PPi_{1}^{\rvvee}\ar[d]_{\cong}\\
(A[1])^{\vvee} \lotimes_{A} \tuD(A)
\ar[d]_{\cong}\ar[r]^{ (-{}^{v}\!\ttheta [1])^{\vvee}_{\tuD(A) }}
&  (A^{\vvee} [1])^{\vvee} \lotimes_{A} \tuD(A) \ar[d]_{\cong}\\
A^{\vvee} \lotimes_{A} \tuD(A)[-1]
\ar[d]_{\cong}\ar[r]^{ (-{}^{v}\!\ttheta)^{\vvee}_{\tuD(A)[-1]   }}
&  (A^{\vvee} )^{\vvee} \lotimes_{A} \tuD(A) [-1]   \ar[d]_{\cong}\\
A^{\vvee} \lotimes_{A} \tuD(A)[-1]
\ar[d]_{\cong}^{\gamma_{\tuD(A)[-1]} } \ar[r]^{ -{}^{v}\!\ttheta_{\tuD(A)[-1]   }}
& \tuD(A) [-1] \ar@{=}[d]\\
\tuD(A)[-1] \lotimes_{A}  A^{\vvee}
\ar[r]^{  - {}_{\tuD(A)[-1] }{}^{\Psi^{-1}(v)}\ttheta }
& \tuD(A) [-1]
\]

where for the commutativity of the first square  we use  a canonical isomorphism

\[
\cong X^{\vvee} \lotimes_{A} \tuD(A)
\]

which is natural  for $X \in \Dbmod{\Pa^{\mre}}$  (see Section <ref>).
For the third square we use (<ref>).
Finally the commutativity of the fourth  square follows from Theorem <ref>.

Since $\RHom_{\Pa^{\op}}(-, \PPi_{1}) \cong \PPi_{1} \lotimes_{\Pa} (-)^{\rvvee}$,
applying $\PPi_{1} \lotimes_{\Pa} -$ to the diagram of Lemma <ref>
we deduce the following lemma.

There exists an isomorphism

\[
{}^{v}\!\sfe: \RHom_{\Pa^{\op}} ( {}^{v}\!\YYM_{1}, \PPi_{1} ) \to {}^{\Psi^{-1}(v)} \!\YYM_{1}
\]

in $\sfD(\Pa^{\mre})$
that completes the  following commutative diagram

\[
\begin{xymatrix}@C=60pt{
(A[1], \PPi_{1}) \ar[r]^{(- {}^{v}\!\ttheta[1], \PPi_{1} ) }\ar[d]_{\cong}  &
(\PPi_{1}, \PPi_{1}) \ar[d]_{\cong} \ar[r]^{({}^{v}\!\ppi_{1}, \PPi_{1} ) } &
({}^{v}\!\YYM_{1}, \PPi_{1}) \ar[d]_{\cong}^{{}^{v}\!\sfe} \ar[r]^{ ({}^{v}\!\rrho, \PPi_{1} ) } &
(A, \PPi_{1}) \ar[d]_{\cong} \\
\ar[r]_{  -{}^{\Psi^{-1}(v)}\!\ttheta}
& A  \ar[r]_{{}^{\Psi^{-1}(v)}\!\rrho }
& {}^{\Psi^{-1}(v)} \!\YYM_{1} \ar[r]_{ -{}^{\Psi^{-1}(v)}\!\ppi_{1} }
& \PPi_{1} .
\]

where we use the  abbreviation  $(-,+) =\RHom_{\Pa^{\op}}(-, +)$.

Using (<ref>), we deduce the following corollary.
We remark that  for the later quotation, in the statement  we use    $\Psi$.

Assume that $v$ is an eigenvector of $\Psi$ with the eigenvalue $\lambda$.
Then, there exists an isomorphism

\[
{}^{v}\!\sfe': \RHom_{\Pa^{\op}} ( {}^{v}\!\YYM_{1}, \PPi_{1} ) \to {}^{v}\!\YYM_{1}
\]

in $\sfD(\Pa^{\mre})$
that completes the  following commutative diagram

\[
\begin{xymatrix}@C=60pt{
(A[1], \PPi_{1}) \ar[r]^{(- {}^{v}\!\ttheta[1], \PPi_{1} ) }\ar[d]_{\cong}  &
(\PPi_{1}, \PPi_{1}) \ar[d]_{\cong} \ar[r]^{({}^{v}\!\ppi_{1}, \PPi_{1} ) } &
({}^{v}\!\YYM_{1}, \PPi_{1}) \ar[d]_{\cong}^{{}^{v}\!\sfe'} \ar[r]^{ ({}^{v}\!\rrho, \PPi_{1} ) } &
(A, \PPi_{1}) \ar[d]_{\cong} \\
\ar[r]_{  -\lambda^{-1} {}^{v}\!\ttheta}
& A  \ar[r]_{{}^{v}\!\rrho }
& {}^{v}\!\YYM_{1} \ar[r]_{ -\lambda {}^{v}\!\ppi_{1} }
& \PPi_{1} .
\]

where we use the  abbreviation  $(-,+) =\RHom_{\Pa^{\op}}(-, +)$.

§ THE DERIVED QUIVER HEISENBERG ALGEBRAS

In this section <ref>,
we introduce the derived quiver Heisenberg algebra ${}^{v}\!\YYM(Q)$ of a quiver $Q$ and establish its basic properties.
It may be worth mentioning that  all the results of this section hold for a finite quiver $Q$ which is not necessarily acyclic.

§.§ The derived preprojective algebras

Before introducing the derived quiver Heisenberg algebra,
we recall the construction of the derived preprojective algebra $\PPi(Q)$ of $Q$.

§.§.§ The preprojective algebra of a quiver

Let $Q$ be a finite quiver and $\overline{Q}$ its double quiver.
We may  identify the arrow module $\kk \overline{Q}_{1}$ with $V \oplus V^{*}$
the path algebra $\kk \overline{Q}$ of $\overline{Q}$ with the tensor algebra  $\sfT_{\Pa_{0}} (V \oplus V^{*})$,
which is isomorphic to
$\sfT_{\Pa}(\Pa V^{*} \Pa)$ (see e.g, <cit.>).

\begin{equation}\label{202006121845}
\kk \overline{Q} = \sfT_{\Pa_{0}}(V \oplus V^{*} ) \cong \sfT_{\Pa}(\Pa V^{*} \Pa)
= \Pa \oplus \Pa V^{*} \Pa \oplus \Pa V^{*} \Pa V^{*} \Pa \oplus \cdots.
\end{equation}

Under this isomorphism, the mesh relation $\rho_{i}$ (<ref>) may be identified with the element below of $\Pa V^{*} \Pa$

\begin{equation}\label{201911262}
\rho_{i} = \sum_{\alpha: t(\alpha) = i} \alpha \otimes \alpha^{*} \otimes e_{i}
- \sum_{\alpha: h(\alpha) =i } e_{i} \otimes \alpha^{*} \otimes \alpha.
\end{equation}

Therefore,  we obtain the following isomorphism by which we identify these two algebras in the sequel.

\[
\Pi(Q) \cong  \sfT_{A} (AV^{*} A)/(\rho).
\]

§.§.§ The derived preprojective algebras

The derived preprojective algebra $\PPi=\PPi(Q)$ of $Q$ is defined to be
the tensor dg-algebra $\sfT_{\Pa}( \PPa^{\vee}[1])$ of $ \PPa^{\vee}[1]$ over $\Pa$,
that is
a dg-algebra whose underlying algebra is
the tensor algebra $\sfT_{\Pa}( \PPa^{\vee}[1])$ of $ \PPa^{\vee}[1]$ over $\Pa$
the differential is that of induced from the differential of the complex $ \PPa^{\vee}[1]$.

For $i \in Q_{0}$, we denote by $s_{i}$ the element of the generator of $\Pa \Pa[1] = \Pa ( \Pa_{0}[1]) \Pa$
corresponding to $e_{i} \in \Pa_{0}$.
We set $S := \Pa_{0}[1] = \bigoplus_{ i \in Q_{0}} \kk s_{i}$ and $s := \sum_{ i \in Q_{0}} s_{i}$.

We give  an explicit description of $\PPi$.
The underlying algebra is
the free algebra over $\Pa_{0}$ generated by $\alpha, \alpha^{*}$ and $s_{i}$.
In other words, it is the tensor algebra
$\sfT_{\Pa_{0}}(V \oplus V^{*} \oplus S) \cong \sfT_{\Pa}(\Pa V^{*} \Pa \oplus \Pa S \Pa)$.
The differential 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
$d(xy) = d(x) y + ( -1)^{|x|}x d(y)$.
Observe that the canonical morphism $\PPi \to \Pi$ induces an isomorphism

\[
\tuH^{0}(\PPi) \xrightarrow{\cong } \Pi.
\]

We equip  $\PPi$ with an extra  grading, which we call the $*$-grading,
in the following way:

\[
\deg^{*} e_{i} : =0, \ \deg^{*} s_{i} := 1 \ (i \in Q_{0}), \
\deg^{*} \alpha := 0, \  \deg^{*} \alpha^{*} := 1 \  ( \alpha \in Q_{1}).
\]

We give the table of the cohomological degrees and the $*$-degrees of the generators:

$e_{i}$     $\alpha$     $\alpha^{*}$     $s_{i}$
$\textup{ch}\deg $    $0$     $0$      $0$     $-1$
$\deg^{*}$     $0$     $0$     $1$     $1$

\[
\begin{xymatrix}{
i \ar@(ld, lu)^{s_{i}}  \ar@/^10pt/[rr]^{\alpha} &&
j  \ar@(rd,ru)_{s_{j}} \ar@/^10pt/[ll]^{\alpha^{*}}
\]

It is clear that the differential $d$ preserves the $*$-grading and
$\PPi$ is a $*$-graded dg-algebra.
It can be checked that the $*$-degree on $\PPi$ coincides with the tensor degree on $\PPi = \sfT_{\Pa}( \PPa^{\vee}[1])$.
if we denote  the $*$-degree $n$ part of $\PPi$ by $\PPi_{n}$,
then $\PPi_{n} =   \PPa^{\vee}[1] \otimes_{\Pa} \cdots \otimes_{\Pa}  \PPa^{\vee}[1]$ ($n$-times).
Thus in particular

\begin{equation}\label{202006122059}
\PPi_{1} = \PPa^{\vee}[1] \cong  \left( \Pa V^{*} \Pa \oplus \Pa \Pa [1], \begin{pmatrix} 0 & -\hat{\rho}\uparrow  \\ 0 & 0 \end{pmatrix}\right)
\end{equation}

where for the second isomorphism we use Lemma <ref>.

§.§ The derived quiver Heisenberg algebras

§.§.§ The derived quiver Heisenberg algebras

We define the derived quiver Heisenberg algebra.

Let $Q$ be a finite quiver and $v \in \kk^{\times} Q_{0}$.
We set

\[
{}^{v}\!\varrho := \sum_{i \in Q_{0}} v_{i}^{-1}\rho_{i},
{}^{v}\!\eta_{a} := [a, {}^{v}\!\varrho]
\]
for $a \in \overline{Q}$.

We define the derived quiver Heisenberg algebra $ {}^{v}\!\YYM(Q)$ in the following way.

We set $A_{0} := \kk Q_{0}, V := \kk Q_{1},  V^{*} := \tuD(V)$ and $A := \kk Q$.
Let $V^{\circ} := V^{*}[1],$  $V^{\circledast }:= V[1]$ and $T := \kk Q_{0}[2]$.
The underlying cohomological graded algebra of ${}^{v}\!\YYM(Q)$ is defined to be

\begin{equation}\label{20191127}
\sfT_{\Pa_{0}}(V \oplus V^{*} \oplus V^{\circ} \oplus V^{\circledast} \oplus T)
\cong \sfT_{\Pa}(\Pa V^{*} \Pa \oplus \Pa V^{\circ} \Pa \oplus \Pa V^{\circledast} \Pa \oplus
\Pa T \Pa ).
\end{equation}

The differential $d$ is  defined in the following way.
We denote by $\alpha^{\circ}, \alpha^{\circledast}$ the elements of $V^{\circ}, V^{\circledast}$ corresponding to $\alpha \in Q_{1}$.
We denote by $t_{i}$ the element of $T$ corresponding to $i \in Q_{0}$.
We set $t := \sum_{ i \in Q_{0}} t_{i}$.
Then the underlying cohomological graded algebra (<ref>)
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{equation}\label{the differentials of derived QHA}
\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 \in Q_{1} } e_{i}[\alpha, \alpha^{\circ}]e_{i} + \sum_{\alpha \in Q_{1}}  e_{i}[\alpha^{*}, \alpha^{\circledast}] e_{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}
\end{equation}

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

From now until the end of this section, we fix an element $v=(v_{i}) \in \kk^{\times}  Q_{0}$
and use abbreviation such as $\YYM = {}^{v}\!\YYM(Q), \varrho = {}^{v}\!\varrho$ and $\eta_{a} = {}^{v}\!\eta_{a}$.
Moreover in the sequel, similarly we omit $v$ and use similar abbreviations.

We equip  $\YYM$ with the  $*$-grading  in the following way.

\[
\begin{split}
\deg^{*} V := 0, \deg^{*} V^{*} = 1, \deg^{*} V^{\circledast} := 1,  \deg^{*} V^{\circ} := 2,  \deg^{*} T:= 2.
\end{split}
\]

We give the table of the cohomological degrees and the $*$-degrees of the generators:

$e_{i}$     $\alpha$     $\alpha^{*}$      $\alpha^{\circledast}$    $\alpha^{\circ} $     $t_{i}$
$\textup{ch}\deg $    $0$     $0$      $0$     $-1$      $-1$     $-2$
$\deg^{*}$     $0$     $0$     $1$     $1$     $2$      $2$

\[
\begin{xymatrix}{
i \ar@(ld, lu)^{t_{i}} \ar@/_20pt/[rr]_{\alpha^{\circledast}} \ar@/^10pt/[rr]_{\alpha} && j  \ar@(rd,ru)_{t_{j}} \ar@/^10pt/[ll]_{\alpha^{*}}
\ar@/_20pt/[ll]_{\alpha^{\circ}}
\]

We denote by $\YYM_{n}$ the component of $\YYM$ having  $*$-degree $n$.
For example, $\YYM_{0} = \Pa = \kk Q$.
It is straightforward to check  that the differential $d$ preserves the $*$-degree and $\YYM$ is a $*$-graded dg-algebra.
Therefore $\YYM_{n}$ has a canonical structure of complex of bimodules over $\Pa$.

The following observation plays an important  role.

The $*$-degree $1$ part $\YYM_{1}$ and the differential $d_{\YYM_{1}}$ are   of the form

\[
\YYM_{1} =\left( \Pa V^{*} \Pa \oplus  \Pa V^{\circledast}\Pa,  \begin{pmatrix} 0 & \hat{\eta}' \\ 0 & 0 \end{pmatrix} \right)
\]

where $\hat{\eta}'$ is the $\Pa^{\mre}$-homomorphism of cohomological degree $1$
such that $\hat{\eta}'(\alpha^{\circledast}) = \eta_{\alpha}$.

We denote by $\sfC(\YYM \Gr), \sfK(\YYM \Gr) $ and $\sfD(\YYM \Gr)$
the category of $*$-graded dg-$\YYM$-modules, its homotopy category and its derived category.
We write $(n)$ to denote $*$-degree shift by $n \in \ZZ$.

The following lemma is clear from the definition.

The canonical map $\YYM \to \YM $
gives an isomorphism $\tuH^{0}(\YYM(Q)) \cong \YM(Q)$
of algebras.

§.§.§ The derived quiver Heisenberg algebras as Ginzburg dg-algebras

In this section <ref>, we assume that $\chara \kk \neq 2$.

We set

\[
W: = -\frac{1}{2} \varrho \rho = -\frac{1}{2} \sum_{i \in Q_{0}} v_{i}^{-1} \rho_{i}^{2}.
\]

By a straightforward calculation (or using  cyclic Leibniz rule <cit.>), we can check  that

\[
\partial_{\alpha}( W) =  -\eta_{\alpha^{*}}, \ \ \ \partial_{\alpha^{*}} (W)=  \eta_{\alpha}.
\]

the quiver Heisenberg algebra $\YM(Q)$ is
the  Jacobi algebra of the double quiver $\overline{Q}$ with the potential $W$.

\[
\YM(Q) = \cP\left (\overline{Q}, W \right).\]

In <cit.>,
another quiver with potential $(\hat{Q}, W')$ such that $\cP(\hat{Q}, W') = \YM(Q)$ is given.

In subsequent work we prove that the derived quiver Heisenberg algebra and the Ginzburg dg-algebras $\cG(\hat{Q}, W')$ of this quiver with potential, are quasi-isomorphic to each other.
In this sense, a point of this paper is that the derived quiver Heisenberg algebra has a smaller number of generators than that of $\cG(\hat{Q}, W')$.

It is straight forward to check that the derived quiver Heisenberg algebra $\YYM(Q)$ is isomorphic to
the Ginzburg dg-algebra $\cG\left(\overline{Q}, W \right)$.

\[
\YYM(Q)  = \cG\left(\overline{Q}, W \right).
\]

The point here is that  although the potential $W =- \frac{1}{2} \varrho\rho$ contains the fraction $\frac{1}{2}$,
the differentials of $\cG\left(\overline{Q}, W \right)$ do not.
Therefore,  the definition of the differentials even works for the case $\chara \kk =2$.

By Ginzburg, Keller and Van den Bergh [23, 35],
the Ginzburg dg-algebras for quivers with potentials  are $3$-Calabi-Yau.
Hence, as a special case, we have

Assume that $\chara \kk \neq 2$.
Then the derived quiver Heisenberg algebra $\YYM$ is $3$-Calabi-Yau.

Later in Theorem <ref>,
we prove that $\YYM$ is $3$-Calabi-Yau even in the case $\chara \kk = 2$.

§.§.§ The  morphism $\tilde{\pi}$

We introduce the elements  $\vars_{i} := v_{i}^{-1} s_{i}$  and
$\vars := \sum_{i \in Q_{0}} \vars_{i} =\sum_{i \in Q_{0}} v_{i}^{-1} s_{i}$ of $\PPi$.
Note that $d(\vars) = -\varrho$.
We also introduce the elements  $\vart_{i} := v_{i}^{-1} t_{i}$  and $\vart := \sum_{i \in Q_{0}}\vart_{i} =\sum_{i \in Q_{0}} v_{i}^{-1} t_{i}$ of $\YYM$.

We define a morphism $\tilde{\pi}: \YYM \to \PPi$ of algebras
over $\Pa_{0}$.
On the generators, $\tilde{\pi}$ is defined by the formula

\[
\begin{split}
&\tilde{\pi}(\alpha) := \alpha, \tilde{\pi}(\alpha^{*}) := \alpha^{*}, \\
\tilde{\pi}(\alpha^{\circledast}) := -[\alpha, \vars] =- v_{h(\alpha)}^{-1} \alpha s_{h(\alpha)} + v_{t(\alpha)}^{-1}s_{t(\alpha)} \alpha,\\
&\tilde{\pi}(\alpha^{\circ}) := [\alpha^{*},\vars] =  v_{t(\alpha)}^{-1} \alpha^{*} s_{t(\alpha)} - v_{h(\alpha)}^{-1} s_{h(\alpha)} \alpha^{*},\\
&\tilde{\pi}(\vart_{i}): = -\vars^{2}_{i} = - v_{i}^{-2}s_{i}^{2}.
\end{split}
\]

Since $\YYM$ is freely generated by these generators, the above formulas defines a morphism $\ppi: \YYM \to \PPi$ of algebras.

Observe that $\tilde{\pi}$ preserves cohomological degree and $*$-degree.
The  $*$-degree $0$ part $\tilde{\pi}_{0}: \YYM_{0} \to \PPi_{0}$ is just the identity map of $\Pa = \YYM_{0} = \PPi_{0}$.

The morphism $\ppi$ is compatible with the differentials and hence is
a morphism of $*$-graded  dg-algebras.

It is enough to check that the equation $\ppi d= d \ppi $ on the generators.

\[
\ppi d( \alpha) = 0 = d \ppi (\alpha), \ppi d( \alpha^{*}) = 0 = d \ppi (\alpha^{*}).
\]

\[
\begin{split}
&d \ppi(\alpha^{\circledast}) =-  d([\alpha, \vars]) =- [d(\alpha), \vars] - [\alpha, d(\vars)] =
[\alpha, \varrho],  \\
& \ppi d(\alpha^{\circledast}) = \ppi(\eta_{\alpha}) = \ppi([\alpha, \varrho]) = [\alpha, \varrho].
\end{split}
\]

\[
\begin{split}
&d \ppi(\alpha^{\circ}) = d([\alpha^{*}, \vars]) = [d(\alpha^{*}), \vars] + [\alpha^{*}, d(\vars)] = -  [\alpha^{*}, \varrho],  \\
& \ppi d(\alpha^{\circ}) = - \ppi(\eta_{\alpha^{*}}) = - \ppi([\alpha^{*}, \varrho]) = - [\alpha^{*}, \varrho].
\end{split}
\]

\[
\begin{split}
& d \ppi(\vart_{i}) = - d(\vars_{i}^{2} ) = -d(\vars_{i})\vars_{i} + \vars_{i}d(\vars_{i}) = \varrho_{i} \vars_{i} - \vars_{i} \varrho_{i} = e_{i}[\rho, \vars]e_{i},\\
& \ppi d(\vart_{i}) =v^{-1}_{i} \ppi\left(  \sum_{\alpha \in Q_{1} } e_{i}[\alpha, \alpha^{\circ}]e_{i} + \sum_{\alpha \in Q_{1}}  e_{i}[\alpha^{*}, \alpha^{\circledast}] e_{i} \right)\\
& \ \ \ \ \
=  v^{-1}_{i} \sum_{\alpha \in Q_{1} } e_{i}[\alpha, [\alpha^{*},\vars] ]e_{i} -  \sum_{\alpha \in Q_{1}}  e_{i}[\alpha^{*}, [\alpha,\vars]] e_{i} \\
& \ \ \ \ \
= v^{-1}_{i}  \sum_{\alpha \in Q_{1} } e_{i}[[\alpha, \alpha^{*}] , \vars]e_{i}
= e_{i}[\varrho, \vars]e_{i}.
\end{split}
\]

It is clear that under the isomorphisms $\tuH^{0} (\YYM) \cong \YM, \tuH^{0}(\PPi) \cong \Pi$,
the $0$-th cohomology morphism $\tuH^{0}(\ppi): \tuH^{0}(\YYM) \to \tuH^{0}(\PPi) $ corresponds to
the canonical projection $\pi: \YM \to \Pi$ of (<ref>).

§.§.§ The exact triangle $\sfU$

In Theorem <ref> below,
we give an exact triangle $\sfU: \YYM(-1)  \xrightarrow{ \sfr_{ \varrho } } \YYM \xrightarrow{ \ppi } \PPi \to \YYM(-1)[1]$
the $0$-th cohomology group of which coincides with
the canonical exact sequence  $\YM(-1)  \xrightarrow{ \sfr_{ \varrho } } \YM \xrightarrow{ \ppi } \Pi \to 0 $.
To state the theorem, first we need to prove the following lemma.

We have $\ppi \sfr_{\varrho} = -d\sfr_{\vars} \ppi -\sfr_{\vars}\ppi d$.
In other words, the morphism $\ppi\sfr_{\varrho}: \YYM \to \PPi$ is homotopic to $0$ via the homotopy

\[
\begin{xymatrix}@C=10mm@R=4mm{
\YYM(-1)  \ar[r]^{\sfr_{\varrho}} \ar@/_2pc/[rr]_{0}   &\YYM \ar[r]^{\ppi} \ar@{=>}[d]^{-\sfr_{\vars}\ppi} & \PPi\\
\]

Let $x \in \YYM$ be a homogeneous element.
Then, using Lemma <ref>, we deduce the following equation

\[
\begin{split}
d (\sfr_{\vars}\ppi(x)) &= ( -1)^{|x|}d(\ppi(x)\vars) = ( -1)^{|x|} (d\ppi(x)) \vars +\ppi(x) d(\vars) \\
&=(-1)^{|x|} \ppi(dx)\vars -  \ppi(x) \varrho\\
&=-\sfr_{\vars} \ppi(dx) -  \sfr_{\varrho} \ppi(x).
\end{split}
\]

By this lemma,  we have the following cochain map of dg-$\YYM$-modules

\[
q_{\sfr_{\varrho}, \ppi, -\sfr_{\vars}\ppi } := ( \ppi, - \sfr_{\vars} \ppi \uparrow): \cone( \sfr_{\varrho}) \to \PPi
\]

where we use the notation in (<ref>).

The map $q_{\sfr_{\varrho}, \ppi, -\sfr_{\vars}\ppi }$ is a quasi-isomorphism.
Therefore we have
an exact triangle  in $\sfD(\YYM\Gr)$

\[
\sfU: \YYM( -1) \xrightarrow{ \sfr_{\varrho}} \YYM \xrightarrow{ \ppi } \PPi \to \YYM( -1)[1].
\]

A proof is given in Section <ref>.
In Section <ref>, we show that there exists an exact triangle
\[
\widehat{\sfU}: \YYM( -1) \xrightarrow{ \widehat{\sfr_{\varrho}} } \YYM \xrightarrow{ \ppi } \PPi \to \YYM( -1)[1].
\]

in $\sfD(\YYM^{\mre}\Gr)$
which is sent to $\sfU$  by the forgetful functor $\sfD(\YYM^{\mre}\Gr) \to \sfD(\YYM\Gr)$.

§.§.§ The homotopy $\sfH$

We define a derivation $\sfH$ of $\YYM$ over $\Pa_{0}$ of cohomological degree $-1$ to be

\[
\sfH(\alpha) := \alpha^{\circledast},
\sfH(\alpha^{*}) := - \alpha^{\circ},
\sfH(\alpha^{\circ}) := - [\alpha^{*}, \vart],
\sfH(\alpha^{\circledast}) := [ \alpha,\vart ],
\sfH(\vart_{i}) := 0
\]

for all $\alpha \in Q_{1}$ and $i \in Q_{0}$.
The values of $\sfH$ for general homogeneous elements are determined from the Leibniz rule
$\sfH(xy) = \sfH(x) y + ( -1)^{|x|} x \sfH(y)$.

We note  that the map $\sfH: \YYM \to \YYM$ increase  the $*$-degree by $1$.

\[
\sfH :\YYM_{n -1} \to \YYM_{n}.
\]

The weighted mesh relation $\varrho$ is not a central element of $\YYM$, but
in the next lemma we see  that it may be said to be  cohomologically central.

The morphism $\sfH$ is a homotopy from
the right multiplication map  $\sfr_{\varrho}$ to
the left multiplication map $\sfl_{\varrho}$.

In particular the weighted mesh relation $\varrho$ is central in the cohomology algebra $\tuH(\YYM)$.

We have to show

\[
d\sfH(x) +\sfH d(x) =-\sfb_{\varrho}(x) = [x, \varrho]
\]

for $x \in \YYM$.
In other words,
we have to prove the following equality  of morphisms $\YYM \to \YYM$

\begin{equation}\label{homotopy equation}
[d, \sfH]_{\Hom} = -\sfb_{\varrho}
\end{equation}

where $[-,+]_{\Hom}$  is the commutator inside the $\Hom$-complex  $\Hom_{\Pa_{0}}^{\bullet} (\YYM, \YYM)$.

Since the both sides of the equation (<ref>)
are derivations of degree $0$,
it is enough to check that
the equality holds on the generators $\alpha, \alpha^{*}, \alpha^{\circ}, \alpha^{\circledast}, t_{i}$.

The  equation  (<ref>) on $\alpha$
is checked as below.

\[
[d, \sfH]_{\Hom}(\alpha) = d\sfH(\alpha) = d \alpha^{\circledast} = \eta_{\alpha}=[\alpha, \rho].
\]

A similar calculation works for $\alpha^{*}$.

We claim $\sfH (\rho_{i}) + d(t_{i})= 0$.

\[
\begin{split}
\sfH(\rho_{i})
& = \sfH \sum_{\alpha \in Q_{1}}e_{i}[\alpha, \alpha^{*}]e_{i}
=  \sum_{\alpha \in Q_{1}}e_{i}[\sfH(\alpha), \alpha^{*}]e_{i} + e_{i}[\alpha, \sfH(\alpha^{*})]e_{i}
=  \sum_{\alpha \in Q_{1}}e_{i}[\alpha^{\circledast}, \alpha^{*}]e_{i} + e_{i}[\alpha, -\alpha^{\circ}] e_{i}\\
& =  \sum_{\alpha \in Q_{1}} - e_{i}[\alpha^{*}, \alpha^{\circledast}]e_{i} -  e_{i}[\alpha, \alpha^{\circ}]e_{i}
=- d( t_{i}).
\end{split}
\]

Using the equation  $\sfH(\varrho) + d(\vart) = 0$, which follows from the claim,
we check (<ref>) on $\alpha^{\circ}$

\[
\begin{split}
[d, \sfH]_{\Hom}(\alpha^{\circ})
& = - d[\alpha^{*}, \vart] -\sfH[\alpha^{*}, \varrho]
= - [d(\alpha^{*}), \vart] - [\alpha^{*}, d(\vart)] -[\sfH (\alpha^{*}), \varrho]- [\alpha^{*}, \sfH(\varrho)]
= [\alpha^{\circ}, \varrho].
\end{split}
\]

A similar calculation works for $\alpha^{\circledast}$.

Finally, we check the equation (<ref>) on $t_{i}$.

\[
\begin{split}
[d, \sfH]_{\Hom}(t_{i})
& = \sfH d (t_{i} ) = \sfH\left(  \sum_{\alpha \in Q_{1}}e_{i}[ \alpha, \alpha^{\circ}]e_{i} +  e_{i} [\alpha^{*}, \alpha^{\circledast}]e_{i}\right)
\\ &
= \sum_{\alpha \in Q_{1}}
e_{i} [\alpha^{\circledast}, \alpha^{\circ}] e_{i}- e_{i}[\alpha, [\alpha^{*}, \vart]]e_{i}
-e_{i}[ \alpha^{\circ}, \alpha^{\circledast}]e_{i} +e_{i} [\alpha^{*}, [\alpha, \vart]] e_{i}
\\&
=e_{i} \left(  \sum_{\alpha \in Q_{1}} [[\vart, \alpha], \alpha^{*}] +  [\alpha, [\vart, \alpha^{*}]] \right) e_{i}
\\&
= e_{i}\left(  \sum_{ \alpha \in Q_{1}}  [\vart, [\alpha, \alpha^{*}] ]\right) e_{i} =[t_{i}, \varrho].
\end{split}
\]

We have $\ppi \sfH = \sfb_{\vars} \ppi$.

By direct computation we can check that
the equation holds on the generators $\alpha, \alpha^{*}, \alpha^{\circ},
\alpha^{\circledast}, \vart_{i}$.

\[
\begin{split}
\ppi\sfH(\alpha) &= \ppi(\alpha^{\circledast}) =\sfb_{\vars}(\alpha) = \sfb_{\vars}\ppi(\alpha), \\
\ppi\sfH(\alpha^{*})& = -\ppi(\alpha^{\circ}) =\sfb_{\vars}(\alpha^{*}) = \sfb_{\vars}\ppi(\alpha^{*}), \\
\ppi\sfH(\alpha^{\circledast}) &= \ppi[\alpha, \vart] = -[ \alpha, \vars^{2} ] = [\vars^{2}, \alpha]  \\
& = \vars[\vars, \alpha] + [\vars, \alpha]\vars = [\vars, [\vars, \alpha] ] =\sfb_{\vars}\ppi(\alpha^{\circledast}) \\
\ppi\sfH(\alpha^{\circ}) & = -\ppi[\alpha^{*}, \vart] = [ \alpha^{*}, \vars^{2}] = - [\vars^{2}, \alpha^{*}] \\
&= -(\vars[\vars, \alpha^{*}] + [\vars, \alpha^{*}]\vars) = -[\vars, [\vars, \alpha^{*}] ] =\sfb_{\vars}\ppi(\alpha^{\circ}) \\
\ppi \sfH(\vart_{i}) & = 0 = \sfb_{\vars}\ppi(\vart_{i}).
\end{split}
\]

Observe that  the both sides of the equation are derivation along $\ppi$ of degree $-1$,
i.e., the morphisms $F = \ppi\sfH, \sfb_{\vars}\ppi$ satisfy
$F(xy) = F(x) \ppi(y) +(-1)^{|x|}\ppi(x) F(y)$.
It follows from  the above observation the equation holds for any $x \in \YYM$.

§.§ The exact triangle $\mathsf{AR}$

We recall from Section <ref>:

\[
\hat{v} := \sum_{ i \in Q_{0} } v_{i} \hat{e}_{i}, \ \ \ttheta := {}^{v}\!\ttheta = \sum_{i \in Q_{0}} v_{i} \tilde{e}_{i}.
\]

\begin{equation}\label{202008141807}
\ttheta: \small
\PPa^{\vee}  = \left( \Pa V^{*} \Pa[-1] \oplus \Pa \Pa,  \begin{pmatrix} 0 & \uparrow \hat{\rho} \\ 0 & 0\end{pmatrix} \right)
\xrightarrow{ \tiny \begin{pmatrix} 0 & \hat{v} \\ 0 & 0 \end{pmatrix}}
\left( \Pa \Pa \oplus \Pa V\Pa [1], \begin{pmatrix} 0 & \hat{\mu} \uparrow \\ 0 & 0\end{pmatrix} \right) =
\PPa.
\end{equation}

Since now we are assuming  $v_{i} \neq 0 $ for all $i \in Q_{0}$,
the morphism $\hat{v}: \Pa\Pa \to \Pa \Pa$ is an isomorphism.
We set $\hat{\eta} := \hat{\rho} \hat{v}^{-1} \hat{\mu}:\Pa V\Pa \to \Pa V^{*} \Pa$.
Then it is straightforward check that
for an arrow $\alpha \in Q_{1}$ we have

\[
\begin{split}
\hat{\eta}( 1 \otimes \alpha\otimes 1) &=
\sum_{\beta \in Q_{1}, h(\beta) = t(\alpha)} v_{h(\alpha)}^{-1}\alpha \beta \otimes \beta^{*} \otimes 1
- \sum_{\beta\in Q_{1},  t(\beta) = t(\alpha)} v_{h(\alpha)}^{-1} \alpha \otimes  \beta^{*}\otimes \beta \\
- \sum_{\beta\in Q_{1}, h(\beta) = h(\alpha)}  v_{t(\alpha)}^{-1} \beta \otimes \beta^{*} \otimes \alpha
+\sum_{\beta\in Q_{1}, t(\beta) = h(\alpha)} v_{t(\alpha)}^{-1} 1 \otimes \beta^{*}\otimes \beta \alpha,
\end{split}\]

where in each term $\beta^{*}$ in the middle is belonging to $V^{*}$.
Therefore, if we regard $\Pa V^{*} \Pa \subset \kk \overline{Q}$ via the isomorphism (<ref>),
then $\hat{\eta}( 1 \otimes \alpha \otimes 1) = \eta_{\alpha}$.
We note that here  the symbol $\otimes$  denotes  tensor products over $\Pa_{0}$ which are suppressed in the sequel.

Comparing this observation  with
Observation <ref>,
we come to the following lemma.

The $*$-degree $1$ part $\YYM_{1}$ is isomorphic to  the
cone $\cone(\hat{\eta})$.
Moreover, the $*$-degree $1$ part $\tilde{\pi}_{1}: \YYM_{1} \to \PPi_{1}$
is identified with

\[
\tilde{\pi}_{1}: \small
\YYM_{1} \cong \left( \Pa V^{*} \Pa \oplus \Pa V \Pa [1],
\begin{pmatrix} 0 & \hat{\eta}\uparrow \\ 0 & 0\end{pmatrix} \right)
\xrightarrow{ \tiny \begin{pmatrix} \id & 0 \\ 0 & - \hat{v}^{-1}\hat{\mu} \end{pmatrix}}
\left( \Pa V^{*} \Pa \oplus \Pa\Pa [1], \begin{pmatrix} 0 & -\hat{\rho}\uparrow \\ 0 & 0\end{pmatrix} \right) =
\PPi_{1}.
\]

We define a morphism $\rrho:  \PPa \to \YYM_{1}$ to be

\[
\rrho:  \PPa = \left( \Pa \Pa \oplus \Pa V \Pa [1], \begin{pmatrix} 0 & \hat{\mu}\uparrow \\ 0 & 0\end{pmatrix} \right)
\xrightarrow{ \tiny \begin{pmatrix} \rrho\hat{v}^{-1}  & 0 \\ 0 & \id \end{pmatrix}}
\left( \Pa V^{*} \Pa \oplus \Pa V\Pa [1], \begin{pmatrix} 0 & \hat{\eta}\uparrow \\ 0 & 0\end{pmatrix} \right) =
\YYM_{1}.
\]

The next lemma says that these morphisms $\rrho, \ppi_{1}$ are representatives of the morphisms in  the exact triangle ${}^{v}\!\sfAR$
given in Section <ref> denoted by the same symbols.

The following diagram gives an exact triangle in the homotopy category $\sfK(\Pa^{\mre})$

\[
\mathsf{AR}:  \PPa \xrightarrow{ \ \ \rrho \ \ } \YYM_{1} \xrightarrow{ \ \ \tilde{\pi}_{1} \ \ } \PPi_{1} \xrightarrow{ -\ttheta[1] }  \PPa[1].
\]

if we define
a morphism
$\sfL:  \PPa \to \PPi_{1}$ of degree $-1$ in $\sfC_{\DG}(\Pa^{\mre})$ as follows

\[
\sfL(e_{i}) := -\vars_{i}, \ \sfL(\downarrow \alpha) : = 0,
\]

for $i \in Q_{0}, \alpha \in Q_{1}$,
then the following statements hold

* It is a homotopy from $\ppi_{1} \rrho$ to $0$

* It induces the homotopy equivalences
$q_{\rrho, \ppi_{1}, \sfL} =(\ppi_{1}, \sfL\hspace{-3pt}\uparrow): \cone(\rrho) \to \PPi_{1}$ and
$j_{\rrho, \ppi_{1}, \sfL} =( \uparrow\hspace{-2pt}\sfL,  -\rrho)^{t}:  \PPa \to \cone(\ppi_{1})[-1]$.

* We have an equality $\sfL = \sfh_{\ppi_{1}} j_{\rrho, \ppi_{1}, \sfL }$ of morphisms $\PPa \to \PPi_{1}$ of degree $-1$
where $\sfh_{\ppi_{1}}$ is the canonical homotopy of (<ref>).

Applying  Lemma <ref> for $f= \hat{\mu}, g = \hat{\rho}, h= \hat{\eta}$ and $H = 0$, we obtain an exact triangle
$\cone(\hat{\mu}) \xrightarrow{\Psi} \cone( \hat{\eta} ) \xrightarrow{\Upsilon} \cone( \hat{\rho}) \xrightarrow{ -\Phi[1]} $.
Modifying this sequence by the isomorphism ${\tiny \begin{pmatrix} \id & 0 \\ 0 & -\hat{v}^{-1} \end{pmatrix}}: \cone ( \hat{\rho}\hat{v}^{-1})
\xrightarrow{\cong}  \cone(- \hat{\rho}) = \PPi_{1}$,
we obtain the desired exact triangle.

\begin{equation}\label{202007311609}
\begin{xymatrix}{
&&\Pa V\Pa  \ar@{=}[rr] \ar[d]_{\hat{\mu} }  &&  \Pa V\Pa \ar[d]^{\hat{\eta} }  && && \\
\cone(\hat{\rho}\hat{v}^{-1} )[-1] \ar[rr] \ar@{=}[d] &&
\Pa\Pa \ar@{=>}[urr]^{0} \ar[d] \ar[rr]^{\hat{\rho}\hat{v}^{-1} } &&
\Pa V^{*} \Pa \ar[d]  \ar[rr] && \cone(\hat{\rho} \hat{v}^{-1}) \ar@{=}[d]
\\
\cone(\hat{\rho}\hat{v}^{-1} )[-1]
\ar[d]_{\tiny \begin{pmatrix} \id & 0\\ 0 & -\hat{v}^{-1} \end{pmatrix} }^{\cong}
\ar[rr]_{\tiny \begin{pmatrix} 0 & - \id \\ 0 & 0 \end{pmatrix} }
&&\cone(\hat{\mu} ) \ar[rr]_{\tiny \begin{pmatrix} \hat{\rho}\hat{v}^{-1} & 0 \\ 0 & \id \end{pmatrix} }
\ar@{=}[d]
&& \cone(\hat{\eta})  \ar@{=}[d] \ar[rr]_{\tiny \begin{pmatrix} \id  & 0 \\ 0 &\hat{\mu}[1] \end{pmatrix} }
\cone(\hat{\rho}\hat{v}^{-1})
\ar[d]^{\tiny \begin{pmatrix} \id & 0\\ 0 & -\hat{v}^{-1} \end{pmatrix} }_{\cong}  \\
\PPi_{1}[-1]  \ar[rr]_{\ttheta} && \PPa \ar[rr]_{\rrho} && \YYM_{1} \ar[rr]_{\ppi_{1}} && \PPi_{1}&&
\end{equation}

In a similar way, we obtain the desired homotopy from Lemma <ref>.

§.§ Exact triangles $\sfV_{n}$ and the morphisms $\eeta^{*}, \zzeta$

§.§.§ The morphism $\zzeta$

We define a morphism $\zzeta: \YYM_{1} \otimes_{\Pa} \YYM \to \YYM$ to be the multiplication map

\[
\zzeta: \YYM_{1} \otimes_{\Pa} \YYM \to \YYM, \ \zzeta(x \otimes y) := xy.
\]

We denote the $*$-graded version  by the same symbol $\zzeta: \YYM_{1} \otimes_{\Pa} \YYM( -1) \to \YYM$.
We denote the $*$-degree $n$-component by $\zzeta_{n}: \YYM_{1} \otimes_{\Pa} \YYM_{n -1}  \to \YYM_{n }$.

§.§.§ The morphism $\eeta_{2}^{*}$

Next we look at the $*$-degree $2$ part $\YYM_{2}$.
For this we introduce several notations.

For $i \in Q_{0}$,
we define elements $\varrho^{\circ}_{i}, \varrho^{\circledast}_{i}$ of $\YYM_{2}$ to be

\[
\varrho^{\circ}_{i} := v_{i}^{-1} \sum_{\alpha \in Q_{0}} e_{i} [\alpha, \alpha^{\circ}]e_{i}, \ \
\varrho^{\circledast}_{i} := v_{i}^{-1} \sum_{\alpha \in Q_{0}} e_{i} [\alpha^{*}, \alpha^{\circledast}] e_{i}.
\]

Observe that  we can  regard $[\alpha^{*}, \alpha^{\circledast}] = \alpha^{*} \alpha^{\circledast} - \alpha^{\circledast}\alpha^{*}$
as an element of $\Pa V^{*}\Pa V^{\circledast} \Pa \oplus \Pa V^{\circledast} \Pa V^{*} \Pa$
which is the cohomological degree $-1$ part of $\YYM_{1} \otimes_{\Pa} \YYM_{1}$.

We set $\varrho^{\circ} := \sum_{i \in Q_{0}} \varrho^{\circ}_{i}$
and $\varrho^{\circledast} := \sum_{i \in Q_{0}} \varrho^{\circledast}_{i}$
so that $d(\vart) = \varrho^{\circ} + \varrho^{\circledast}$.

We define  morphisms

\[
\hat{\varrho}^{\circ}: \Pa T \Pa \to \Pa V^{\circ} \Pa, \
\hat{\varrho}^{\circledast}: \Pa T \Pa \to \YYM_{1} \otimes_{\Pa} \YYM_{1}, \
\hat{\eta}^{\circ}: \Pa V^{\circ} \Pa \to \YYM_{1} \otimes_{\Pa} \YYM_{1}\]

of cohomological degree $1$ in $\sfC_{\DG}(\Pa^{\mre})$
by  the formulas

\[
\hat{\varrho}^{\circ}(\vart_{i}) := \varrho^{\circ}_{i}, \
\hat{\varrho}^{\circledast}(\vart_{i}) := \rho^{\circledast}_{i}, \
\hat{\eta}^{\circ}(\alpha^{\circ}) := \eta_{\alpha^{*}}.
\]

Then we obtain the following description of $\YYM_{2}$

\[
\YYM_{2} = \left( ( \YYM_{1} \otimes_{\Pa} \YYM_{1})  \oplus \Pa V^{\circ} \Pa \oplus \Pa T \Pa,
{\small \begin{pmatrix} d_{\YYM_{1} \otimes \YYM_{1}} & - \hat{\eta}^{\circ} & \hat{\varrho}^{\circledast} \\ 0 & 0& \hat{\varrho}^{\circ} \\
0 & 0& 0 \end{pmatrix}}\right).
\]

Observe that
the subcomplex $\PPi_{1}^{\circ}[1] :=\left( \Pa V^{\circ} \Pa \oplus \Pa T \Pa,
\begin{pmatrix} 0 & \hat{\varrho}^{\circ} \\ 0 & 0 \end{pmatrix}\right)$
is isomorphic to  $\PPi_{1}[1]$.
We fix an identification as follows

\begin{equation}\label{202102041839}
\mathsf{idn}:  \PPi_{1}[1] \xrightarrow{ \cong} \PPi^{\circ}_{1}[1],  \alpha^{*} \mapsto - \alpha^{\circ}, \ \vars_{i} \mapsto - \vart_{i}.
\end{equation}

We define a morphism $\eeta^{*}_{2}: \PPi_{1} \to \YYM_{1} \otimes_{\Pa} \YYM_{1}$ in $\sfC(\Pa^{\mre})$
by the formulas

\[
\eeta^{*}_{2}(\alpha^{*}) := \eta_{\alpha^{*}}, \eeta^{*}_{2}(\vars) := - \varrho^{\circledast}.
\]

Then we may identify  $\YYM_{2}$ with  the cone of $\eeta^{*}_{2}$

\[
\YYM_{2} \cong \cone(\eeta^{*}_{2}) =
\left( (\YYM_{1} \otimes_{\Pa} \YYM_{1}) \oplus \PPi_{1}[1],
\begin{pmatrix} d_{\YYM_{1} \otimes_{\Pa} \YYM_{1}} & \eeta^{*}_{2}\uparrow  \\ 0 & d_{\PPi[1]} \end{pmatrix} \right).
\]

Observe that the $*$-degree $2$-component  $\zzeta_{2} : \YYM_{1} \otimes_{\Pa} \YYM_{1} \to \YYM_{2}$
is the cone morphism $i_{1}^{\eeta^{*}_{2}}$ of $\eeta^{*}_{2}$.
Therefore we obtain the  sequence below in $\sfC(\Pa^{\mre})$
that gives an exact triangle in $\sfK(\Pa^{\mre})$

\[
\begin{xymatrix}@C=10mm@R=4mm{
\sfV_{2} :&
\PPi_{1}  \ar[r]^{\eeta^{*}_{n}} \ar@/_2pc/[rr]_{0} &
\YYM_{1} \otimes_{\Pa} \YYM_{1} \ar[r]^{\zzeta_{2}}  \ar@{=>}[d]_{\sfg_{\eeta^{*}_{2}}}&
\YYM_{2}  &\\
\]

where $\sfg_{\eeta^{*}_{2}}= (0, \downarrow)^{t}$ as given in (<ref>).

§.§.§ The exact triangle $\sfV_{n}$

Let $ n \geq 3$.
We set  $\eeta^{*}_{n} := ({}_{\YYM_{1} }\zzeta_{ n-1}) (\eeta^{*}_{2,\YYM_{ n-2} })$ in $\sfC(\Pa^{\mre})$.

\[
\eeta^{*}_{n}:  \PPi_{1} \otimes_{A} \YYM_{ n-2} \xrightarrow{ \eeta^{*}_{2, \YYM_{ n-2} } }
\YYM_{1} \otimes_{A} \YYM_{1} \otimes_{A} \YYM_{n -1}
\xrightarrow{ {}_{\YYM_{1} }\zzeta_{ n-1}  }
\YYM_{1} \otimes_{A} \YYM_{n}.
\]

Observe that as cohomological graded modules, we have

\[
\begin{split}
\YYM_{n } &
= \left( \YYM_{1}  \otimes_{\Pa} \YYM_{n-1} \right)  \oplus
\left( \Pa V^{\circ} \Pa \otimes_{\Pa} \YYM_{ n-2} \right)  \oplus
\left( \Pa T \Pa \otimes_{\Pa} \YYM_{ n-2}\right)\\
& = \left( \YYM_{1}  \otimes_{\Pa} \YYM_{n-1} \right)  \oplus
\left( \PPi^{\circ}_{1} \otimes_{\Pa} \YYM_{ n-2}[1] \right)
\end{split}
\]

Using the identification (<ref>), we obtain

\[
\YYM_{n} =
\left( \left( \YYM_{1}  \otimes_{\Pa} \YYM_{n-1} \right) \oplus
\left(\PPi_{1}\otimes_{\Pa} \YYM_{ n-2}[1]\right) , \begin{pmatrix} d & \eeta_{n}^{*}\uparrow\\
0 & d
\end{pmatrix}\right).
\]

Thus we may identify $\YYM_{n}$ with the cone $\cone(\eeta^{*}_{n})$
and the cone morphism $i_{1}^{\eeta^{*}_{n}}$ with $\zzeta_{n}$.
We  obtain  a diagram in $\sfC_{\DG}(\Pa^{\mre})$ below
which becomes an exact triangle in $\sfK(\Pa^{\mre})$

\[
\begin{xymatrix}@C=10mm@R=4mm{
\sfV_{n} :&
\PPi_{1} \otimes_{\Pa} \YYM_{n -2} \ar[r]^{\eeta^{*}_{n}} \ar@/_2pc/[rr]_{0} &
\YYM_{1} \otimes_{\Pa} \YYM_{n -1} \ar[r]^{\zzeta_{n}}  \ar@{=>}[d]_{\sfg_{\eeta^{*}_{n}}}&
\YYM_{n} &\\
\]

where $\sfg_{\eeta^{*}_{n}}= (0, \downarrow)^{t}$ as given in (<ref>).

By convention, we set $\zzeta_{1} := \id_{\YYM_{1}}$ and $\zzeta_{n -1}^{*} := 0, \eeta^{*}_{n } := 0$ for $n \leq 1$.
Taking the direct sums of $\zzeta_{n}$ and $\eeta^{*}_{n}$,
we obtain the  diagram below in $\sfC_{\DG}(\Pa \otimes \YYM^{op}\Gr)$ below
which becomes an exact triangle in $\sfK(\Pa \otimes \YYM^{\op}\Gr)$

\begin{equation}\label{20191202}
\begin{xymatrix}@C=10mm@R=4mm{
\sfV :&
\PPi_{1} \otimes_{\Pa} \YYM(-2) \ar[r]^{\eeta^{*}} \ar@/_2pc/[rr]_{0} &
\YYM_{1} \otimes_{\Pa} \YYM(-1) \ar[r]^{\zzeta}  \ar@{=>}[d]_{\sfg_{\eeta^{*}}}&
\YYM_{\geq 1}. &\\

§.§.§ Koszul resolution

Let $\iota: \YYM_{\geq 1} \to \YYM$ be a canonical injection and $\epsilon: \YYM \to \Pa$ be a canonical projection.
Then, we have an exact sequence $0 \to \YYM_{\geq 1}\xrightarrow{ \iota} \YYM \xrightarrow{ \epsilon } \Pa \to 0$.
We define an object $P \in \sfC_{\DG}(\Pa\otimes \YYM^{\op} \Gr)$ to be

\[
P :=
\left( \YYM \oplus  ( \YYM_{1} \otimes_{\Pa} \YYM(-1)[1] )\oplus ( \PPi_{1} \otimes_{\Pa}\YYM(-2)[2]),
\begin{pmatrix} d & \iota \zzeta\uparrow &\iota \sfg \uparrow^{2} \\
0 & d &- \downarrow \eeta^{*}\uparrow^{2}\\
0& 0& d
\end{pmatrix}
\right).
\]

Combining the above exact sequence  and the diagram (<ref>), we see that the morphism
\epsilon_{P}:=(\epsilon, 0,0): P \to \Pa
is a quasi-isomorphism in $\sfC_{\DG}(\Pa \otimes \YYM^{\op} \Gr)$.

§.§.§ Remark on Koszul duality

The above observations show that
the derived quiver Heisenberg algebra $\YYM$ has the following description:

\[
\YYM = \left( \sfT_{\Pa}(\YYM_{1} \oplus \PPi_{1}[1]), d_{\sfT} + \delta\right)
\]

where $d_{\sfT}$ is the differential of the dg-tensor algebra $\sfT_{\Pa}(\YYM_{1} \oplus \PPi_{1}[1])$
$\delta$ is a morphism of cohomological degree $1$ induced from $\eeta^{*}_{2}$.
This description says that $\YYM$ is the Kosuzl dual of the graded dg-coring
$C = \Pa \oplus \YYM_{1}[1] \oplus \PPi_{1}[2]$ over $\Pa$
whose comultiplication is essentially $\eeta^{*}_{2}$.

In view of Koszul duality,
the quasi-isomorphism $\epsilon_{P} : P \to \Pa$ may be regarded as  the Koszul resolution of $\Pa$

\[
\begin{xymatrix}{
\PPi_{1} \otimes_{\Pa} \YYM( -2)  \ar[rr]^{\eeta^{*}} &&
\YYM_{1} \otimes_{\Pa} \YYM( -1) \ar[rr] \ar[dr]^{\zzeta} &&
\YYM \ar[rr]^{\epsilon}  && \Pa \ar[dlll]^{[1]} \\
&&& \YYM_{\geq 1}  \ar[ulll]^{[1]} \ar[ur]^{\iota}
\]

This point will be studied in a subsequent work and in particular give a construction of Calabi-Yau algebras which generalize Keller's Calabi-Yau completion [35].

§.§.§ An explicit formula of $\eeta^{*}$

We give  an explicit formula of $\eeta^{*}: \PPi \lotimes_{\Pa} \YYM \to \YYM_{1} \lotimes_{\Pa} \YYM$
which uses  the following  identifications of cohomological graded modules over $\Pa^{\mre}$

\begin{equation}\label{202105211917}
\PPi_{1} \otimes_{\Pa} \YYM \cong   \Pa V^{*}  \YYM \oplus \Pa S  \YYM, \ \
\YYM_{1} \otimes_{\Pa} \YYM \cong   \Pa V^{*}\YYM \oplus \Pa V^{\circ}\YYM.
\end{equation}

Although the formula is obtained by a straightforward calculation, we explain it in detail,  since  similar explicit formulas for other morphisms play key roles in the sequel.

In  the identification of $\PPi_{1} \otimes_{\Pa} \YYM$ with $\Pa V^{*}  \YYM \oplus \Pa S  \YYM$,
an element of $\PPi_{1} \otimes_{\Pa} \YYM$ is given by a sum of elements of the forms
$p\alpha^{*} x, p\vars_{i} x$ where $p \in \Pa, x \in \YYM, \alpha \in Q_{1}, i \in Q_{0}$.
Thus in particular, $\PPi_{1} \otimes_{\Pa} \YYM$ is generated over $\Pa^{\mre}$
by elements of the forms $\alpha^{*}x, \vars_{i} x$ for $ x \in \YYM, \alpha \in Q_{1}, i \in Q_{0}$.
The values of the morphism $\eeta^{*}$ over $\Pa^{\mre}$ on these generators are given as below

\[
\eeta^{*}(\alpha^{*}  x) = [\alpha^{*}, \varrho]  x = \alpha^{*}\varrho   x - \varrho \alpha^{*} x ,
\ \eeta^{*}(\vars_{i}  x) = -\varrho_{i}^{\circledast} x
\]

where we use the identification $\YYM_{1} \otimes_{\Pa} \YYM \cong   \Pa V^{*}\YYM \oplus \Pa V^{\circ}\YYM$
to write the right hand sides. More precisely, the right hand sides should be interpreted as follows.
For the first term $\alpha^{*}\varrho  x = \alpha^{*} (\varrho x)$ of the first equation,
we   regard  the first factor $\alpha^{*}$ as an element of $V^{*}$
and the second factor  $\varrho  x$ as an element $\YYM$.
Then using the identification  $\Pa V^{*}  \YYM \subset \YYM_{1} \otimes_{\Pa} \YYM$,
we regard $\alpha^{*} \varrho x$ as an element of $\YYM_{1} \otimes_{\Pa} \YYM$.

As for the second  term $\varrho \alpha^{*}  x= \varrho (\alpha^{*} x)$ of the first equation,
we   regard  the first factor $\varrho $ as an element of $\YYM_{1}$
and the second factor $\alpha^{*}  x$ as an element $\YYM$.
Then we write $\varrho \otimes \alpha^{*} x$ as $\varrho \alpha^{*}x$.
In a similar way, we regard $\varrho_{i}^{\circledast} p x$ as an element of $\YYM_{1} \otimes_{\Pa} \YYM$.

We note that under the identification (<ref>), the element  $\varrho \alpha^{*}  x $
corresponds to the element

\[
\sum_{i \in Q_{0}} v_{i}^{-1}\left(\sum_{\beta: t(\beta) = i } \beta\beta^{*} \alpha^{*}x
-\sum_{\beta: h(\beta) = i} \beta^{*} \beta \alpha^{*} x\right)
\]

where the first term $\beta\beta^{*} \alpha^{*}x$ is regarded  as an element of $\Pa V^{*} \YYM$ by viewing
$\beta\beta^{*} \in \Pa V^{*}$ and $\alpha^{*} x \in \YYM$
and the second term $\beta^{*} \beta\alpha x$
is regarded  as an element of $\Pa V^{*} \YYM$ by viewing
$\beta^{*} \in \Pa V^{*}$ and $\beta \alpha^{*} x \in \YYM$.
In a similar way, we can write the  element of $\Pa V^{\circledast} \YYM$ which corresponds to  $\varrho_{i}^{\circledast} p x$.

§.§.§ The homotopy $\sfK$

Since $\ppi: \YYM \to \PPi$ is a dg-algebra homomorphism,
we see that  the  square in the following diagram is commutative.

\[
\begin{xymatrix}@C=30pt{
\PPi_{1} \otimes_{A} \YYM (-2) \ar@/_20pt/[drr]_{0} \ar[rr]^{\eeta^{*}} &&
\YYM_{1} \otimes_{A} \YYM ( -1) \ar[rr]^-{\zzeta} \ar[d]^{\ppi_{1} \otimes \ppi(-1)} \ar@{=>}[dl]_{\sfK}&&
\YYM_{ \geq 1} \ar[d]^{\ppi_{\geq 1}} \\
&& \PPi_{1} \otimes_{A} \PPi ( -1) \ar@{=}[rr] && \PPi_{\geq 1}
\]

Decomposing the morphism $\ppi_{\geq 1}$ according to the decomposition
$\YYM_{\geq 1} = \YYM_{1} \otimes_{A} \YYM(-1) \oplus \PPi_{1} \otimes_{A} \YYM( -2)[1]$ of underlying cohomological graded modules,
we deduce the following lemma.

We define a morphism $\sfK: \PPi_{1} \otimes_{\Pa} \YYM( -2) \to \PPi_{\geq 1}$ in $\sfC(\Pa\otimes \YYM^{\op}\Gr^{*})$
of cohomological degree $-1$ by the formula

\[
\sfK(\alpha^{*} p x) := [\vars, \alpha^{*}] p \ppi(x), \ \sfK(\vars_{i}  p x) := \vars_{i}^{2}p \ppi(x).
\]

for $i \in Q_{0}, \alpha \in Q_{1}, p \in \Pa,   x \in \YYM$
where we use the identification $\PPi_{1}\otimes_{\Pa} \YYM \cong \Pa V^{*} \YYM \oplus \Pa S \YYM$
to exhibit elements in the domain of $\sfK$.

Then it is a homotopy from $(\ppi_{1} \otimes \ppi(-1)) \eeta^{*}$ to $0$
and we have $\ppi_{\geq 1} = q_{\eeta^{*}, \ppi_{1} \otimes \ppi( -1),  \sfK}$
as morphisms from $\YYM_{\geq 1} = \cone(\eeta^{*}) $ to $\PPi_{ \geq 1}$.

§.§ The exact triangle $\sfU$

The aim of this section <ref> is to prove Theorem <ref>
and Theorem <ref> below.

Taking  the tensor product $\sfAR \otimes_{A} \YYM$, we obtain the upper row of the following diagram.
It is clear that the right square is commutative.
We set  $\brho: = \zzeta \rrho_{\YYM}$ to make the left square commutative.

\begin{equation}\label{202007312226}
\begin{xymatrix}@C=60pt@R=6mm{
\PPa \otimes_{\Pa}  \YYM  \ar@/^30pt/[rr]^{0}  \ar[r]^{\rrho_{\YYM} } \ar@{=}[d] &
\YYM_{1} \otimes_{\Pa} \YYM \ar@{=>}[u]_{\sfL_{\YYM}} \ar[r]^{\ppi_{1, \YYM} } \ar[d]_{\zzeta} &
\PPi_{1} \otimes_{\Pa} \YYM \ar[d]^{{}_{\PPi_{1} }\ppi} \\
\PPa \otimes_{\Pa} \YYM  \ar[r]_{\brho} &\YYM \ar[r]_{\ppi} & \PPi
\end{equation}

We set
$\sfM := ({}_{\PPi_{1}} \pi)(  \sfL_{ \YYM}): \PPa \otimes \YYM \to \PPi$.
It  is a homotopy from $\ppi \brho $ to $0$ in $\sfC_{\DG}(\Pa\otimes \YYM^{\op})$ by Lemma <ref>.

\[
\begin{xymatrix}@C=10mm@R=4mm{
\PPa \otimes_{\Pa} \YYM   \ar[r]^{\brho } \ar@/_2pc/[rr]_{0}   &\YYM \ar[r]^{\ppi} \ar@{=>}[d]^{\sfM} & \PPi\\
\]

We  give explicit formulas for these morphisms.
For this we use the identification $\PPa \otimes_{\Pa} \YYM \cong \Pa \Pa_{0} \YYM \oplus \Pa (V[1] )\YYM$.
Observe that  $\PPa \otimes_{\Pa} \YYM$ is generated over $\Pa^{\mre}$ by elements
of the forms $e_{i} x, \downarrow \alpha x$ where $i \in Q_{0}, \alpha \in Q_{1} $ and $x \in \YYM$.
On these generators, the morphisms $\brho$ and $\sfM$ take the following values

\[
\brho(e_{i} x) = \varrho_{i} x, \ \brho(\downarrow \alpha x)= \alpha^{\circ} x, \ \sfM(e_{i}x ) = -\vars_{i} \ppi(x),  \ \sfM(\downarrow \alpha x) = 0.
\]

We remark that the  module $\Pa \Pa_{0} \YYM$ that appeared in the above identification has the simpler expression  $\Pa \YYM$.
However if we use the latter, then the expression $px \ (p \in \Pa, x \in \YYM)$ of an element of $\Pa\YYM$
could have two meanings that are  $ e_{t(p)} \otimes_{\Pa_{0}} px$ and $p \otimes_{\Pa_{0}} x$.
To avoid this confusion, we insert $\Pa_{0}$ in between $\Pa$ and $\YYM$.

For example, if we apply the differential $d$ to a homogeneous element of $\PPa \otimes_{\Pa} \YYM$ which has the form
$\downarrow \alpha x$,
then  the result is written as $d(\downarrow \alpha x) = \alpha e_{h(\alpha)} x - e_{t(\alpha)} \alpha x - \downarrow\alpha d(x)$.

The  main result of Section <ref> is the following, which is a two-sided version of Theorem <ref>.

The induced morphism
$q :=q_{\brho, \ppi_{1}, \sfM}: \cone(\brho) \to \PPi$ is a quasi-isomorphism in $\sfC(\Pa \otimes \YYM^{\op})$.

we have an exact triangle of the following form in $\sfD(\Pa \otimes \YYM^{\op}\Gr)$

\begin{equation}\label{202008141923}
\YYM(-1) \xrightarrow{\brho} \YYM \xrightarrow{\ppi} \PPi  \to \YYM(-1)[1].
\end{equation}

Before proceeding  a proof, we discuss its consequence.
Let $n \geq 1$.
We note that looking $*$-degree $n$ part of the exact triangle (<ref>)
we obtain the following exact triangle in $\sfD(\Pa^{\mre})$,

\begin{equation}\label{202008141924}
\YYM_{n -1} \xrightarrow{ \brho_{n} } \YYM_{n } \xrightarrow{ \ppi_{n} } \PPi_{n } \to \YYM_{n -1}[1].
\end{equation}

Thus taking the tensor product with $M \in \Dbmod{\Pa}$, we obtain the following  exact triangle in $\Dbmod{\Pa}$

\begin{equation}\label{2020081419241}
\YYM_{n -1} \lotimes_{\Pa} M \xrightarrow{ \brho_{n, M } } \YYM_{n } \lotimes_{\Pa} M \xrightarrow{ \ppi_{n, M} } \PPi_{n } \lotimes_{\Pa} M \to \YYM_{n -1} \lotimes_{\Pa} M[1].
\end{equation}

Since the weighted Euler characteristic is additive for an exact triangle, we deduce the following corollary.

Let $u \in \kk Q, \ M \in \Dbmod{\Pa}, \ n \geq 0$. Then we have the following equality

\[
{}^{u}\! \Euch(\YYM_{n} \lotimes_{\Pa} M) = \sum_{i = 0}^{n} {}^{u}\!\Euch(\PPi_{i} \lotimes_{\Pa} M).
\]

In particular in the case where $u $ is an eigenvector of $\Psi = \Phi^{-t}$ (the inverse of the transpose of  the Coxeter matrix) with the eigenvalue $\lambda$,
we have

\[
\frac{ {}^{u}\! \Euch(\YYM_{n} \lotimes_{\Pa} M) }{ {}^{u}\! \Euch(M)} =  \sum_{i = 0}^{n} \lambda^{i}.
\]

§.§.§ The morphism $\brho'$

To prove Theorem <ref>, first we need to introduce
a morphism $\brho'  : \PPa \otimes_{\Pa} \YYM \to \YYM$ related to the morphism $\brho$.

We consider the co-cone $\cone(\ppi_{1}) \otimes_{A}\YYM[ -1]$ of $\ppi_{1, \YYM}$.
We set $p := - p_{2}^{\ppi_{1}}$ and  $\brho' := \zzeta  p_{\YYM} : \PPa \otimes_{\Pa} \YYM \to \YYM$.

\begin{equation}\label{2020073122261}
\begin{xymatrix}@C=60pt@R=6mm{
\cone(\ppi_{1}) \otimes_{\Pa}  \YYM [-1]  \ar@/^30pt/[rr]^{0} \ar[r]^{p_{\YYM} } \ar@{=}[d] &
\YYM_{1} \otimes_{\Pa} \YYM \ar@{=>}[u]_{(\sfh_{\pi_{1}})_{\YYM}} \ar[r]^{\ppi_{1, \YYM} } \ar[d]_{\zzeta} &
\PPi_{1} \otimes_{\Pa} \YYM \ar[d]^{{}_{\PPi_{1} }\ppi} \\
\cone(\ppi_{1}) \otimes_{\Pa} \YYM [-1] \ar[r]_{\brho'} &\YYM \ar[r]_{\ppi} & \PPi
\end{equation}

Then setting
$\sfM' = {}_{\PPi_{1} } \pi (\sfh_{\ppi_{1}})_{ \YYM}$,
we obtain the following diagram in $\sfC_{\DG}(\Pa \otimes\YYM^{\op})$ by Lemma <ref>.

\[
\begin{xymatrix}@C=10mm@R=4mm{
\cone(\ppi_{1}) \otimes_{\Pa} \YYM [-1]  \ar[r]^-{\brho' } \ar@/_2pc/[rr]_{0}   &\YYM \ar[r]^{\ppi} \ar@{=>}[d]^{\sfM'} & \PPi.\\
\]

The  morphism
$q :=q_{\brho, \ppi_{1}, \sfM}: \cone(\brho) \to \PPi$ is a quasi-isomorphism
if and only if so is
the induced morphism
$q' :=q_{\brho', \ppi_{1}, \sfM}: \cone(\brho') \to \PPi$.

We set $j: = j_{\rrho, \ppi_{1}, \sfL}$.
First we claim  $\brho'  j_{\YYM} = \brho$.
Indeed, we have $pj =\rrho$
and hence $p_{\YYM} j_{\YYM} = \rrho_{\YYM}$.

\begin{equation}\label{2020073122263}
\begin{xymatrix}@C=60pt{
\PPa \otimes_{\Pa} \YYM \ar[dr]^{\rrho_{\YYM} } \ar[d]_{j_{ \YYM} }& & \\
\cone(\ppi_{1}) \otimes_{\Pa} \YYM[-1] \ar[r]_{p_{\YYM}} &
\YYM_{1} \otimes_{\Pa} \YYM
\end{equation}

Thus we have $\brho' j_{\YYM} = \zzeta p_{\YYM}j_{\YYM} = \zzeta  \rrho_{\YYM} = \rrho$.

From the claim, we obtain the homotopy equivalence
$\sfJ := \begin{pmatrix} \id_{\YYM} & 0 \\ 0 & j_{ \YYM }\end{pmatrix} : \cone(\brho) \to \cone(\brho')$.
It follows from the equation $\sfh_{\ppi_{1}} j = \sfL$ that
$q_{\brho', \ppi_{1}, \sfM'} \sfJ =  q_{\brho, \ppi_{1}, \sfM}$.

\begin{equation}\label{202007312234}
\begin{xymatrix}@C=40pt{
\cone( \brho) \ar[r]^{q_{\brho, \ppi_{1}, \sfM} } \ar[d]_{\sfJ}^{\wr}  & \PPi \\
\cone( \brho') \ar[ur]_{q_{\brho', \ppi_{1}, \sfM'} }
\end{equation}

Therefore we conclude that $q=q_{\brho, \ppi_{1}, \sfM}$ is quasi-isomorphism
if and only if so is $q' = q_{\brho', \ppi_{1}, \sfM'}$.

§.§.§ The homotopy $\sfN$

We define a morphism $\sfN: \PPi_{1} \otimes_{\Pa} \PPa \otimes_{\Pa}  \YYM_{ n-2}
\to \PPi_{1} \otimes_{\Pa} \YYM_{n -1}$ in $\sfC_{\DG}(\Pa^{\mre} \Gr)$ of cohomological  degree $-1$
by the formulas

\[
\begin{split}
&\sfN( \alpha^{*} p e_{i}  x  ) :=\vars \alpha^{*} px - \alpha^{*} \sfH(p)x, \
\sfN(\vars p e_{i} x ) :=  \vars \sfH(p)x, \\
&\sfN(\alpha^{*} p \downarrow \beta x) := 0, \
\sfN(\vars p \downarrow \beta x) := 0
\end{split}
\]

for $i \in Q_{0}, \alpha, \beta  \in Q_{1}, p \in \Pa$ and $x  \in \YYM_{ n-2}$
to which we use the identifications

\begin{equation}\label{202105211909}
\begin{split}
\PPi_{1} \otimes_{\Pa} \PPa \otimes_{\Pa}  \YYM_{ n-2}
& \cong \Pa V^{*} \Pa\Pa_{0} \YYM \oplus \Pa S \Pa \Pa_{0} \YYM \oplus  \Pa V^{*} \Pa(V[1]) \YYM \oplus \Pa S \Pa(V[1]) \YYM, \\
\PPi_{1} \otimes_{\Pa} \YYM_{n -1} &\cong \Pa V^{*} \YYM \oplus \Pa S \YYM.
\end{split}
\end{equation}

Then $\sfN$ is a homotopy from $(\ppi_{1,\YYM_{n -1}} ) \eeta^{*}_{n}({}_{\PPi_{1}} \mu_{\YYM_{n-2}} ) $ to ${}_{\PPi_{1}}\brho_{n -1}$.

\begin{equation}\label{202007302239}
\begin{xymatrix}{
\PPi_{1} \otimes_{\Pa} \PPa \otimes_{\Pa} \YYM_{n -2}
\ar[d]_{{}_{\PPi_{1}}\mu_{\YYM_{ n-2}}  }\ar@{=}[rrrr] &&&&
\PPi_{1} \otimes_{\Pa} \PPa \otimes_{\Pa} \YYM_{n -2}
\ar[d]^{ {}_{\PPi_{1}}\brho_{n -1}  }
\\
\PPi_{1} \otimes_{\Pa} \YYM_{n -2}
\ar@{=>}[urrrr]^{\sfN}
\ar[rr]_{\eeta^{*}_{n}} && \YYM_{1} \otimes_{\Pa} \YYM_{n - 1}
\ar[rr]_{\ppi_{1, \YYM_{ n-1}} } & &
\PPi_{1} \otimes_{\Pa} \YYM_{ n-1}  \\
\end{equation}

Thus, we have the following commutative diagram in $\sfD(\Pa^{\mre})$

\[
\begin{xymatrix}@C=60pt{
\PPi_{1}\lotimes_{\Pa} \YYM_{n -2}\ar[d]_{\eeta^{*}_{n}} \ar@{=}[r] & \PPi_{1}\lotimes_{\Pa} \YYM_{n -2}  \ar[d]^{{}_{\PPi_{1} }\brho_{n -1} } & \\
\YYM_{1} \lotimes_{\Pa} \YYM_{n -1} \ar[r]_-{\ppi_{1, \YYM_{n -1} }} & \PPi_{1} \lotimes_{\Pa} \YYM_{ n-1}.
\]

We check the equation
$(\ppi_{1,\YYM_{n -1}} ) \eeta^{*}_{n}({}_{\PPi_{1}} \mu_{\YYM_{n-2}} ) - {}_{\PPi_{1}}\brho_{n -1}= d\sfN + \sfN d$
on the generators by direct calculation.
For simplicity, we denote the left hand side and the right hand side of the equation by $L$ and $R$.
We point out a key equation $\sfH(p) - \sfH(\beta p) =-  \sfH(\beta) p = - \beta^{\circledast}p$
for $p \in \Pa, \beta\in Q_{1}$. Also note that in the $\stackrel{(1)}{=}, \stackrel{(2)}{=}$ below we use Lemma <ref>.

\[
\begin{split}
L (\alpha^{*} p e_{i} x)
&=  \ppi_{1,  \YYM_{n -1}}  \eeta^{*}_{n}(\alpha^{*}px) - \alpha^{*}p \varrho x \\
&=  \ppi_{1, \YYM_{n -1}}  ( [\alpha^{*}, \varrho] px ) - \alpha^{*}p \varrho x \\
&= \alpha^{*} \varrho p x - \varrho \alpha^{*}p x - \alpha^{*}p \varrho x  \\
R (\alpha^{*} p e_{i} x)
& =d( \vars\alpha^{*} px - \alpha^{*} \sfH(p) x ) + \sfN ( \alpha^{*} p e_{i} d(x) ) \\
& =- \varrho \alpha^{*} px - \vars\alpha^{*} p d(x) - \alpha^{*} d(\sfH(p)) x + \alpha^{*}\sfH(p) d(x)
+ \vars \alpha^{*} p d(x) - \alpha^{*}\sfH(p)d(x)  \\
& = - \varrho \alpha^{*} px  - \alpha^{*} d(\sfH(p)) x  \\
& \stackrel{(1)}{=} - \varrho \alpha^{*} px  + \alpha^{*} [\varrho, p] x  \\
&= \alpha^{*} \varrho p x - \varrho \alpha^{*}p x - \alpha^{*}p \varrho x.  \\
\end{split}
\]

\[
\begin{split}
L(\vars_{j} pe_{i}x)
& = (\ppi_{1, \YYM_{n -1}}) \eeta^{*}_{n}(\vars_{j} px) -\vars_{j} p\varrho x\\
& = (\ppi_{1,\YYM_{n-1}}) (- \varrho^{\circledast}_{j} px) - \vars_{j} p\varrho x\\
& = (\ppi_{1,\YYM_{n-1}}) \left(v_{j}^{-1} \sum_{\beta} \beta^{\circledast}\beta^{*} px-  v_{j}^{-1} \sum_{\beta} \beta^{*}\beta^{\circledast} px\right) -\vars_{j} p\varrho x\\
& = v_{j}^{-1} \sum_{\beta} [\vars,\beta] \beta^{*} px- v_{j}^{-1} \sum_{\beta} \beta^{*}\beta^{\circledast} px -\vars_{j} p\varrho x\\
\end{split}
\]

\[
\begin{split}
R(\vars_{j} pe_{i} x)
d (\vars \sfH(p) x) + \sfN(-\varrho_{j} pe_{i}x - \vars_{j} pe_{i} d(x)) \\
& =- \varrho_{j}  \sfH(p)x - \vars_{j} d\sfH(p) x  + \vars_{j} \sfH(p)d(x)
+ \sfN\left(v_{j}^{-1}  \sum_{\beta}\beta^{*} \beta pe_{i}x - v_{j}^{-1} \sum_{\beta} \beta\beta^{*}pe_{i}x  - \vars_{j} pe_{i} d(x)\right) \\
& = v_{j}^{-1}  \sum_{\beta} \beta^{*} \beta\sfH(p)x  - v_{j}^{-1}  \sum_{\beta} \beta \beta^{*}   \sfH(p)x - \vars_{j} d\sfH(p) x   + \vars_{j} \sfH(p)d(x)  \\
&\ \ \ \ +
v_{j}^{-1}  \sum_{\beta}\left( \vars_{j}  \beta^{*} \beta px- \beta^{*}\sfH(\beta p) x \right)
- v_{j}^{-1}  \sum_{\beta} \left( \beta \vars_{j}  \beta^{*}px - \beta \beta^{*} \sfH(p) x\right)   - \vars_{j} \sfH(p) d(x) \\
& = v_{j}^{-1}  \sum_{\beta} \left( \beta^{*} \beta\sfH(p)x - \beta^{*}\sfH(\beta p) x\right)   - \vars_{j} d\sfH(p) x +
v_{j}^{-1}  \sum_{\beta} \vars_{j}  \beta^{*} \beta px -v_{j}^{-1}  \sum_{\beta} \beta \vars_{j}  \beta^{*}px   \\
& \stackrel{(2)}{=} - v_{j}^{-1} \sum_{\beta}   \beta^{*} \beta^{\circledast} px  +  \vars_{j} [\varrho ,p] x +
v_{j}^{-1}  \vars_{j} \left( \sum_{\beta} \beta\beta^{*}\right)   px - \vars_{j}  \varrho px  - v_{j}^{-1}  \sum_{\beta} \beta \vars_{j}  \beta^{*}px   \\
& = - v_{j}^{-1}  \sum_{\beta}   \beta^{*} \beta^{\circledast}p x  -  \vars_{j} p\varrho x +
v_{j}^{-1}  \vars_{j} \left( \sum_{\beta} \beta\beta^{*}\right)   px   -  v_{j}^{-1} \sum_{\beta} \beta \vars_{j}  \beta^{*}px   \\
& = v_{j}^{-1}  \sum_{\beta} [\vars_{j},\beta] \beta^{*} px- v_{j}^{-1} \sum_{\beta} \beta^{*}\beta^{\circledast} px -\vars_{j} p\varrho x\\
\end{split}
\]

\[
\begin{split}
L(\alpha^{*} p \downarrow \beta x) & = - \alpha^{*} p \beta^{\circledast} x \\
R(\alpha^{*} p \downarrow \beta x) &
= \sfN(\alpha^{*} p \beta e_{t(\beta)} x - \alpha^{*} p e_{h(\beta)} \beta x - \alpha^{*} \downarrow \beta dx)  \\
&= \vars  \alpha^{*} p x- \alpha^{*} \sfH(p\beta)x  - \vars  \alpha^{*} p \beta x + \alpha^{*}\sfH(p)\beta x   \\
& = -\alpha^{*}p\beta^{\circledast} x\\
\end{split}
\]

\[
\begin{split}
L(\vars  p \downarrow \beta x) & = - \vars  p \beta^{\circledast} x \\
R(\vars  p \downarrow \beta x)
& =  \sfN(-\varrho p \downarrow \beta x - \vars p \beta e_{t(\beta)} x + \vars  p e_{h(\beta)} \beta x + \vars  \downarrow \beta dx)\\
& = -\vars \sfH(p\beta)x + \vars  \sfH(p) \beta x\\
& = - \vars  p \beta^{\circledast} x.
\end{split}
\]

Now we obtain the following diagram that appeared in Lemma <ref>.

\begin{equation}\label{202008021750}
\begin{xymatrix}{
\PPi_{1} \otimes_{\Pa} \PPa \otimes_{\Pa} \YYM_{n -2}
\ar[d]_{{}_{\PPi_{1}}\mu_{\YYM_{n -2}} }\ar@{=}[rrrr] &&&&
\PPi_{1} \otimes_{\Pa} \PPa \otimes_{\Pa} \YYM_{n -2}
\ar[d]^{ {}_{\PPi_{1}}\brho } \ar@/^70pt/[dd]^{0} &&\\
\PPi_{1} \otimes_{\Pa} \YYM_{n -2}
\ar@{=>}[urrrr]^{\sfN}
\ar[rr]_{\eeta^{*}}
\ar@/_20pt/[drrr]_{0} && \YYM_{1} \otimes_{\Pa} \YYM_{n - 1}
\ar[rr]_{\ppi_{1, \YYM_{ n-1}} } & &
\PPi_{1} \otimes_{\Pa} \YYM_{ n-1}  \ar[d]^{{}_{\PPi_{1} }\ppi_{n -1} } \ar@{=>}[dll]^{\sfK} \ar@{=>}[r]^{{}_{\PPi_{1}}\sfM}&&\\
&&& \PPi_{n} \ar@{=}[r]^-{\sim} & \PPi_{1} \otimes_{\Pa} \PPi_{ n-1}&&
\end{equation}

A key step is to prove that the homotopies in the diagram satisfy the condition of Lemma <ref>.

We have the following equality.

\[
\sfK({}_{\PPi_{1}}\mu_{\YYM_{n -2}} ) =  {}_{\PPi_{1}} \sfM  + ({}_{\PPi_{1} }\ppi_{n-1}) \sfN
\]

For simplicity, we denote the left hand side and the right hand side of the equation by $L$ and $R$.
We check $L = R$ on  generators of $\PPi_{1} \otimes_{\Pa} \PPa \otimes_{\Pa} \YYM_{n -2}$ by direct calculation.
We use the identification (<ref>) to exhibit  generators of $\PPi_{1} \otimes_{\Pa} \PPa \otimes_{\Pa}  \YYM_{n -2}$.

First let $i \in Q_{0}, \alpha \in Q_{1}, p \in \Pa$ and $x \in \YYM_{ n-2}$.
we have

\[
\begin{split}
L(\alpha^{*}p e_{i} x ) &= [\vars, \alpha^{*}] p\pi(x),\\
R(\alpha^{*} pe_{i} x)
& = - \alpha^{*} p\vars  \ppi(x) + \vars \alpha^{*} p \ppi(x) - \alpha^{*} \ppi ( \sfH(p))\ppi(x)\\
& \stackrel{(1)}{=} - \alpha^{*} p\vars  \ppi(x) + \vars \alpha^{*} p \ppi(x) - \alpha^{*} [\vars , p]\ppi(x)\\
& = \vars  \alpha^{*} p \ppi(x) - \alpha^{*}\vars p \ppi(x) \\
& = [\vars , \alpha^{*}]p \ppi(x). \\
L(\vars p e_{i} x) &=\vars ^{2} p \ppi(x)\\
R(\vars  pe_{i} x) &  \stackrel{(2)}{=} \vars p \vars \ppi(x) + \vars  \ppi\sfH(p) \ppi(x) \\
& \stackrel{(3)}{=} \vars  p \vars  \ppi(x) + \vars [\vars ,p] \ppi(x) \\
& = \vars ^{2} \ppi(x)
\end{split}
\]

where for $\stackrel{(1)}{=}, \stackrel{(3)}{=}$ we use Lemma <ref>.
For  the first term of  $\stackrel{(2)}{=}$ we use Koszul sign rule as below.

\[
({}_{\PPi_{1}}\sfM )( \vars pe_{i} x) = - \vars p \sfM(e_{i} x) = -\vars p( -\vars \ppi(x)) = \vars p \vars \ppi(x).
\]

We can easily check that $L$ and $R$ both vanish
in the generators of the forms $\alpha^{*} p \downarrow \beta x, s p \downarrow \beta x$.

We use the  notations of Section <ref>.
By definition, we have the equality $\Phi_{\eeta^{*}, \ppi_{1} \otimes \YYM } = \brho'$ of morphisms from $\cone(\ppi_{1}) \otimes_{A} \YYM[-1]$
to $\YYM$.

The next step is to identify the morphism $ \acute{q}_{\eeta^{*}, \ppi_{1}\otimes \YYM, \PPi_{1} \otimes \ppi, \sfK}:
\cone(\Phi_{\eeta^{*}, \ppi_{1} \otimes \YYM }) \to \PPi$ with
$q': \cone(\brho') \to \PPi$.

We have
\[
\acute{q}_{\eeta^{*}, \ppi_{1}\otimes \YYM, \PPi_{1} \otimes \ppi, \sfK} = q'.
\]

Recall that

\[
q' = (\ppi, \sfM'\uparrow): \cone(\brho') = \YYM \oplus ( \cone(\ppi_{1} ) \otimes_{\Pa} \YYM) \to \PPi.
\]

by definition $\sfM' = ({}_{\PPi_{1} } \ppi) (\sfh_{\ppi_{1}})_{ \YYM}$
\[
(\sfh_{\ppi_{1}})_{ \YYM} = (\downarrow, 0) : \cone(\ppi_{1}) \otimes \YYM [-1] = \
(\PPi_{1} \otimes_{\Pa} \YYM [-1])\oplus ( \YYM_{1} \otimes_{\Pa} \YYM)
\to
\PPi_{1} \otimes \YYM,
\]

we obtain the following  description of the second component $\sfM'\uparrow: \cone(\ppi_{1}) \otimes_{\Pa} \YYM \to \PPi$.

\[
\sfM' \uparrow =({}_{\PPi_{1} }\ppi, 0) :
\cone(\ppi_{1} ) \otimes_{\Pa} \YYM = (\PPi_{1} \otimes_{\Pa} \YYM )\oplus ( \YYM_{1} \otimes_{\Pa} \YYM[1])
\to  \PPi_{1} \otimes_{\Pa} \PPi = \PPi_{\geq 1}.
\]

Consequently, we come to the desired equation.

\[
\begin{split}
q' &
= (\ppi, \sfM'\uparrow) = (\ppi_{1} \otimes \ppi, \sfK\uparrow, {}_{\PPi_{1} }\ppi, 0) \\
( ({}_{\PPi_{1}} \ppi) (\ppi_{1, \YYM}), \sfK\uparrow, {}_{\PPi_{1} } \ppi, 0)\\
&= \acute{q}_{\eeta^{*}, \ppi_{1} \otimes \YYM, \PPi_{1} \otimes\ppi, \sfK}.
\end{split}
\]

Thanks to Lemma <ref>, we can apply Lemma <ref>
to the diagram (<ref>).
By Lemma <ref> we obtain the following commutative diagram

\begin{equation}\label{202008021755}
\begin{xymatrix}{
\cone(\brho'_{n} ) \ar[dd]_{q'_{n}} & \cone(\Phi_{\eeta^{*}, \ppi_{1}\otimes \YYM_{n -1}} )
\ar[l]_-{\sfind} \ar[r]^-{\dot{q}}  & \cone(\PPi_{1} \otimes \brho_{n -1} ) \ar[d]^{\sfiso} \\
&& \PPi_{1} \otimes \cone(\brho_{n-1} ) \ar[d]^{{}_{\PPi_{1} } q_{n -1}} \\
\PPi_{n} \ar@{=}[rr] && \PPi_{1} \otimes_{\Pa} \PPi_{ n-1}
\end{equation}

where $\dot{q}$ is a homotopy equivalence and $\sfiso$ denotes a canonical isomorphism.
The morphism $\sfind$ is induced from a quasi-isomorphism ${}_{\PPi_{1}} \mu_{\YYM_{n-2}}: \PPi_{1} \otimes_{\Pa} \PPa \otimes_{\Pa} \YYM_{ n-2}
\to \PPi_{1} \otimes_{\Pa} \YYM_{n -2}$.
Therefore it  is a quasi-isomorphism.

§.§.§ Proof of Theorem <ref>

We prove that $q_{n}$ and $q'_{n}$
are quasi-isomorphisms by induction on $n \geq 0$.

Since $\brho_{0} =0, \brho'_{0} = 0, \ppi_{0} = \id_{\Pa}, \sfM_{0} = 0$, we see that $q_{0} = q'_{0} = \id_{\Pa}$ and
hence they are quasi-isomorphisms.

The morphism $q_{1}$ and $q'_{1}$ are quasi-isomorphism by Lemma <ref>.

We deal with the case $n \geq 2$.
We assume $q_{n-1}$ is a quasi-isomorphism.
all the morphisms in  the diagram (<ref>) are quasi-isomorphisms
except $q'_{n}$.
$q'_{n}$ is also a quasi-isomorphism.
Thanks to Lemma <ref> that $q_{n}$ is also a quasi-isomorphism.

The commutative diagram given below plays a key role later.

For $n \geq 2$, we have the following commutative diagram in $\sfD(\Pa^{\mre})$.

\[\begin{xymatrix}@C=60pt{
\PPi_{1} \lotimes_{\Pa} \YYM_{n -2} \ar@{=}[r] \ar[d]_{\eeta^{*}_{n}}
\PPi_{1} \lotimes_{\Pa} \YYM_{n -2}  \ar[d]^{ {}_{\PPi_{1}} \brho_{n -1}}
\\
\YYM_{n -1}   \ar@{=}[d]  \ar[r]^{ \rrho_{\YYM_{n -1} } }
\YYM_{1} \lotimes_{\Pa} \YYM_{n -1}  \ar[r]^{\ppi_{1, \YYM_{n -1} }  } \ar[d]_{\zzeta_{n} }
\PPi_{1} \lotimes_{\Pa} \YYM_{n- 1} \ar[d]^{ {}_{ \PPi_{1} }\ppi_{n -1}} \ar[r]^{- \ttheta[1]_{ \YYM_{n -1}}}
& \YYM_{n -1} [1] \ar@{=}[d]\\
\YYM_{n-1}   \ar[r]_{\brho_{n}}   &
\YYM_{n} \ar[r]_{\ppi_{n}}  \ar[d]
& \PPi_{n} \ar[r] \ar[d]  & \YYM_{ n-1}[1]\\
& \PPi_{1} \lotimes_{\Pa} \YYM_{n -2}[1] \ar@{=}[r] & \PPi_{1} \lotimes_{\Pa} \YYM_{n -2}[1] &
\]

where two middle rows and two middle columns are exact triangles.

It only remains to show that the two middle columns give a morphism of exact triangles.
Recall that we are identifying $\YYM_{n}$ with $ \cone(\eeta^{*}_{n})$ and $\zzeta_{n}$ with a canonical morphism $i_{1}^{\eeta^{*}_{n}}$.
For simplicity we set $\phi :=  (\eeta^{*}_{n})({}_{\PPi_{1}} \mu_{\YYM_{n-2}})$,
then we have that following commutative diagram in $\sfC(\Pa)$

\[\begin{xymatrix}@C=40pt{
\PPi_{1} \otimes_{\Pa} \YYM_{n -2} \ar[d]_{\eeta^{*}_{n}}
\PPi_{1} \otimes_{\Pa} \PPa \otimes_{\Pa}  \YYM_{n -2} \ar[l]_{{}_{\PPi_{1}} \mu_{\YYM_{n-2}} } \ar@{=}[r]
\ar[d]_{ \phi }
\PPi_{1} \otimes_{\Pa} \PPa \otimes_{\Pa} \YYM_{n -2}  \ar[d]^{ {}_{\PPi_{1}} \brho_{n -1} } \ar@{=}[r]
&  \PPi_{1} \otimes_{\Pa} \PPa \otimes_{\Pa} \YYM_{n -2} \ar[d]^{ {}_{\PPi_{1}} \brho_{n -1} }
\\
\YYM_{1} \otimes_{\Pa} \YYM_{n -1}   \ar@{=}[r]  \ar[d]_{ \zzeta_{n}  }
\YYM_{1} \otimes_{\Pa} \YYM_{n -1}  \ar[r]_{\ppi_{1, \YYM_{n -1} }  } \ar[d]_{i_{1}^{\phi}} \ar@{=>}[ur]^{\sfN}
\PPi_{1} \otimes_{\Pa} \YYM_{n- 1}
\ar[d]^{i_{1}^{{}_{\PPi_{1} } \brho_{n-1} }}   \ar@{=}[r]
& \PPi_{1} \otimes_{\Pa} \YYM_{n- 1} \ar[d]^{ {}_{ \PPi_{1} }\ppi_{n -1}}
\\
\YYM_{n}  &
\cone(\phi)  \ar[l]^{\Psi_{1}}  \ar[r]_{\Psi_{2}}
& \cone({}_{\PPi_{1}} \brho_{n -1} ) \ar[r]_{q}  & \PPi_{n}
\]

where we set

\[
\Psi_{1} = \begin{pmatrix} \id & 0 \\ 0 & {}_{\PPi_{1}}\mu_{\YYM_{n -2}} \end{pmatrix}, \quad
\Psi_{2} = \begin{pmatrix} \ppi_{1, \YYM_{n -1}} & \sfN\uparrow \\ 0 & \id \end{pmatrix}, \quad
q = ({}_{\PPi_{1}} \ppi_{n -1}. \sfM\uparrow)
\]

Observe that all columns extend to exact triangles and morphisms between them induce morphisms between exact triangles.
Moreover the morphisms from the second column to the first column are quasi-isomorphisms.
Thus it is enough to verify the equality $\ppi_{n} = q \Psi_{2} \Psi_{1}^{-1}$ in $\sfD(\Pa)$.
The last equality follows from the equality $\ppi_{n} \Psi_{1} = q \Psi_{2}$
that can be checked from Lemma <ref> and Lemma <ref>.

§.§.§ Proof of Theorem <ref>

We prove Theorem <ref> by reducing it to Theorem <ref>.

Define a morphism $\sfO:  \PPa \otimes_{\Pa} \YYM( -1) \to \YYM$ in $\sfC_{\DG}(\Pa \Gr)$ of cohomological  degree $-1$
by the formula

\[
\sfO(e_{i} x) := -e_{i} \sfH(x), \ \sfO(\downarrow \alpha x ) := 0
\]

for $i \in Q_{0}, \alpha \in Q_{1}$ and $x \in \YYM$ to which we use the identification
$\PPa \otimes_{\Pa} \YYM \cong \Pa \Pa_{0} \YYM \oplus \Pa (V[1]) \YYM$.

Then the following statements hold.

$\sfO$ is a homotopy from
$\brho $ to $\sfr_{\varrho} ( \mu\otimes \YYM(-1))$.

* We have the following equality

\[
\sfM = \ppi \sfO - \sfr_{\vars} \ppi(\mu_{ \YYM( -1)})
\]

of morphisms from $\PPa \otimes_{\Pa} \YYM(-1)$ to $\PPi$ of degree $-1$.

Note that
we have the following diagram

\begin{equation}\label{202102051520}
\begin{xymatrix}@C=50pt@R=10pt{
\PPa \otimes_{\Pa} \YYM(-1)
\ar@/^30pt/[rr]^{0}
\ar[dd]_{\mu_{\YYM(-1)}}
\ar[r]^{\brho} &
\YYM \ar@{=}[dd] \ar[r]^{\ppi} \ar@{=>}[u]_{\sfM} \ar@{=>}[ddl]_{\sfO}
& \PPi \ar@{=}[dd]\\
\YYM(-1) \ar@/_30pt/[rr]_{0}
\ar[r]^{\sfr_{\varrho}} & \YYM \ar[r]_{\ppi} \ar@{=>}[d]^{- \sfr_{\vars} \ppi} & \PPi \\
&& .
\end{equation}

It is enough to show that the equation
$ \zzeta (\rrho_{\YYM(-1)})  - \sfr_{\varrho} (\mu_{\YYM(-1)} ) =  d \sfO + \sfO d$.
holds on the generators $e_{i} x$ and $\downarrow \alpha  x$.
We check this by direct calculation.
For simplicity, we denote by $L$ and $R$ the left hand side and the right had side of the equation.

\[
\begin{split}
L(e_{i}  x)  & = \varrho_{i} x -  e_{i} x \varrho, \\
R(e_{i}  x)  & = - e_{i} ((d \sfH + \sfH d)(x)) = -e_{i} (x \varrho - \varrho x) = \varrho_{i}x - e_{i} x \varrho, \\
L(\downarrow \alpha   x) & = \alpha^{\circledast} x,  \\
R(\downarrow \alpha  x) & = \sfO d( \downarrow \alpha x) =
\sfO(\alpha e_{h(\alpha)}  x - e_{t(\alpha)}\alpha x- \downarrow \alpha  dx) \\
& = - \alpha \sfH(x) + \sfH(\alpha x)   = \sfH(\alpha) x = \alpha^{\circledast} x.
\end{split}
\]

We also check the equation holds on the generators $e_{i} x$ and $\downarrow \alpha  x$ by direct calculation.

\[
\begin{split}
(\ppi \sfO - \sfr_{\vars} \ppi(\mu_{ \YYM( -1) })) ( e_{i} x)
&= - e_{i} \ppi \sfH(x) - \sfr_{\vars}\ppi(x) \\
&= -[\vars, \ppi(x)]- \sfr_{\vars} \ppi(x) \\
& = - \vars\ppi(x) = \sfM(x), \\
(\ppi \sfO - \sfr_{\vars} \ppi(\mu_{\YYM( -1)} ))(\downarrow \alpha  x) & = 0 = \sfM(\downarrow\alpha   x)
\end{split}
\]

We proceed a proof of Theorem <ref>.

By Lemma <ref>, the  diagram (<ref>) induces the following commutative diagram

\[
\begin{xymatrix}@C=50pt@R=20pt{
\cone(\brho) \ar[r]^{q_{\brho, \ppi, \sfM}} \ar[d]_{\sfind} & \PPi \ar@{=}[d] \\
\cone(\sfr_{\rho}) \ar[r]_{q_{\sfr_{\varrho}, \ppi, - \sfr_{\vars} \ppi}} & \PPi
\]

where $\sfind$ is the morphism induced from $\mu_{ \YYM( -1)}$.
By Theorem <ref>, the upper arrow $q_{\brho, \ppi, \sfM}$ is a quasi-isomorphism.
Since $\mu_{\YYM( -1)}$ is a quasi-isomorphism, so is the induced morphism $\sfind$.
Thus we conclude that $q_{\sfr_{\varrho}, \ppi,-\sfr_{\vars}\ppi}$ is a quasi-isomorphism.

§.§ Lemmas

We collect two lemmas  for the later quotations

It follows from Lemma <ref> that the composite $\zzeta_{2}({}_{\YYM_{1}}\rrho- \rrho_{\YYM_{1}})$ becomes   a zero morphism
$\YYM_{1} \to \YYM_{2} $ in $\sfD(\Pa_{0}^{\mre})$.
Since $\eeta^{*}_{2}$ is a co-cone morphism of $\zzeta_{2}$,
there exists a morphism $f: \YYM_{1} \to \PPi_{1}$ such that ${}_{\YYM_{1}}\rrho- \rrho_{\YYM_{1}} = \eeta^{*}_{2} f$.
The next lemma says that we can take $f$ to be $\ppi_{1}$.

We define a morphism $\sfP: \PPa \otimes_{\Pa} \YYM_{1} \otimes_{\Pa} \PPa \to \YYM_{1}\otimes_{\Pa} \YYM_{1}$ of degree $-1$ in $\sfC_{\DG}(\Pa^{\mre})$ by the formula

\[
\begin{split}
\sfP(e_{i} p\alpha^{*} q e_{j}) & := \sfH(p) \alpha^{*} q + p \alpha^{*} \sfH(q), \
\sfP(e_{i} p\alpha^{\circledast} q e_{j} )  := \sfH(p) \alpha^{\circledast} q -p \alpha^{\circledast} \sfH(q), \\
\sfP(\downarrow \beta p\alpha^{*} q e_{j} ) & :=0,\  \sfP(\downarrow \beta p\alpha^{\circledast} q e_{j} ) :=0,
\sfP(e_{j} p\alpha^{*} q \downarrow \gamma  )  :=0,\  \sfP(e_{i}p\alpha^{\circledast} q \downarrow \gamma )  :=0, \\
\sfP(\downarrow \beta p\alpha^{*} q \downarrow \gamma  ) & :=0,\  \sfP(\downarrow \beta p\alpha^{\circledast} q \downarrow \gamma ) :=0
\end{split}
\]

for $i,j \in Q_{0}, \ \alpha,\beta, \gamma \in Q_{1}, \ p,q \in \Pa$
to which we use the identifications

\[
\begin{split}
\PPa \otimes_{\Pa} \YYM_{1} \otimes_{\Pa} \PPa  &\cong
\Pa\Pa_{0} \Pa V^{*} \Pa\Pa_{0} \Pa \oplus \cdots \oplus \Pa(V[1]) \Pa V^{\circ} \Pa(V[1]) \Pa,\\
\YYM_{1}\otimes_{\Pa} \YYM_{1}  &\cong
\Pa V^{*} \Pa V^{*} \Pa \oplus \cdots \oplus \Pa V^{\circ } \Pa V^{\circ } \Pa\end{split}\]

induced from
$\PPa \cong \Pa\Pa \oplus \Pa (V[1])\Pa, \YYM_{1} \cong \Pa V^{*} \Pa \oplus \Pa V^{\circledast} \Pa$.
Then $\sfP$ is a homotopy from $\mu\otimes \YYM_{1} \otimes \rrho-\rrho \otimes \YYM_{1} \otimes \mu$ to
$\eta^{*}_{2} \ppi_{1} (\mu \otimes \YYM_{1} \otimes \mu)$.

\[
\begin{xymatrix}@C=50pt{
\YYM_{1} \ar[rr]^-{\ppi_{1}} && \PPi_{1} \ar[d]^{\eta^{*}_{2}} \\
\PPa \otimes_{\Pa} \YYM_{1} \otimes_{\Pa} \PPa \ar[u]^{\mu\otimes \YYM_{1} \otimes \mu}
\ar[rr]_-{\mu\otimes \YYM_{1} \otimes \rrho-\rrho \otimes \YYM_{1} \otimes \mu }&
\ar@{=>}[u]_{\sfP} &
\YYM_{1}\otimes_{\Pa} \YYM_{1}

Therefore, we have the following commutative diagram in $\sfD(\Pa^{\mre})$.

\[
\begin{xymatrix}@C=50pt{
& \PPi_{1} \ar[d]^{\eta^{*}_{2}} \\
\YYM_{1} \ar[ur]^-{\ppi_{1}}
\ar[r]_-{ {}_{\YYM_{1}}\rrho- \rrho_{\YYM_{1}} } & \YYM_{1}\otimes_{\Pa} \YYM_{1}

Since it is verified by straightforward calculation, we leave the proof to the readers.

We may construct  a right version $\eeta^{*\textup{right}}_{n}: \YYM_{n -2} \lotimes_{\Pa} \PPi_{1} \to \YYM_{n-1} \lotimes_{\Pa} \YYM_{1}$ of the morphism $\eeta^{*}_{n}$
as below

\[
\eeta^{*\textup{right}}_{n}: \YYM_{n -2} \lotimes_{\Pa} \PPi_{1} \xrightarrow{{}_{\YYM_{n-2}}\eeta^{*}_{2} }
\YYM_{n-2} \lotimes_{\Pa} \YYM_{1} \lotimes_{\Pa} \YYM_{1}
\xrightarrow{ \ \ \ \ \ }
\YYM_{n-1} \lotimes_{\Pa} \YYM_{1}
\]

where the second morphism is  the multiplication morphism.
The following is a right version of Lemma <ref>.

We have the following commutative diagram in $\sfD(\Pa^{\mre})$

\[
\begin{xymatrix}@C=60pt{
\YYM_{n -2} \lotimes_{\Pa}  \PPi_{1}\ar[d]
_{\eeta^{*, \textup{right} }_{n} } \ar@{=}[r]
\YYM_{n -2} \lotimes_{\Pa}  \PPi_{1} \ar[d]^{ -\brho_{n -1, \PPi_{1} } } & \\
\YYM_{n-1} \lotimes_{\Pa} \YYM_{1} \ar[r]_-{{}_{\YYM_{n-1}}\ppi_{1 }} &
\YYM_{n -1} \lotimes_{\Pa} \PPi_{1} .
\]

We use the case $n =2$ later.
In this case, we have
$\eeta^{*, \textup{right} }_{2} = \eeta^{*}_{2}$
hence  the equality ${}_{\YYM_{1}}\ppi_{1}\eeta^{*}_{2} = -\brho_{1, \PPi_{1}}$.

We can verify this lemma in a similar way of Lemma <ref> a left version of it.
Here we take a different way. We show that we can deduce this lemma from
Lemma <ref> for the opposite quiver $Q^{\op}$.

We may  identify $(\kk Q)^{\op} \cong \kk (Q^{\op})$ and $\kk Q \cong (\kk (Q^{\op}))^{\op}$.
We introduce an isomorphism
$f: \Pa^{\mre} = \kk Q \otimes (\kk Q)^{\op} \to \kk ( Q^{\op}) \otimes (\kk ( Q^{\op}))^{\op} = \kk (Q^{\op})^{\mre}$
to be $f(p\otimes q)= q \otimes p$.
Let $f^{*}: \sfC_{\DG}(\kk (Q^{\op})^{\mre}) \to \sfC_{\DG}(\Pa^{\mre})$ be the induced isomorphism.
Then we have $f^{*}(X \otimes_{\kk(Q^{\op})} Y) \cong Y \otimes_{\kk Q} X$.
The anti-algebra isomorphism  $\phi: \kk (Q^{\op}) \to \kk Q, \phi(\alpha_{1} \alpha_{2}\cdots, \alpha_{n})
:= \alpha_{n} \cdots \alpha_{2} \alpha_{1}$ induces an isomorphism $f^{*}\kk (Q^{\op}) \cong \kk Q$.
Similar anti-$*$-graded-dg-algebra isomorphisms $\phi: \PPi(Q^{\op}) \to \PPi(Q), \phi: \YYM(Q^{\op}) \to \YYM(Q)$ denoted by the same symbol induces isomorphisms $f^{*}\PPi(Q^{\op}) \to \PPi(Q), f^{*} \YYM(Q^{\op}) \to \YYM(Q)$.
We note that $\phi(\rho) = -\rho$ and $\phi(s_{i}) = -s_{i}$.
We can check that under these isomorphisms $\phi$, the morphisms
$f^{*}(\eeta^{*}_{Q^{\op}})$ and $f^{*}(\ppi_{Q^{\op}})$ correspond to $\eeta^{*, \textup{right}}$ and $\ppi$.
Thus, from Lemma <ref> for $Q^{\op}$, we can deduce the desired commutativity.

§.§ $3$-Calabi-Yau property

§.§.§ Calabi-Yau algebras

Let $n, l\in \ZZ$.
Recall that a DG-algebra $R = (\bigoplus_{i,j } R_{i}^{j}, d_{R})$ with an additional grading
is called $n$-Calabi-Yau algebras of Gorenstein parameter $l$
if it is homologically smooth and there exists an isomorphism
$R^{\vvee}[n](-l) \cong R$ in $\sfD(R^{\mre})$.

The aim of this theorem is to prove the following theorem.
Note that the case $\chara \kk \neq 2$ is already shown in Proposition <ref>.
We give an alternative proof which works for arbitrary characteristic.

The derived quiver Heisenberg algbebra $\YYM$ is a $3$-Calabi-Yau
algebra of Gorenstein parameter $2$.

§.§.§ A preparation

Let $M=\bigoplus_{i\in\ZZ} M^{i}$ be a cohomologically graded $\Pa_{0}$-$\Pa_{0}$-bimodule.
Then by a DG-version of Lemma <ref>,
the map

\[
\begin{split}
F_{M}: & \YYM \tuD(M) \YYM \to ( \YYM M\YYM)^{\vee},
F_{M}(x \otimes f \otimes y) ( z \otimes m \otimes w)
:= (-1)^{\epsilon }f(m) zy \otimes xw \\
\epsilon & := |x|(|f|+|y| + |z|+|m|) + |f|(|y|+|z|) +|y||z|
\end{split}
\]

is an isomorphism of cohomologically graded  $\YYM^{\mre}$-bimodules.

Let $M=\bigoplus_{i \in \ZZ} M^{i}, N= \bigoplus_{i \in \ZZ} N^{i}$ be cohomologically graded $\Pa_{0}^{\mre}$-modules
and $\{m_{i}\}, \{n_{i}\}$ homogeneous basis of $M$ and $N$.
We identify $(\YYM M \YYM)^{\vee}, (\YYM N \YYM)^{\vee}$ with $\YYM \tuD(M) \YYM, \YYM\tuD(N)\YYM$
via $F_{M}, F_{N}$.
If a homogeneous morphism
$\phi: \YYM M \YYM \to \YYM N \YYM$ is given by

\[
\phi(m_{j} ) = \sum_{i} a_{ij} n_{i} b_{ij}
\]

for some $a_{ij}, b_{ij} \in \YYM$,
then the $\YYM^{\mre}$-dual $\phi^{\vee}: (\YYM N \YYM)^{\vee} \to (\YYM M \YYM)^{\vee}$ of $\phi$ is given by

\[
\phi^{\vee}(n_{i}^{\vee} ) = \sum_{j} ( -1)^{|\phi||n_{i}| + |a_{ij}|(|m_{j}| +|n_{i}| + |b_{ij}| ) } b_{ij}m_{j}^{\vee} a_{ij}
\]

where  $\{ m_{j}^{\vee}\}, \{ n_{i}^{\vee}\}$ denote the dual basis of $\tuD(M)$ and $\tuD(N)$.

§.§.§ Proof of Theorem <ref>

We prove the theorem by constructing a projectively cofibrant resolution $Q \xrightarrow{\sim} \YYM$ of $\YYM$ in
$\sfC(\YYM^{\mre}\Gr)$ such that $Q$ is perfect and  $Q^{\vee}[3](-2) \cong Q$.

Recall that
$\PPi_{1} \cong \Pa V^{*} \Pa \oplus \Pa S \Pa, \ \YYM_{1} \cong \Pa V^{*} \Pa \oplus \Pa V^{\circledast} \Pa,
\PPa \cong \Pa \Pa \oplus \Pa V[1] \Pa$
as cohomological graded modules over $\Pa^{\mre}$.
It is straightforward to check  that the canonical isomorphisms $\tuD(S) =\Pa_{0}, \tuD(V^{*}) = V, \tuD(V^{*}) =V^{\circledast}$
induces  isomorphisms

\begin{equation}\label{202112161145}
\PPi_{1}^{\vee}[1] \xrightarrow{\cong} \PPa,  \ \YYM^{\vee}_{1}[1] \xrightarrow{\cong}  \YYM_{1}, \
\PPa^{\vee}[1] \xrightarrow{\cong} \PPi_{1}
\end{equation}

(where $(-)^{\vee}$ denotes $\Pa^{\mre}$-duality) in $\sfC(\Pa^{\mre})$.

We set $Q_{(2)}:= \YYM\otimes_{\Pa} \PPi_{1} \otimes_{\Pa} \YYM(-2),
Q_{(1)} := \YYM \otimes_{\Pa} \YYM_{1} \otimes_{\Pa} \YYM(-1), \
Q_{(0)} := \YYM \otimes_{\Pa} \PPa \otimes_{\Pa} \YYM$.
As cohomological graded modules over $\YYM^{\mre}$, we have
$Q_{(2)}  \cong \YYM V^{*} \YYM \oplus \YYM S \YYM,
\ Q_{(1)}\cong \YYM V^{*} \YYM \oplus \YYM V^{\circledast} \YYM,
Q_{(0)} \cong \YYM  \YYM \oplus \YYM V[1] \YYM$.

We define a morphism $Y: Q_{(2)} \to Q_{(1)}$ in $\sfC(\YYM^{\mre}\Gr)$
in the following way. For $\alpha \in Q_{1}$ and $i \in Q_{0}$, we set

\[
\begin{split}
Y(\alpha^{*} )&
:= \alpha^{*} \otimes \varrho \otimes 1  - \varrho \otimes \alpha^{*} \otimes 1
+ 1\otimes \alpha^{*} \otimes \varrho - 1 \otimes \varrho \otimes \alpha^{*},\\
Y(\vars_{i})& := v_{i}^{-1} \sum_{\beta: t(\beta) = i} \beta^{\circledast} \otimes \beta^{*} \otimes 1 + 1 \otimes \beta^{\circledast} \otimes \beta^{*}- v^{-1}_{i} \sum_{\beta: h(\beta) = i} \beta^{*} \otimes \beta^{\circledast} \otimes 1 + 1\otimes \beta^{*} \otimes \beta^{\circledast}
\end{split}
\]

where $\otimes$ denotes $\otimes_{\Pa}$.

For $i \in Q_{0}$,
we define an element $\varrho'_{i} \in Q_{(0)}$ of cohomological degree $-1$
to be

\[
\varrho'_{i} := v^{-1}_{i}\sum_{\beta: t(\beta) = i} 1 \otimes \downarrow \beta \otimes \beta^{*}  - v^{-1}_{i}
\sum_{\beta: h(\beta) = i} \beta^{*} \otimes \downarrow \beta \otimes 1.
\]

We define a morphism $Z: Q_{(1)} \to Q_{(0)}$ in $\sfC(\YYM^{\mre}\Gr)$
in the following way. For $\alpha \in Q_{1}$ and $i \in Q_{0}$, we set

\[
\begin{split}
Z(\alpha^{*})  &:= \alpha^{*} \otimes 1 \otimes 1 - 1 \otimes 1 \otimes \alpha^{*} \\
Z(\alpha^{\circledast})  & :=
\varrho' \alpha - \alpha \varrho' + \varrho \otimes \downarrow \alpha \otimes 1 - 1 \otimes \downarrow \alpha \otimes \varrho
+ \alpha^{\circledast} \otimes 1 \otimes 1 - 1 \otimes 1 \otimes \alpha^{\circledast}
\end{split}
\]

where $\otimes$ denotes $\otimes_{\Pa}$.

We define a morphism $\sfG: Q_{(2) } \to Q_{(0)} $
in $\sfC_{\DG}(\YYM^{\mre}\Gr)$ of cohomological degree $-1$ in the following way.
For $\alpha \in Q_{1}$ and $i \in Q_{0}$, we set

\[
\begin{split}
\sfG(\alpha^{*}) & := -\alpha^{\circ} \otimes 1 \otimes 1 + 1 \otimes 1 \otimes \alpha^{\circ}- \alpha^{*}\varrho' + \varrho' \alpha^{*}\\
\sfG(\vars_{i} ) & := -\vart_{i} \otimes 1\otimes 1 + 1 \otimes 1\otimes \vart_{i}+ v^{-1}_{i}\sum_{\beta: t(\beta) = i} 1 \otimes \downarrow \beta \otimes \beta^{\circ}  +v^{-1}_{i}
\sum_{\beta: h(\beta) = i} \beta^{\circ} \otimes \downarrow \beta \otimes 1.
\end{split}
\]

where $\otimes$ denotes $\otimes_{\Pa}$.

The isomorphisms (<ref>)
induces isomorphisms

\begin{equation}\label{202112152209}
Q_{(0)}^{\vee} (-2)[1] \cong Q_{2}, \ Q_{(1)}^{\vee} (-2)[1] \cong Q_{1},
\ Q_{(2)}^{\vee}(-2)[1] \cong Q_{0}
\end{equation}

(where $(-)^{\vee}$ denotes $\YYM^{\mre}$-duality) in $\sfC(\YYM^{\mre}\Gr)$.

We leave it to prove the following lemma to the readers.

* $\sfG$ is a homotopy from $ZY $ to $0$.

* The morphism  $Y^{\vee}(-2)[1]: Q_{(1)}^{\vee} (-2)[1]
\to Q_{(2)}^{\vee} (-2)[1]
$ corresponds to
$Z$ under the isomorphisms (<ref>).

* The morphism  $Z^{\vee}(-2)[1]: Q_{(0)}^{\vee} (-2)[1]
\to Q_{(1)}^{\vee} (-2)[1]
$ corresponds to
$Y$ under the isomorphisms (<ref>).

* The morphism  $\sfG^{\vee}(-2)[1]:
Q_{(0)}^{\vee}(-2)[1] \to Q_{(2)}^{\vee}(-2)[1]$ corresponds to
$\sfG$ under the isomorphisms (<ref>).

In other   words,
we have the following diagram in $\sfC(\YYM^{\mre}Gr)$

\begin{equation}\label{20200731222611}
\begin{xymatrix}@C=60pt@R=6mm{
Q_{(0)}^{\vee}(-2)[1]\ar@/^30pt/[rr]^{0} \ar[r]^{Z^{\vee}(-2)[1]} \ar[d]_{\cong} &
Q_{(1)}^{\vee}(-2)[1] \ar@{=>}[u]_{\sfG^{\vee}(-2)[1]}
\ar[r]^{Y^{\vee}(-2)[1]} \ar[d]_{\cong}
Q_{(0)}^{\vee}(-2)[1] \ar[d]^{\cong}
\\
Q_{(2)} \ar[r]_{Y} \ar@/_30pt/[rr]_{0} & Q_{(1) }\ar[r]_{Z} \ar@{=>}[d]^{\sfG} & Q_{(0)}  \\
\end{equation}

Let $Q$ be the totalization of the lower half part of the above diagram (<ref>).
Namely, we set

\[
Q:= \left(
Q_{(0)}\oplus Q_{(1)}[1] \oplus Q_{(2)}[2],
\begin{pmatrix}
d & Z\uparrow & G\uparrow^{2} \\
0 & d & -\downarrow Y \uparrow^{2} \\
0 & 0 & d
\end{pmatrix}
\right).
\]

It follows from Lemma <ref> that the isomorphism isomorphisms (<ref>)
induces an isomorphism $Q^{\vee}(-2)[3] \cong Q$ in $\sfC(\YYM^{\mre})$.
It is clear that $Q$ is projectively cofibrant in $\sfC(\YYM^{\mre})$.
Therefore, to prove Theorem <ref>, it is enough to show that $Q$ is quasi-isomorphic to $\YYM$ in $\sfC(\YYM^{\mre})$.

We set $Q'_{(0)} :=\YYM \YYM= \YYM \otimes_{\Pa} \Pa\otimes_{\Pa} \YYM$ and
denote by $\mu': Q_{(0)} \to Q'_{(0)}$ the morphism induced from $\mu: \PPa \to \Pa$.
We set $Z' := \mu' Z, \ \sfG' := \mu' \sfG$
and $Q'$ to be the totalization of the diagram below.

\[
\begin{xymatrix}@C=60pt@R=6mm{
Q_{(2)} \ar[r]_{Y} \ar@/_30pt/[rr]_{0} & Q_{(1) }\ar[r]_{Z'} \ar@{=>}[d]^{\sfG'} & Q'_{(0)}  \\
\]

Since $\mu$ is a quasi-isomorphism of projectively cofibrant DG-modules over $\Pa^{\mre}$,
$\mu'$ is a quasi-isomorphism over $\YYM^{\mre}$
and hence  so is the morphism $Q \to Q'$ induced from $\mu'$.

Let $\epsilon: \YYM \YYM \to \YYM, \ x\otimes y \mapsto xy$ be the multiplication map.
Observe that $\epsilon Z' = 0, \epsilon \sfG = 0$ and hence that $\epsilon$ induces
a  morphism $\epsilon_{Q'}:=(\epsilon, 0,0): Q' \to \YYM$ in $\sfC(\YYM^{\mre})$.
We finish the proof by showing $\epsilon_{Q'}$ is a quasi-isomorphism.

We define increasing  filtrations on $Q'$ and $\YYM$ as follows.
For $i \in \ZZ$, we set

\[
F_{i}(Q_{(2)}) = \YYM_{\geq -i} \otimes_{\Pa} \PPi_{1}\otimes_{\Pa} \YYM,
F_{i}(Q_{(1)}) = \YYM_{\geq -i} \otimes_{\Pa} \YYM_{1}\otimes_{\Pa} \YYM,
F_{i}(Q'_{(0)}) = \YYM_{\geq -i}  \YYM,
\]

and $F_{i}(Q')$ denotes the induced filtration.
We set $F_{i}(\YYM) = \YYM_{\geq -i}$ for $i \in \ZZ$.
Then $\epsilon_{Q'}$ preserves filtrations.
For $i > 0$, the graded quotients $G_{i}(Q'), G_{i}(\YYM)$ are zero.
It is clear that $G_{0}(\YYM) \cong \Pa$ as  objects of $\sfC(\Pa\otimes\YYM^{\mre}\Gr)$.
On the other hand,  $0$-th graded quotient $G_{0}(Q')$ is isomorphic to the object $P$ of $\sfC(\Pa\otimes \YYM^{\mre}\Gr)$ defined in
Section <ref> and that the induced morphism $G_{0}(\epsilon_{Q'})$ corresponds to the quasi-isomorphism
$\epsilon_{P}: P \to \Pa$.
Observes that  for $i > 0$, we have an isomorphisms $G_{-i}(Q') \cong \YYM_{i} \otimes_{\Pa} P$ and $G_{-i}(\YYM) \cong \YYM_{i}$
in $\sfC(\Pa\otimes \YYM^{\mre}\Gr)$
and that the induced morphism $G_{-i}(\epsilon_{Q'})$ corresponds to $\YYM_{i} \otimes_{\Pa} \epsilon_{P}$.
Since $\YYM_{i}$ is projectively cofibrant, $\YYM_{i} \otimes_{\Pa} \epsilon_{P}$ is a quasi-isomorphism and so is $G_{-i}(\epsilon_{Q'})$.
Finally, since the filtration is exhausted and bounded in each $*$-degree,
we conclude that the morphism $\epsilon_{Q'}: Q'\to \YYM$ is
a quasi-isomorphism.

§.§ The exact triangle $\widehat{\sfU}$

The aim of this section is to show that there exists an exact triangle

\[
\widehat{\sfU}: \YYM( -1) \xrightarrow{ \widehat{\sfr_{\varrho}} } \YYM \xrightarrow{ \ppi } \PPi \to
\]

in $\sfD(\YYM^{\mre}\Gr)$ which is sent to $\sfU$  by the forgetful functor $\sfD(\YYM^{\mre}\Gr) \to \sfD(\YYM\Gr)$.
The reader can postpone this section until Section <ref>.

The problem here is that the morphism $\sfr_{\varrho}: \YYM( -1) \to \YYM$ dose not commute with the
right action of $\YYM$ on $\YYM$
and hence it is not a morphism of dg-modules over $\YYM^{\mre}$.
To overcome this problem, we use theory of A${}_{\infty}$-algebras for which we refer [37].

For simplicity we set $R := \YYM^{\mre}$.
We denote the category of $*$-graded A${}_{\infty}$-$R$-modules by $\sfC_{A_{\infty}}(R\Gr)$.
Recall that we regard a  $*$-graded dg-$R$-module $M$ as a $*$-graded A${}_{\infty}$-$R$-modules
by setting higher multiplication $\{ m^{M}_{n}\}_{n \geq 1}$ as

\[
m^{M}_{n} :=
\begin{cases}
d \ (\textup{the differential of $M$}) & ( n=1), \\
\textup{the left action map}: R\otimes M \to M & (n =2), \\
0 & (n \geq 3).
\end{cases}
\]

This assignment yields    a functor $\sfcan: \sfC(R\Gr) \to \sfC_{A_{\infty}}(R\Gr)$,
which induces an equivalence $\sfD(R\Gr) \xrightarrow{\cong} \sfD_{A_{\infty}}(R\Gr)$ of derived categories.
Thus we may identify the derived category $\sfD_{A_{\infty}}(R\Gr)$ of $*$-graded A${}_{\infty}$-$R$-modules
with the derived category $\sfD(R\Gr)$ of $*$-graded dg-$R$-modules.

A point here is that the functor $\sfcan$ is faithful, but  not full.
In other words,
even for $*$-graded dg-$R$-modules $M$ and $N$,
there exists a morphism $g:=\{g_{n}\}_{n \geq 1} : M \to N$ of $*$-graded A${}_{\infty}$-$R$-modules
which does not come from morphisms of $*$-graded dg-$R$-modules, i.e., $g_{n} \neq 0$ for some $n \geq 2$.

We define a morphism $\wsv=\{ f_{n}\}_{n \geq 1} : \YYM(-1) \to \YYM$ of $*$-graded A${}_{\infty}$-$R$-modules
whose $1$-st component  $f_{1}$ equals to $\sfr_{\varrho}$ in the following way.
So, first we set $f_{1}: = \sfr_{\varrho}$.
Next, we define a morphism $f_{2}: R \otimes \YYM(-1) \to \YYM$ of $\sfC_{\DG}(\kk\Gr)$ of cohomological degree $-1$ to be

\[
f_{2}((r \otimes s) \otimes x) := ( -1)^{|r|+ (|s| + 1)|x|} rx\sfH(s)
\]

for $ r \otimes s \in R$ and $x \in \YYM$.
Finally, for $n \geq 3$, we set $f_{n} := 0$.

The collection $\widehat{\sfr_{\varrho}} := \{f_{n}\}_{n \geq 1}$ is a morphism of $*$-graded A${}_{\infty}$-$R$-modules.

\[
\widehat{\sfr_{\varrho}}: \YYM(-1) \to \YYM.
\]

Since almost all the relevant morphisms are zero,
we only have to check the following three equations

\[
\begin{split}
f_{1}m^{\YYM}_{1} - m^{\YYM}_{1}f_{1}  &= 0,\\
f_{1} m^{\YYM}_{2} -m^{\YYM}_{2}({}_{R} f_{1}) &=  m^{\YYM}_{1} f_{2}+f_{2} (\sfd_{\YYM}) + f_{2}({}_{R}m^{\YYM}_{1}),\\
f_{2}({}_{R} m^{\YYM}_{2}) - f_{2} \sfm_{\YYM} &= -m^{\YYM}_{2} ({}_{R}f_{2})
\end{split}
\]

where  $\sfd: R \to R$ is the differential and $\sfm: R\otimes R \to R$ is the multiplication map.

The first one is just says that $f_{1} = \sfr_{\varrho}$ is a cochain map.
Thus it is already verified.

We set the left hand side and the right hand side of the second equation by $L$ and $R$.

\[
\begin{split}
L((r\otimes s) \otimes x) & = (-1)^{|s||x|}rxs \varrho - (-1)^{|s||x|}rx\varrho s= (-1)^{|s||x|} rx[s, \varrho], \\
R((r\otimes s) \otimes x)
& = d(( -1)^{|r|+ (|s| + 1)|x|} rx\sfH(s)) +\\
& \ \ \ \ \ \ \  f_{2}\left((dr \otimes s) \otimes x + (-1)^{|r|}(r \otimes ds) \otimes x +(-1)^{|r|+ |s|}(r \otimes s ) \otimes x \right)\\
&  = (-1)^{|s||x|}rx(d\sfH(s) +\sfH d(s))\\
& =(-1)^{|s||x|} rx[s, \varrho]
\end{split}
\]

This shows $L=R$.

We set the left hand side and the right hand side of the third equation by $L$ and $R$.

\[
\begin{split}
L((r\otimes s) \otimes (t \otimes u) \otimes x)
& = (-1)^{|x||u|} f_{2}((r\otimes s) \otimes txu) -(-1)^{|s||t|+|u||s|} f_{2}((rt \otimes us ) \otimes x) \\
& = (-1)^{|x||u|+|r| + (|t|+ |x| + |u|)(|s| +1) } r txu\sfH(s)  \\
& \ \ \ \ \ \ \  -(-1)^{|s||t|+ |u||s| + |r| + |t| + |x|(|s| +|u| + 1) } rt  x\sfH(us) \\
& = (-1)^{|s||t|+ |u||s| + |r| + |t| + |x|(|s| +|u| + 1) }\left(( -1)^{|u|} r txu\sfH(s) - rt  x\sfH(us) \right) \\
& = (-1)^{|s||t|+ |u||s| + |r| + |t| + |x|(|s| +|u| + 1) +1} rt  x\sfH(u)s,  \\
R((r\otimes s) \otimes (t \otimes u) \otimes x)
& =-(-1)^{|t| + (|u|+1)|x|+|r|+|s| } m_{2} \left( (r\otimes s) t x \sfH(u)\right) \\
& =(-1)^{|t| + (|u|+1)|x| +|r| + |s| +1+(|t| + |x| + |u| -1)|s| } rt x \sfH(u)s
\end{split}
\]
We check that the exponents of $(-1)$ of both equations coincide and verify $L =R$.

By the definition of composition of morphisms of A${}_{\infty}$-modules,
the composition
$\ppi \widehat{\sfr_{\varrho}} :\YYM \to \PPi$ is the collection  $\ppi \widehat{\sfr_{\varrho}} =\{ \ppi f_{n}\}_{n \geq 1}$.

Next  we construct a homotopy $\cH$ from $\ppi\wsv$ the composition  to 0.
We define morphisms $\cH_{n}: R^{\otimes n -1} \otimes \YYM \to \YYM$ in $\sfC_{\DG}(\kk\Gr)$ of cohomological  degree $ -n$
to be

\[
\cH_{n}: =
\begin{cases}
- \sfr_{\vars}\ppi & ( n = 1), \\
0 & ( n \geq 2).
\end{cases}
\]

The collection $\cH := \{ \cH_{n} \}_{n \geq 1}$
is a homotopy from $\ppi \wsv$ to $0$ of morphisms of A${}_{\infty}$-$R$-modules.

Since almost all the relevant morphisms are zero,
we only have to check the following two equations

\[
\begin{split}
\ppi f_{1}&= m^{\PPi}_{1}\cH_{1} + \cH_{1} m^{\YYM}_{1}, \\
\ppi f_{2}  &=  -m^{\PPi}_{2} ({}_{R} \cH_{1}) + \cH_{1} m^{\YYM}_{2}.\\
\end{split}
\]

The first one is already verified in Lemma <ref>.

We set the left hand side and the right hand side of the second equation by $L$ and $R$.
Then, we have

\[
\begin{split}
L((r\otimes s) \otimes x) &=
(-1)^{|r| +(|s| +1)|x|} \ppi(r) \ppi(x) \ppi(\sfH(s)) \\
& \stackrel{(*)}{=} (-1)^{|r| +(|s| +1)|x|} \ppi(rx) [\vars, \ppi(s)]\\
R((r\otimes s) \otimes x)
& = -(-1)^{|r|+ |s|}m^{\PPi}_{2}((r\otimes s) \otimes \cH_{1}(x) ) + (-1)^{|s||x|} \cH_{1}(r xs)\\
& = (-1)^{|r|+ |s|}m^{\PPi}_{2}((r\otimes s)\otimes  \sfr_{\vars}\ppi(x)) + (-1)^{|s||x|+1} \sfr_{\vars}\ppi(r xs)\\
& = (-1)^{|r|+|s| +|x| +(|x|+1)|s|}\ppi(r)\ppi(x)\vars  \ppi(s) + (-1)^{|s||x| +1 + |r| + |x| + |s|} \ppi(rxs) \vars\\
& = (-1)^{|r|+|x|+|s||x|}\left( \ppi(rx)\vars \ppi(s) - ( -1)^{|s|}\ppi(rx) \ppi(s) \vars\right)\\
& =  (-1)^{|r| +(|s| +1)|x|} \ppi(rx) [\vars, \ppi(s)]
\end{split}
\]

where for $\stackrel{(*)}{=}$ we use Lemma <ref>.
This shows $L =R$.

Let $C := \cone(\wsv)$ be the cone of $\wsv=\{ f_{n}\}_{n \geq 1} : \YYM \to \YYM$.
Namely, it is a $*$-graded A${}_{\infty}$-$R$-module (which is actually a $*$-graded dg-$R$-module) whose underlying cohomological graded object
is $\YYM \oplus \YYM[1]$ with the higher multiplications $\{m^{C}_{n} \}_{n \geq 1}$ given as below

\[
m^{C}_{1} = \begin{pmatrix} d_{\YYM} & f_{1} \uparrow\\ 0 & d_{\YYM[1]} \end{pmatrix}, \
m^{C}_{2} = \begin{pmatrix} m_{2}  &- f_{2} \uparrow \\ 0 & m_{2} \end{pmatrix}, \
m^{C}_{n } = 0  \ \ \ (n \geq 3).
\]

The homotopy $\cH$ induces a morphism $\hat{q}=\{ \hat{q}_{n}\}_{n \geq 1} : C\to \PPi$ of A${}_{\infty}$-$R$-modules,
which is defined by

\[
\hat{q}_{1} := (\ppi, \cH_{1} \uparrow), \ \hat{q}_{n} = 0  \ \ \ (n \geq 2).
\]

The morphism $\hat{q}: C \to \PPi$ is a quasi-isomorphism of A${}_{\infty}$-$\YYM^{\mre}$-modules.
Hence we have an exact triangle

\[
\widehat{\sfU}: \YYM( -1) \xrightarrow{ \widehat{\sfr_{\varrho}} } \YYM \xrightarrow{ \ppi } \PPi \to \YYM( -1)[1].
\]

in $\sfD(\YYM^{\mre}\Gr)$
which is sent to the exact triangle $\sfU$ by the forgetful functor $\sfD(\YYM^{\mre}\Gr) \to \sfD(\YYM\Gr)$.

Recall that a morphism $g=\{ g_{n}\}_{n \geq 1}: M \to N$ of A${}_{\infty}$-modules is called a quasi-isomorphism
if the $1$-st component $g_{1}$  is a quasi-isomorphism of complexes.
Since  $\hat{q}_{1} = q_{\sfr_{\varrho}, \ppi, - \sfr_{\vars} \ppi}$  is a quasi-isomorphism by Theorem <ref>,
the morphism $\hat{q}: C \to \PPi$ is a quasi-isomorphism.

The forgetful functor $\sfD(\YYM^{\mre}\Gr) \to \sfD(\YYM\Gr)$ is the restriction functor along the morphism
$a: \YYM \to \YYM^{\mre}, r \mapsto r \otimes 1$.
It follows that the morphism $\wsv: \YYM( -1) \to \YYM$ of $\sfC_{A_{\infty}}(\YYM^{\mre})$
is sent to $\sfr_{\varrho}: \YYM( -1) \to \YYM$.
Thus we conclude that the exact triangle $\widehat{\sfU}$ is sent to $\sfU$ by the forgetful functor $\sfD(\YYM^{\mre}\Gr) \to \sfD(\YYM\Gr)$.

§ QHA OF NON-DYNKIN TYPE  FOR NON-SINCERE WEIGHT

In this section, we study the  QHA ${}^{v}\!\YM$ of a non-Dynkin quiver $Q$ for arbitrary weight $v \in \kk Q_{0}$.
So throughout this section $Q$ denotes a non-Dynkin quiver.

As we point out in Proposition <ref> below,
the derived QHA ${}^{v}\!\YYM$ defined only for a sincere weight $v$,
is quasi-isomorphic to the (non-derived) QHA ${}^{v}\!\YM$.
The aim of this section is to show that almost all results about ${}^{v}\!\YM \simeq {}^{v}\!\YYM$ given in the previous section, holds true for ${}^{v}\!\YM$ even in the case where the weight $v$ is not sincere.

§.§ Preparations

First we point out that the derived QHA and the non-derived QHA are quasi-isomorphic.

Let $v \in \kk^{\times}Q_{0}$ be a sincere weight.
Then the derived QHA ${}^{v}\!\YYM$ is concentrated in the $0$-th cohomological grading and
consequently the canonical morphism ${}^{v}\!\YYM \to {}^{v}\!\YM$ is a quasi-isomorphism.

By Theorem <ref>, there exists an exact triangle
${}^{v}\!\YYM_{n} \xrightarrow{ {}^{v}\!\varrho} {}^{v}\!\YYM_{n+1} \xrightarrow{ {}^{v}\ppi } \PPi_{n+1} \to$
for $n \geq 0$.
Since $Q$ is non-Dynkin, $\PPi_{n}$ is concentrated in $0$-th cohomological degree.
Thus by induction on $n$, we can show that ${}^{v}\!\YYM_{n}$ is concentrated in $0$-th cohomological degree.

Thanks to  Theorem <ref>, we deduce the following theorem.

Let $Q$ be a non-Dynkin quiver and $v\in \kk^{\times} Q_{0}$ is sincere.
Then the QHA ${}^{v}\!\YM(Q)$ is $3$-Calabi-Yau.

§.§.§ Flatness of the deformation family $\Pi_{\bullet}$ of the preprojective algebras over $\kk Q_{0}$

Recall that the deformation family of the preprojective algebras is defined to be

\[
\Pi_{\bullet} :=\Pi(Q)_{\bullet} := \frac{\kk [x_{1}, \ldots, x_{r}]\overline{Q}}{ (\rho_{i} -x_{i}e_{i} \mid i \in Q_{0}) }
\]

where $r = \# Q_{0}$.
Let $R:=\kk [x_{1}, \dots, x_{r}]$  be the coordinate ring of $\kk Q_{0}$.
Then the deformation family $\Pi_{\bullet}$  is an algebra over $R$.

We give a proof of  the following well-known result.

The algebra $\Pi_{\bullet}$ is flat over $R$.

We may assume that $\kk$ is algebraically closed.
In this proof, we equip $\overline{Q}$ with the path length grading and $R$ is regraded as in $0$-th degree with respect to this grading.
Then the grading gives a filtration $\{\Pi_{\bullet, \leq n} \mid n \geq 0 \}$ on $\Pi_{\bullet}$.
Since it is exhaustive, it is enough to show that $\Pi_{\bullet, \leq n}$ is flat over $R$ for any $n \geq 0$.

For $\lambda=(\lambda_{i})_{i} \in  \kk Q_{0}$,
we denote by  $\Pi_{\lambda} := \kk \overline{Q}/(\rho_{i} -\lambda_{i} \mid i \in Q_{0})$ the deformed preprojective algebra.
Then we have $\Pi_{\bullet}\otimes_{R} \kappa(\lambda)  \cong \Pi_{\lambda}$
where $\kappa(\lambda) = R/(x_{i} -\lambda_{i} \mid i \in Q)$ is the residue field and we identify it with $\kk$.
The grading also gives a filtration  $\{ \Pi_{\lambda, \leq n} \mid n \geq 0\}$ on $\Pi_{\lambda}$ and
the above isomorphism compatible with filtrations.
Namely we have  $\Pi_{\bullet, \leq n }\otimes_{R} \kappa(\lambda)  \cong \Pi_{\lambda, \leq n}$ for $n \geq 0$.

By <cit.>, the $n$-th graded quotient $\Pi_{\lambda, \leq n}/\Pi_{\lambda, \leq n -1}$
is isomorphic to the $n$-th degree part $\Pi_{n}$ of $\Pi$ with respect to the path length grading.
It follows that  $\dim_{\kk} \Pi_{\lambda, \leq n}$ is independent on $\lambda$
and hence $\Pi_{\lambda, \leq n}$ is flat over $R$ by Lemma <ref>.

We recall from Lemma <ref> that
if a weight $v \in \kk^{\times} Q_{0}$ is  sincere, then

\[
{}^{v}\!\YM(Q) \cong \frac{\kk[z]\overline{Q}}{( \rho_{i} -v_{i}z e_{i} \mid i \in Q_{0} )}.
\]

We identify these two algebras.
In the case $v \in \kk Q_{0}$ is  not necessarily sincere, we set

\[
{}^{v}\!\YM(Q) := \frac{\kk[z]\overline{Q}}{( \rho_{i} -v_{i}z e_{i} \mid i \in Q_{0} )}.\]

We regard this algebra as $*$-graded algebra by setting
$\deg^{*} \alpha := 0, \ \deg^{*} \alpha^{*} :=1 \textup{ for } \alpha \in Q_{1}$ and $\deg^{*} z: =1$.