microscopic thing as the rate of elementary reaction, and the set of reactions
serves just for the approximation of the right hand side of the kinetic
equation. We cannot fully disprove this point of view but can just say that in
some cases the collision-based approach with physically distinguished
elementary reactions is based on the solid experimental and theoretical
background. If the elementary reactions physically exist then the detailed
balance for $T$-invariant systems is proved.

##  8 Conclusions

We present the general formalism of the Quasiequilibrium approximation (QE)
with the proof of the persistence of entropy production in the QE
approximation (Section 2).

We demonstrate how to apply this formalism to chemical kinetics and give
several examples for the Mass Action law kinetic equation. We discuss the
difference between QE and Quasi-Steady-State (QSS) approximations and analyze
the classical Michaelis-Menten and Briggs-Haldane model reduction approaches
(Section 3). After that, we use ideas of Michaelis, Menten and Stueckelberg to
create a general approach to kinetics.

Let us summarize the main results of our discussion. First of all, we believe
that this is the finish of the Michaelis-Menten-Stueckelberg program. The
approach to modeling of the reaction kinetics proposed by Michaelis and Menten
in 1913 MichaelisMenten1913  for enzyme reactions was independently in 1952
applied by Stueckelberg Stueckelberg1952  to the Boltzmann equation.

The idea of the complex balance (cyclic balance) relations was proposed by
Boltzmann as an answer to the Lorentz objections against Boltzmann’s proof of
the $H$-theorem. Lorentz stated that the collisions of the polyatomic
molecules may have no inverse collisions. Cercignani and Lampis
CercignaniLamp1981  demonstrated that the Boltzmann $H$-theorem based on the
detailed balance conditions is valid for the polyatomic molecules under the
microreversibility conditions and this new Boltzmann’s idea was not needed.
Nevertheless, this seminal idea was studied further by many authors
Heitler1944 ; Coester1951 ; Watanabe1955  mostly for linear systems.
Stueckelberg Stueckelberg1952  proved these conditions for the Boltzmann
equation. He used in his proof the $S$-matrix representation of the micro-
kinetics.

Some consequences of the Stueckelberg approach were rediscovered for the Mass
Action law kinetics by Horn and Jackson in 1972 HornJackson1972  and
supplemented by the “zero deficiency theorem” Feinberg1972 . This is the
history.

In our work, we develop the Michaelis-Menten-Stueckelberg approach to general
kinetics. This is a combination of the QE (fast equilibria) and the QSS (small
amounts) approaches to the real or hypothetical intermediate states. These
intermediate states (compounds) are included in all elementary reactions (46)
as it is illustrated in Figure 1. Because of the small amount, the free energy
for these compounds $B_{i}$ is perfect (48), the kinetics of compounds is the
first order Markov kinetics and satisfies the master equation.

After that, we use the combination of QE and QSS approximations and exclude
the concentrations of compounds. For the general kinetics the main result of
this approach is the general kinetic law (74). Earlier, we just postulated
this law because of its convenient and natural form G1 ; BykGOrYab1982 , now
we have the physical framework where this law can be proved.

We do not assume anything about reaction rates of the main reactions (17). We
use only thermodynamic equilibrium, the hypothesis about fast equilibrium with
compounds and the smallness of concentration of compounds. This smallness
implies the perfect entropy and the first order kinetics for compounds. After
that, we get the reaction rate functions from the qualitative assumptions
about compounds and the equilibrium thermodynamic data.

For example, if we relax the assumption about fast equilibrium and use just
smallness of compound concentrations (the Briggs-Haldane QSS approach
BriggsHaldane1925 ; Aris1965 ; Segel89 ) then we immediately need the formulas
for reaction rates of compound production. Equilibrium data become
insufficient. If we relax the assumption about smallness of concentrations
then we lose the perfect entropy and the first order Markov kinetics. So, only
the combination of QE and QSS gives the desired result.

For the kinetics of rarefied gases the mass action law for elastic collisions
(the Boltzmann equation) or for inelastic processes like chemical reactions
follows from the “molecular chaos” hypothesis and the low density limits. The
Michaelis-Menten-Stueckelberg approach substitutes low density of all
components by low density of the elementary events (or of the correspondent
compounds) together with the QE assumption.

The general kinetic law has a simple form: For an elementary reaction

$\sum_{i}\alpha_{i}A_{i}\to\sum_{i}\beta_{i}A_{i}$

the reaction rate is $r=\varphi\Omega$, where $\Omega>0$ is the Boltzmann
factor, $\Omega=\exp\left(\sum_{i}\alpha_{i}\check{\mu}_{i}\right)$,
$\check{\mu}_{i}=-\partial S/\partial N_{i}$ is the chemical potential $\mu$
divided by $RT$, and $\varphi\geq 0$ is the kinetic factor. Kinetic factors
for different reactions should satisfy some conditions. Two of them are
connected to the basic physics:

  * •

The detailed balance: The kinetic factors for mutually reverse reaction should
coincide, $\varphi^{+}=\varphi^{-}$. This identity is proven for systems with
microreversibility (Section 7.5).

  * •

The complex balance: The sum of the kinetic factors for all elementary
reactions of the form $\sum_{i}\alpha_{i}A_{i}\to\ldots$ is equal to the sum
of the kinetic factors for all elementary reactions of the form
$\ldots\to\sum_{i}\alpha_{i}A_{i}$ (91). This identity is proven for all
systems under the Michaelis-Menten-Stueckelberg assumptions about existence of
intermediate compounds which are in fast equilibria with other components and
are present in small amounts.

For the general kinetic law we studied several sufficient conditions of
accordance between thermodynamics and kinetics: Detailed balance, complex
balance and $G$-inequality.

In the practice of modeling, a kinetic model may, initially, do not respect
thermodynamic conditions. For these cases, we solved the problem of whether it
is possible to add a linear function to entropy in order to provide agreement
with the given kinetic model and deformed thermodynamics. The answer is
constructive (Section 7) and allows us to prove the general algebraic
conditions for the detailed and complex balance.

Finally, we have to mention that Michaelis, Menten and Stueckelberg did not
prove their “big theorem”. Michaelis and Menten did not recognize that their
beautiful result of mass action law produced from the equilibrium relations
between substrates and compounds, the assumption about smallness of compound
concentrations and the natural hypothesis about linearity of compound kinetics
is a general theorem. Stueckelberg had much more and fully recognized that his
approach decouples the Boltzmann $H$-theorem and the microreversibility
(detailed balance). This is important because for every professional in
theoretical physics it is obvious that the microreversibility cannot be
important necessary condition for the $H$-theorem. Entropy production should
be positive without any relation to detailed balance (the proof of the
$H$-theorem for systems with detailed balance is much simpler but it does not
matter: Just the Markov microkinetics is sufficient for it). Nevertheless,
Stueckelberg did not produce the generalized mass action law and did not
analyze the general kinetic equation. Later, Horn, Jackson and Feinberg
approached the complex balance conditions again and studied the generalized
mass action law but had no significant interest in the microscopic assumptions
behind these properties. Therefore, this paper is the first publication of the
Michaelis-Menten-Stueckelberg theorem.

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## Appendix

## 1\. Quasiequilibrium Approximation

### 1.1. Quasiequilibrium Manifold

Let us consider a system in a domain $U$ of a real vector space $E$ given by
differential equations

$\frac{{\mathrm{d}}x}{{\mathrm{d}}t}=F(x)$ (112)

We assume that for any $x_{0}\in U$, solution $x(t;x_{0})$ to the initial
problem $x(0)=x_{0}$ for (112) exists for all $t>0$ and belongs to $U$. Shifts
in time, $x_{0}\mapsto x(t;x_{0})$ ($t>0$), form a semigroup in $U$.

We do not specify the space $E$ here. In general, it may be any Banach or even
more general space. For nonlinear operators we will use the Fréshet
differentials: For an operator $\Psi(x)$ the differential at point $x$ is a
linear operator $(D\Psi)_{x}$:

$(D\Psi)_{x}(y)=\left.\frac{{\mathrm{d}}\Psi(x+\alpha
y)}{{\mathrm{d}}y}\right|_{\alpha=0}$

We use also notation $(D_{x}\Psi)$ when it is necessary to stress the choice
of independent variable. The choice of variables is not obvious.

The QE approximation for (112) uses two basic entities: entropy and slow
variables.

Entropy $S$ is a concave Lyapunov function with non-degenerated Hessian for
(3) which increases in time:

$\frac{{\mathrm{d}}S}{{\mathrm{d}}t}\geq 0\,$ (113)

In this approach, the increase of the entropy in time is exploited (the Second
Law in the form (4)).

Formally, any Lyapunov function may be used. Nevertheless, most of famous
entropies, like the relative Boltzmann-Gibbs-Shannon entropy, the Rényi
entropy, the Burg entropy, the Cressie-Read and the Tsallis entropies could be
defined as universal Lyapunov functions for Markov chains which satisfy some
natural additivity conditions GorbanGorbanJudge2010 .

“Universal” means that they do not depend on kinetic coefficients directly but
only on the equilibrium point. The “natural additivity conditions” require
that these entropies can be represented by sums (or integrals) over states
maybe after some monotonic transformation of the entropy scale, and, at the
same time, are additive with respect to the joining of statistically
independent systems (maybe, after some monotonic rescaling as well).

Slow variables $M$ are defined as some differentiable functions of variables
$x$: $M=m(x)$. We use notation $E_{M}$ for the space of slow variables, $M\in
E_{M}$. Selection of the slow variables implies a hypothesis about separation
of fast and slow motion. In its strongest form it consists of two assumptions:
The slaving assumption and the assumption of small fast-slow projection.

The slaving assumption. For any admissible initial state $x_{0}\in U$ after
some relatively small time $\tau$ (initial layer), solution $x(t;x_{0})$
becomes a function of $M$ (up to a given accuracy $\epsilon$) and can be
represented in a slaving form:

$x(t)=x^{*}_{M(t)}+\delta(t)\;\;{\rm for}\;\;t>\tau\,,\;\;{\rm
where}\;\;M(t)=m(x(t)),\;\|\delta(t)\|<\epsilon\,$ (114)

This means that everything is a function of slow variables, after some initial
time and up to a given accuracy.

The smallness of $\tau$ is essential. If there is no restriction on $\tau$
then every globally stable system will satisfy this assumption because after
some time it will arrive into a small vicinity of equilibrium.

The second assumption requires that the slow variables (almost) do not change
during the fast motion: During the initial layer $\tau$, the state $x$ can
change significantly because of fast motion, but the change in $M=m(x)$ during
$\tau$ are small with $\tau$: The assumption of small fast-slow projection.

The QE approximation defines the functions $x^{*}_{M}$ as solutions to the
following MaxEnt optimization problem:

$S(x)\to\max\;\;\mbox{subject to}\;\;m(x)=M\,$ (115)

The reasoning behind this approximation is simple: During the fast initial
layer motion, entropy increases and $M$ almost does not change. Therefore, it
is natural to assume (and even to prove using smallness of $\tau$ and
$\epsilon$ if the entropy gradient in fast directions is separated from zero)
that $x^{*}_{M}$ in (114) is close to solution to the MaxEnt optimization
problem (115). Further, $x^{*}_{M}$ denotes a solution to the MaxEnt problem.

Some additional conditions on $m$ and $S$ are needed for the regularity of the
dependence $x^{*}_{M}$ on $M$. It is more convenient to discuss these
conditions separately for more specific systems. In general settings, let us
just assume that for given $S$ and $m$ the dependencies $m(x)$ and $x^{*}_{M}$
are differentiable. For their differentials we use the notations $(Dm)_{x}$
and $(Dx^{*}_{M})_{M}$. The differentials are linear operators: $(Dm)_{x}:E\to
E_{M}$ and $(Dx^{*}_{M})_{M}:E_{M}\to E$.

A solution to (115), $x^{*}_{M}$, is the QE state, the corresponding value of
the entropy

$S^{*}(M)=S(x^{*}_{M})$ (116)

is the QE entropy and the equation

$\frac{{\mathrm{d}}M}{{\mathrm{d}}t}=(Dm)_{x^{*}_{M}}(F(x^{*}_{M}))$ (117)

represents the QE dynamics.

Remark. The strong form of the slaving assumption, “everything becomes a
function of the slow variables”, is too strong for practical needs. In
practice, we need just to have a “good” dependence on $M$ for the time
derivative ${\mathrm{d}}M/{\mathrm{d}}t$. Moreover, the short-time
fluctuations of ${\mathrm{d}}M/{\mathrm{d}}t$ do not affect the dependence
$M(t)$ too much, and only the average values

$\langle\dot{M}\rangle_{\theta}(t)=\frac{1}{\theta}\int_{t}^{t+\theta}\frac{{\mathrm{d}}M}{{\mathrm{d}}t}$

for sufficiently small time scale $\theta$ are important.

### 1.2. Preservation of Entropy Production

Theorem about preservation of entropy production. Let us calculate
${\mathrm{d}}S^{*}(M)/{\mathrm{d}}t$ at point $M$ according to the QE dynamics
(117) and find ${\mathrm{d}}S(x)/{\mathrm{d}}t$ at point $x=x^{*}_{M}$ due to
the initial system (3). The results always coincide:

$\frac{{\mathrm{d}}S^{*}(M)}{{\mathrm{d}}t}=\frac{{\mathrm{d}}S(x)}{{\mathrm{d}}t}\,$
(118)

The left hand side in (118) is computed due to the QE approximation (117) and
the right hand side corresponds to the initial system (112). Here, this
theorem is formulated in more general setting than in Section 2: The slow
variables $M=m(x)$ may be nonlinear functions of $x$. For more details about
QE approximation with nonlinear dependencies $M=m(x)$ we refer to papers
GorKarQE2006 ; GorKarProjector2004 ; GorKarPRE1996 . The general theorem about
preservation of entropy production and thermodynamic projector is presented in
GorKarProjector2004 .

###### Proof.

To prove this identity let us mention that

$\frac{{\mathrm{d}}S^{*}(M)}{{\mathrm{d}}t}=(DS^{*})_{M}\left(\frac{{\mathrm{d}}M}{{\mathrm{d}}t}\right)=(DS^{*})_{M}\circ(Dm)_{x^{*}_{M}}(F(x^{*}_{M}))\,$
(119)

where $\circ$ stands for superposition. On the other hand, just from the
definitions of the differential and of the time derivative of a function due
to a system of differential equations, we get

$\frac{{\mathrm{d}}S(x)}{{\mathrm{d}}t}=(DS)_{x}(F(x))\,$ (120)

To finalize the proof, we need an identity

$(DS^{*})_{M}\circ(Dm)_{x^{*}_{M}}=(DS)_{x^{*}_{M}}\,$ (121)

Let us use the Lagrange multipliers representation of the MaxEnt problem:

$(DS)_{x}=\Lambda_{M}\circ(Dm)_{x}\,,\;\;m(x)=M\,$ (122)

This system of two equations has two unknowns: The vector of state $x$ and the
linear functional $\Lambda_{M}$ on the space of slow variables (the Lagrange
multiplier), which depends on $M$ as on a parameter.

By differentiation of the second equation $m(x)=M$, we get an identity

$(Dm)_{x^{*}_{M}}\circ(Dx^{*}_{M})_{M}={\rm id}_{E_{M}}\,$ (123)

where ${\rm id}$ is the unit operator.

Lagrange multiplier $\Lambda_{M}$ is the differential of the QE entropy:

$(DS^{*})_{M}=\Lambda_{M}\,$ (124)

Indeed, due to the chain rule,
$(DS^{*})_{M}=(DS)_{x^{*}_{M}}\circ(Dx^{*}_{M})_{M}$, due to (122),
$(DS)_{x}=\Lambda_{M}(Dm)_{x}$ and, finally

$\begin{split}(DS^{*})_{M}=&(DS)_{x^{*}_{M}}\circ(Dx^{*}_{M})_{M}=\Lambda_{M}\circ(Dm)_{x}\circ(Dx^{*}_{M})_{M}\\\
=&\Lambda_{M}\circ{\rm id}_{E_{M}}=\Lambda_{M}\,\end{split}$

Now we can prove the identity (121):

$(DS^{*})_{M}\circ(Dm)_{x^{*}_{M}}=\Lambda_{M}\circ(Dm)_{x^{*}_{M}}=(DS)_{x^{*}_{M}}$

(here we use the Lagrange multiplier form (122) again). ∎

The preservation of the entropy production leads to the preservation of the
type of dynamics: If for the initial system (112) entropy production is non-
negative, ${\mathrm{d}}S/{\mathrm{d}}t\geq 0$, then for the QE approximation
(117) the production of the QE entropy is also non-negative,
${\mathrm{d}}S^{*}/{\mathrm{d}}t\geq 0$.

In addition, if for the initial system
$({{\mathrm{d}}S}/{{\mathrm{d}}t})_{x}=0$ if and only if $F(x)=0$ then the
same property holds in the QE approximation.

## 2\. First Order Kinetics and Markov Chains

First-order kinetics form the simplest and well-studied class of kinetic
systems. It includes the continuous-time Markov chains (the master equation
VanKampen1981 ), kinetics of monomolecular and pseudomonomolecular reactions
LumpWei2 , and has many other applications.

We consider a general network of linear reactions. This network is represented
as a directed graph (digraph) (Yablonskii1991 ; Temkin1996 ): Vertices
correspond to components $B_{j}$ ($1\leq j\leq q$), edges correspond to
reactions $B_{j}\to B_{l}$. For each vertex $B_{j}$ a positive real variable
$\varsigma_{j}$ (concentration) is defined. For each reaction $B_{j}\to B_{l}$
a rate constant $\kappa_{lj}>0$ is given. To follow the standard notation of
the matrix multiplication, the order of indexes in $\kappa_{ji}$ is always
inverse with respect to reaction: It is $\kappa_{j\leftarrow i}$, where the
arrow shows the direction of the reaction. The kinetic equations for
concentrations $\varsigma_{j}$ have the form

$\frac{{\mathrm{d}}\varsigma_{j}}{{\mathrm{d}}t}=\sum_{l,\,l\neq
j}\left(\kappa_{jl}\varsigma_{l}-\kappa_{lj}\varsigma_{j}\right)$ (125)

The linear conservation law (for the Markov chains this is the conservation of
the total probability) is:

$\sum_{j}\frac{{\mathrm{d}}\varsigma_{j}}{{\mathrm{d}}t}=0\;\;\mbox{{\em
i.e.},}\;\;\sum_{j,l,\,l\neq
j}\left(\kappa_{jl}\varsigma_{l}-\kappa_{lj}\varsigma_{j}\right)=0\,.$ (126)

Let a positive vector $\varsigma^{*}$ ($\varsigma^{*}_{j}>0$) be an
equilibrium for the system (125): For every $j=1,\ldots,q$

$\sum_{l,\,l\neq j}\kappa_{jl}\varsigma_{l}^{*}=\sum_{l,\,l\neq
j}\kappa_{lj}\varsigma_{j}^{*}$ (127)

An equivalent form of (125) is convenient. Let us use the equilibrium
condition (127) and write

$\sum_{l,\,l\neq j}\kappa_{lj}\varsigma_{j}=\left(\sum_{l,\,l\neq
j}\kappa_{lj}\varsigma_{j}^{*}\right)\frac{\varsigma_{j}}{\varsigma_{j}^{*}}=\sum_{l,\,l\neq
j}\kappa_{jl}\varsigma_{l}^{*}\frac{\varsigma_{j}}{\varsigma_{j}^{*}}$

Therefore, under condition (127) the master equation (125) has the equivalent
form:

$\frac{{\mathrm{d}}\varsigma_{j}}{{\mathrm{d}}t}=\sum_{l,\,l\neq
j}\kappa_{jl}\varsigma_{l}^{*}\left(\frac{\varsigma_{l}}{\varsigma_{l}^{*}}-\frac{\varsigma_{j}}{\varsigma_{j}^{*}}\right)$
(128)

The following theorem Morimoto1963  describes the large class of the Lyapunov
functions for the first order kinetics. Let $h(x)$ be a smooth convex function
on the positive real axis. A Csiszár–Morimoto function $H_{h}(\varsigma)$ is
(see the review GorbanGorbanJudge2010 ):

$H_{h}(\varsigma)=\sum_{l}\varsigma_{l}^{*}h\left(\frac{\varsigma_{l}}{\varsigma_{l}^{*}}\right)$

The Morimoto $H$-theorem. The time derivative of $H_{h}(\varsigma)$ due to
(125) under condition (127) is nonpositive:

$\frac{{\mathrm{d}}H_{h}(\varsigma)}{{\mathrm{d}}t}=\sum_{l,j,\,j\neq
l}h^{\prime}\left(\frac{\varsigma_{j}}{\varsigma_{j}^{*}}\right)\kappa_{jl}\varsigma^{*}_{l}\left(\frac{\varsigma_{l}}{\varsigma_{l}^{*}}-\frac{\varsigma_{j}}{\varsigma_{j}^{*}}\right)\leq
0$ (129)

###### Proof.

Let us mention that for any $q$ numbers $h_{i}$, $\sum_{i,j,\,j\neq
i}\kappa_{ij}\varsigma^{*}_{j}(h_{j}-h_{i})=0$. Indeed, for
$h_{i}=p_{i}/p_{i}^{*}$ this is precisely the condition of conservation of the
total probability for equations (128). The extension from a simplex of the
$h_{i}$ values ($\sum_{i}p^{*}_{i}h_{i}=1$, $h_{i}\geq 0$) to the positive
orthant $\mathbb{R}^{q}_{+}$ is trivial because of uniformity of the identity.
Finally, if a linear identity holds in a positive orthant then it holds in the
whole space $\mathbb{R}^{q}$. Therefore,

$\sum_{l,j,\,j\neq
l}h^{\prime}\left(\frac{\varsigma_{j}}{\varsigma_{j}^{*}}\right)\kappa_{jl}\varsigma^{*}_{l}\left(\frac{\varsigma_{l}}{\varsigma_{l}^{*}}-\frac{\varsigma_{j}}{\varsigma_{j}^{*}}\right)=\sum_{l,j,\,j\neq
l}\kappa_{jl}\varsigma^{*}_{l}\left[h\left(\frac{\varsigma_{j}}{\varsigma_{j}^{*}}\right)-h\left(\frac{\varsigma_{l}}{\varsigma_{l}^{*}}\right)+h^{\prime}\left(\frac{\varsigma_{j}}{\varsigma_{j}^{*}}\right)\left(\frac{\varsigma_{l}}{\varsigma_{l}^{*}}-\frac{\varsigma_{j}}{\varsigma_{j}^{*}}\right)\right]\leq
0$

The last inequality holds because of the convexity of $h(x)$:
$h^{\prime}(x)(y-x)\leq h(y)-h(x)$ (Jensen’s inequality).∎

For example, for the convex function $h(x)=x(\ln x-1)$ the Csiszár–Morimoto
function is:

$H_{h}(\varsigma)=\sum_{l}\varsigma_{l}\left(\ln\left(\frac{\varsigma_{l}}{\varsigma^{*}_{l}}\right)-1\right)$
(130)

This expression coincides with the perfect component of the free energy (50)
(to be more precise, $f=RTH_{h}(\varsigma)$).

Each positive equilibrium $\varsigma^{*}$ belongs to the ray of positive
equilibria $\lambda\varsigma^{*}$ ($\lambda>0$). We can select a
$l_{1}$-normalized direction vector and write for an equilibrium
$\varsigma^{\rm eq}$ from this ray:

$\varsigma^{\rm eq}=\beta\frac{\varsigma^{*}}{\sum_{j}\varsigma^{*}_{j}}\,,$

where $\beta=\sum_{j}\varsigma^{\rm eq}_{j}$.

The kinetic equations (125) allow one and only one ray of positive equilibria
if and only if the digraph of reactions is strongly connected: It is possible
to reach any vertex starting from any other vertex by traversing edges in the
directions in which they point. Such continuous-time Markov chains are called
ergodic chains MeynMarkCh2009 .

Let us assume that the system is weakly reversible: For any two vertices
$B_{i}$ and $B_{j}$, the existence of an oriented path from $B_{i}$ to $B_{j}$
implies the existence of an oriented path from $B_{j}$ to $B_{i}$. Under this
assumption the graph of reactions is a unit of strongly connected subgraphs
without connections between them. Let the graph of reactions $B_{j}\to B_{l}$
have $d$ strongly connected components $C_{s}$ and let $V_{s}$ be the set of
indexes of those $B_{j}$ which belong to $C_{s}$: $B_{j}\in C_{s}$ if and only
if $j\in V_{s}$. For each $s=1,\ldots,d$ there exists a conservation law

$\beta_{s}(\varsigma)=\sum_{j\in V_{s}}\varsigma_{j}=const$ (131)

For any set of positive values of $\beta_{s}>0$ ($s=1,\ldots,d$) there exists
a unique equilibrium of (125), $\varsigma^{\rm eq}$, which is positive
($\varsigma^{\rm eq}_{j}>0$). This equilibrium can be expressed through any
positive equilibrium $\varsigma^{*}$:

$\varsigma^{\rm eq}_{j}=\beta_{s}\frac{\varsigma_{j}^{*}(c,T)}{\sum_{l\in
V_{s}}\varsigma_{j}^{*}(c,T)}$ (132)

For positive $\beta_{s}$, logarithms of $\varsigma^{\rm eq}$ form a
$d$-dimensional linear manifold in $R^{q}$:

$\ln\varsigma^{\rm
eq}_{j}=\ln\beta_{s}+\ln\left(\frac{\varsigma_{j}^{*}(c,T)}{\sum_{l\in
V_{s}}\varsigma_{l}^{*}(c,T)}\right)$ (133)

The natural coordinates on this manifold are $\ln\beta_{s}$.

Remark. In the construction of the free energy (130) any positive equilibrium
state $\varsigma^{\rm eq}$ can be used. The correspondent functions differs in
an additive constant. Let us calculate the difference between two “free
energies”:

$\begin{split}&\sum_{j=1}^{q}\varsigma_{j}\left(\ln\left(\frac{\varsigma_{j}}{\varsigma_{j}^{*}}\right)-1\right)-\sum_{j=1}^{q}\varsigma_{j}\left(\ln\left(\frac{\varsigma_{j}}{\varsigma_{j}^{\rm
eq}}\right)-1\right)\\\
&=\sum_{j=1}^{q}\varsigma_{j}\ln\left(\frac{\varsigma_{j}^{\rm
eq}}{\varsigma_{j}^{*}}\right)=\sum_{s=1}^{d}\beta_{s}(\varsigma)\ln\left(\frac{\beta_{s}(\varsigma^{\rm
eq})}{\beta_{s}(\varsigma^{*})}\right)\end{split}$ (134)

The result is constant in time for the solutions of the master equation (125),
hence, these functions are equivalent: They both are the Lyapunov functions
for (125) and have the same conditional minimizers for given values of
$\beta_{s}>0$ ($s=1,\ldots,d$).


1.5pt

# Middle terms of AR-sequences of graded Kronecker modules

Jie Liu  SUSTech International Center For Mathematics, Southern University of
Science and Technology, shenzhen 518055, China  Christian-Albrechts-
Universität zu Kiel, Heinrich-Hecht-Platz 6, 24118 Kiel, Germany

###### Abstract.

Let $(T(n),\Omega)$ be the covering of the generalized Kronecker quiver
$K(n)$, where $\Omega$ is a bipartite orientation. Then there exists a
reflection functor $\sigma$ on the category $\operatorname{mod}(T(n),\Omega)$.
Suppose that $0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0$ is an AR-
sequence in the regular component $\mathcal{D}$ of
$\operatorname{mod}(T(n),\Omega)$, and $b(Z)$ is the number of flow modules in
the $\sigma$-orbit of $Z$. Then the middle term $Y$ is a sink (source or flow)
module if and only if $\sigma Z$ is a sink (source or flow) module. Moreover,
their radii and centers satisfy $r(Y)=r(\sigma Z)+1$ and $C(Y)=C(\sigma Z)$.

###### Key words and phrases:

reflection functor; sink module; radii

<EMAIL_ADDRESS>
Tel.:+86 0755-88011089 Fax: +86 0755-88010224

2020 Mathematics subject classification: 16G20; 16G70

##  1\. introduction

Let $k$ be a field. Recall that the generalized Kronecker quiver $K(n)$ is
just the quiver with two vertices $1$, $2$, and $n$ arrows
$\gamma_{1},\cdots,\gamma_{n}:$ $1\rightarrow 2$. Let $\mathcal{K}_{n}=kK(n)$,
and let $\operatorname{rep}_{k}(K(n))$ denote the finite-dimensional
representations of $K(n)$. If we use $\operatorname{mod}\mathcal{K}_{n}$ to
denote the category of finite-dimensional modules of $\mathcal{K}_{n}$, then
there exists an equivalence between categories $\operatorname{rep}_{k}(K(n))$
and $\operatorname{mod}\mathcal{K}_{n}$. We will frequently identify these two
subcategories.

Since the representation type of $K(n)$ is wild [Kerner, 1.3], it is hopeless
to classify all the indecomposable modules. Hence it is prevalent to
investigate the module category of Kronecker quiver concerning the invariants.
If we let $(T(n),\Omega)$ be the universal covering of $K(n)$, then one
practical way is to find invariants for those modules of $K(n)$ which can be
lifted to its covering $(T(n),\Omega)$, where $\Omega$ is a fixed bipartite
orientation (see (2.2)). In fact, $T(n)$ is an $n$-regular tree. In general,
we can take $k$-modules of the quiver $T(n)$ as graded Kronecker modules
(cf.[Claus]). We write $\operatorname{mod}(T(n),\Omega)$ to denote the
category of finite-dimensional graded Kronecker modules (or simply modules).
There exists a reflection (or shift) functor
$\sigma:\mod(T(n),\Omega)\rightarrow\mod(T(n),\sigma\Omega)$ at all sinks,
where $\sigma\Omega$ is the orientation under the action of $\sigma$.
Similarly, we have reflection functor $\sigma^{\prime}$ for
$\operatorname{mod}\mathcal{K}_{n}$. Using the push-down functor
$\pi_{\lambda}:\operatorname{mod}(T(n),\Omega)\rightarrow\operatorname{mod}\mathcal{K}_{n}$
[Bongartz, 3.2], we show that
$\pi_{\lambda}\circ\sigma(M)=\sigma^{\prime}\circ\pi_{\lambda}(M)$ for all
$M\in\operatorname{mod}(T(n),\Omega)$. Let
$M\in\operatorname{mod}(T(n),\Omega)$ be an indecomposable module. We say that
$M$ is regular, provided $\sigma^{t}(M)\neq 0$ for all $t\in\mathbb{Z}$.

Let $M\in\operatorname{mod}(T(n),\Omega)$ be a regular indecomposable module,
and let $T(M)$ be the minimal tree containing the set $\\{y\in T(n)_{0}\mid
M_{y}\neq 0\\}$. In 2018, Claus Michael Ringel introduces three types of
modules: sink module, source module and flow module for $T(M)$ and presents
two invariants: the smallest possible radius and the center path in the
$\sigma$-orbit of module $M$ (see (2.5) and Section 3). Moreover, a module
$Y\in\operatorname{mod}(T(n),\Omega)$ is said to be a sink (source, or flow)
module if all its indecomposable direct summands are sink (source, or flow)
modules with the same center. In the end of the paper [Claus], Ringel asks
readers a question: for an AR-sequence $0\rightarrow X\rightarrow Y\rightarrow
Z\rightarrow 0$ consisting of regular modules, what the precise relationship
is between the middle term $Y$ and $\sigma Z$. Since we know a little about
the reflection functor $\sigma$ in his paper [Claus], it is hard to answer
that question. In this article, we follow his work and build a bigger ball
(see (2.5)) that could let us see clearly the properties of reflection functor
$\sigma$ and finally give a solution to Ringel’s problem. In particular, we
can write down the functor $\sigma$ individually for every indecomposable
module.

We show that the functor $\sigma$ induces an equivalence between $k$-linear
full subcategories of $\operatorname{mod}(T(n),\Omega)$ and
$\operatorname{mod}(T(n),\sigma\Omega)$ with quasi-inverse $\sigma^{-}$. This
conclusion guarantees the indecomposability of regular modules of the category
$\operatorname{mod}(T(n),\Omega)$ under the action of $\sigma$. Furthermore,
it allows us to consider AR-sequences in the regular components and give the
clues to solve Ringel’s problem directly.

In Section 3, we focus on a regular indecomposable module $M$ of
mod$(T(n),\Omega)$. It is also well-known that the regular components (or AR-
components) of $K(n)$ are components of type $\mathbb{Z}A_{\infty}$
(cf.[Claus1]), and it is the same for the covering $T(n)$. Suppose that
$\mathcal{D}$ is a regular component of mod$(T(n),\Omega)$. Then for an AR-
sequence $0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0$ in
$\mathcal{D}$, the module $Y$ is a sink (source, or flow) module if and only
if $\sigma Z$ is a sink (source, or flow) module.

##  2\. preliminaries

In this section, we present a few concepts and some basic results. For
convenience, we will give some definitions in a short way. A thorough
introduction to this part can be found in [Assem1, 2-7], [Claus] or [Daniel].
Throughout, $k$ will denote a field.

(2.1) Auslander-Reiten theory Given a finite-dimensional algebra $\mathcal{A}$
over $k$, let $\operatorname{mod}\mathcal{A}$ denote the category of finite-
dimensional right $\mathcal{A}$-modules. Let
$X,Y\in\operatorname{mod}\mathcal{A}$. We say that a homomorphism
$f:X\rightarrow Y$ is irreducible, provided $f$ is neither a split
monomorphism nor a split epimorphism, and if $f=f_{1}\circ f_{2}$, then either
$f_{1}$ is a split epimorphism or $f_{2}$ is a split monomorphism. Let
$\operatorname{rad}_{\mathcal{A}}$ be the radical of
$\operatorname{mod}\mathcal{A}$. Then we can define the set

$\operatorname{Irr}(X,Y):=\operatorname{rad}_{\mathcal{A}}(X,Y)/\operatorname{rad}^{2}_{\mathcal{A}}(X,Y)$

and call it the space of irreducible morphisms [Assem1, 4.1.6].

Let $\operatorname{mod}\mathcal{A}^{op}$ denote the $\mathcal{A}$-dual
category of $\operatorname{mod}\mathcal{A}$. We introduce the
$\mathcal{A}$-dual functor

$(-)^{t}=$
Hom${}_{\mathcal{A}}(-,\mathcal{A}):\operatorname{mod}\mathcal{A}\rightarrow\operatorname{mod}\mathcal{A}^{op}$.

Let $M\in\operatorname{mod}\mathcal{A}$, and let
$P_{1}\xrightarrow{p_{1}}P_{0}\xrightarrow{p_{0}}M\rightarrow 0$ be a minimal
projective representation of $M$. Using functor $(-)^{t}$, we can get an exact
sequence of left $\mathcal{A}$-modules

$0\rightarrow
M^{t}\xrightarrow{p^{t}_{0}}P^{t}_{0}\xrightarrow{p^{t}_{1}}P^{t}_{1}\rightarrow\text{Coker
}p^{t}_{1}\rightarrow 0$.

We define $\operatorname{Tr}M:=$ Coker $p^{t}_{1}$ and call it the transpose
of $M$. When $M$ is indecomposable but not projective, we have
$\operatorname{Tr}(\operatorname{Tr}M)\cong M$ [Assem1, 4.2.1$(c)$]. Let
$D=\operatorname{Hom}_{k}(-,k)$. The Auslanter-Reiten translation in
$\operatorname{mod}\mathcal{A}$ is defined by the composition of
$\operatorname{Tr}$ and $D$, that is, $\tau=D\circ\operatorname{Tr}$ and
$\tau^{-1}=\operatorname{Tr}\circ D$.

Let $N\in\operatorname{mod}\mathcal{A}$ be an indecomposable module. We call
$N$ preinjective provided $\tau^{-t}N=0$ for some $t\in\mathbb{N}_{0}$.
Dually, module $N$ is called preprojective provided $\tau^{t}N=0$ for some
$t\in\mathbb{N}_{0}$. Finally, module $N$ is called regular if $N$ is not
preinjective or preprojective.

In fact, there exists no nonzero morphism from preinjecitve modules to
preprojective or regular modules. When $N$ is decomposable, we sometimes say
that $N$ is still regular, provided there does not exist indecomposable direct
summand which is preinjective or preprojective in $N$.

  * Definition.

The Auslander-Reiten quiver $\Gamma(\mathcal{A})$ of $\mathcal{A}$ is defined
as follows:

    1. $(a)$

The points $\Gamma(\mathcal{A})_{0}$ correspond to the isomorphism classes
$[X]$ of indecomposable modules $X$ in $\operatorname{mod}\mathcal{A}$.

    2. $(b)$

Let $[X],[Y]$ be two points in $\Gamma(\mathcal{A})_{0}$. Then there are
$\dim_{k}\operatorname{Irr}(X,Y)$ arrows from $[X]$ to $[Y]$ in
$\Gamma(\mathcal{A})_{1}$.

Let $X,Y\in\operatorname{mod}\mathcal{A}$ be indecomposable modules. We say
that $X$ is a predecessor of $Y$, provided there exists a directed path from
$[X]$ to $[Y]$ in $\Gamma(\mathcal{A})$, i.e. a chain of irreducible maps from
$X$ to $Y$. We say that $X$ is a sucessor of $Y$, provided there exists a
directed path from $[Y]$ to $[X]$ in $\Gamma(\mathcal{A})$, i.e. a chain of
irreducible maps from $Y$ to $X$.

  * Definition.

A connected component $\mathcal{D}$ of $\Gamma(\mathcal{A})$ is called
preprojective, preinjective, or regular, provided modules contained in it are
all preprojective, preinjective, or regular.

We use $(\rightarrow Y)$ to denote the subset of $\Gamma(\mathcal{A})_{0}$
consisting of $Y$ and all its predecessors, and $(Y\rightarrow)$ to denote the
subset of $\Gamma(\mathcal{A})_{0}$ consisting of $Y$ and all its sucessors,
respectively.

Let $X$ be a non-projective indecomposable module (or let $Y$ be a non-
injective indecomposable module). Then there exists a uniquely determined
short exact sequence, called Auslander-Reiten sequence, or simply AR-sequence
(or almost split sequence)

$0\rightarrow
Y\xrightarrow{f}\oplus^{t}_{i=1}M^{n_{i}}_{i}\xrightarrow{g}X\rightarrow 0$,

where $Y$ (or $X$) is indecomposable, modules $M_{i}$ are pairwise non-
isomorphic and indecomposable. Suppose that $f=\begin{bmatrix}f_{1}\\\
\vdots\\\
f_{t}\end{bmatrix},g=\begin{bmatrix}g_{1},\cdots,g_{t}\end{bmatrix},$ where
$f_{i}=\begin{bmatrix}f_{i_{1}}\\\ \vdots\\\
f_{in_{i}}\end{bmatrix},g_{i}=\begin{bmatrix}g_{i1},\cdots,g_{in_{i}}\end{bmatrix}$.
Then the maps $f_{i_{1}},f_{i_{2}},\cdots,f_{i_{n_{i}}}:Y\rightarrow M_{i}$
and $g_{i_{1}},\cdots,g_{i_{n_{i}}}:M_{i}\rightarrow X$ correspond to the
bases of Irr$(Y,M_{i})$ and Irr$(M_{i},X)$, respectively. We write $Y=\tau X$
(or $X=\tau^{-}Y$) and we denote this in $\Gamma(\mathcal{A})$ by
$[Y]\dashleftarrow[X]$. Note that we sometimes don’t distinguish $X$ (or $Y$)
and its isomorphism class $[X]$ (or $[Y]$) in $\Gamma(\mathcal{A})$. It is
well-known that the regular components of $\Gamma(\mathcal{K}_{n})$ are the
type of $\mathbb{Z}A_{\infty}$ (cf.[Claus1]), and they are of the following
form (see FIGURE 1).

$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$$\cdots$$\cdots$$\vdots$$\cdots$$\cdots$$\vdots$
Figure 1. Regular component $\mathbb{Z}A_{\infty}.$

A module $M$ is called  quasi-simple if it is located in the bottom row of a
$\mathbb{Z}A_{\infty}$ component. Let $M$ be a module in a regular
$\mathbb{Z}A_{\infty}$-component $\mathcal{C}$. Then there exists a quasi-
simple module $X\in\mathcal{C}$ (or $Y\in\mathcal{C}$) such that we can find a
chain of irreducible monomorphisms $X=X_{1}\rightarrow\cdots\rightarrow
X_{s-1}\rightarrow X_{s}=M$ (or irreducible epimorphisms $M=Y_{s}\rightarrow
Y_{s-1}\rightarrow\cdots\rightarrow Y_{1}=Y$). We call $s$ the quasi-length of
$M$ and the module $X$ (or $Y$) quasi-socle (or quasi-top ) of $M$. We call
the factors $X_{i}/X_{i-1}$ or $Y_{i}/Y_{i-1}$ the quasi-composition factors
of $M$. It can be shown that $M$ is uniquely determined by its quasi-length
and quasi-socle (or quasi-top), whence we can define $ql(M):=s=$ quasi-length
of $M$.

(2.2) Graphs A graph is a pair $G=(G_{0},G_{1}):$ $G_{0}$ (whose elements
called the vertices) and $G_{1}$ (whose elements called the edges). If
$\\{a,b\\}$ is an edge, then $a,b$ are called neighbours.

For a graph $G$, there exsits a map $\Omega$, called orientation
$:G\rightarrow G_{0}\times G_{0}$, such that $\Omega(\\{a,a^{\prime}\\})$ is
either ($a,a^{\prime}$) or ($a^{\prime},a$). We call $(G,\Omega)$ an oriented
graph. We write $a\rightarrow a^{\prime}$ if
$\Omega(\\{a,a^{\prime}\\})=(a,a^{\prime})$ and call $a$ the start,
$a^{\prime}$ the target. Meanwhile we call $a$ a sink (or a source) if it is
not a start (or the target, respectively) of any arrow. If any vertex is a
sink or a source, then the orientation $\Omega$ will be called bipartite. A
subgraph of a graph $G=(G_{0},G_{1})$ is a graph
$G^{\prime}=(G^{\prime}_{0},G^{\prime}_{1})$ such that
$G^{\prime}_{0}\subseteq G_{0},G^{\prime}_{1}\subseteq G_{1}$.

A path of length $t\geq 0$ in a graph $G$ is a finite sequence
$(a_{0},a_{1},\cdots,a_{t})$ of vertices such that $a_{i-1},a_{i}$ are
neighbours for $1\leq i\leq t$, and $a_{j-1}\neq a_{j+1}$ for $1\leq j<t$. If
$t=2r$, then $a_{r}$ is called its center and $r$ its radius [Claus, 2.1]. If
$t=2r+1$, then $\\{a_{r},a_{r+1}\\}$ is called its center and $r$ its radius.
Suppose that for any pair $a,b$ of vertices in $G$, there always exists a path
connecting $a$ and $b$. Then $G$ is said to be connected. We call a path of
length $t\geq 0$ a cycle whenever its source and target coincide. A cycle of
length $1$ is said to be a loop. A graph $G$ is called acyclic if it contains
no cycles, and it is called finite if $G_{0}$ and $G_{1}$ are finite sets.

A graph $G$ is said to be a tree if it is connected and paths connecting a
vertex with itself are always length $0$. Let $G$ be a tree. Then we define

$d(a,b):=$ the length of a path connecting $a$ and $b$.

This is well-defined since there exists only one path from $a$ to $b$ in the
tree $G$. If $G^{\prime}$ is a subset of tree $G$ and $x\in G$, we define

$d(x,G^{\prime}):=$min$\\{d(x,a)\mid a\in G^{\prime}\\}$.

For every finite tree $G$, the paths of maximal possible length $d(G)$ are
called diameters of $G$. Let $r(G)$ denote the radius of $G$. It can be shown
that for a finite tree $G$, all diameter paths of $G$ have the same center
$C(G)$ [Claus, 2.1]. Suppose that $(a_{0},a_{1},\cdots,a_{t})$ is a diameter
path of $G$. We define,

$C(G)=\begin{cases}a_{r}&t=2r,\\\ \\{a_{r},a_{r+1}\\}&t=2r+1.\end{cases}$

(2.3) Quivers A quiver is just an oriented graph (loops and multiple arrows
are allowed), usually denoted by $Q=(Q_{0},Q_{1},s,t)$, where
$s,t:Q_{1}\rightarrow Q_{0}$. For every $x\in Q_{0}$, we define a set
$\mathcal{N}(x):=\\{y\in Q_{0}\mid\exists\alpha\in Q_{1}:t(\alpha)=y\text{
when }s(\alpha)=x\text{ or }s(\alpha)=y\text{ when }t(\alpha)=x\\}$ and call
it the neighbourhood of $x$. When $\mid\mathcal{N}(x)\mid\leq 1$, We say that
$x$ is a leaf. A finite-dimensional representation $M=((M_{x})_{x\in
Q_{0}},(M(\alpha))_{\alpha\in Q_{1}})$ over $Q$ consists of vector spaces
$M_{x}$ and $k$-linear maps $M(\alpha):M_{s(\alpha)}\rightarrow M_{t(\alpha)}$
such that $\dim_{k}M:=\sum_{x\in Q_{0}}\dim_{k}M_{x}$ is finite. We thus
define a category Rep${}_{k}(Q)$ of $k$-linear representations of $Q$. We
denote by $\operatorname{rep}_{k}(Q)$ the full subcategory of Rep${}_{k}(Q)$
consisting of finite-dimensional representations.

Let $Q=(Q_{0},Q_{1},s,t)$ be a finite, connected, and acyclic quiver [Assem1,
7.5], and let $n=|Q_{0}|$. For every point $a\in Q_{0}$, we define a new
quiver $\sigma_{a}Q=(Q^{\prime}_{0},Q^{\prime}_{1},s^{\prime},t^{\prime})$ as
follows: all the arrows of $Q$ having a as source or as target are reversed,
all other arrows remain unchanged. Hence there exists a reflection functor
$\mathcal{S}^{+}_{a}$ (or $\mathcal{S}^{-}_{a}$) : rep${}_{k}(Q)\rightarrow$
rep${}_{k}(Q^{\prime})$ when $a$ is a sink (or source). In fact, the functors
$\mathcal{S}^{+}_{a}$ and $\mathcal{S}^{-}_{a}$ induce equivalences between
the $k$-linear full subcategories of $\operatorname{mod}(kQ)$ and
$\operatorname{mod}(kQ^{\prime})$ consisting of regular modules [Assem1, 7.
Corollary 5.7]. In particular, functors $\mathcal{S}^{+}_{a}$ and
$\mathcal{S}^{-}_{a}$ send regular indecomposable modules to regular
indecomposable modules. If $M\in$ mod$(kQ)$ is a regular indecomposable
module, then $\mathcal{S}^{-}_{a}\mathcal{S}^{+}_{a}(M)\cong M$.

(2.4) Push-down functor and graded Kronecker modules Now we focus on the
generalized Kronecker quiver $K(n)$ ($n\geq 3$).

${1\circ}$${\circ
2.}$$\scriptstyle{\vdots}$$\scriptstyle{\gamma_{1}}$$\scriptstyle{\gamma_{n}}$

Let $\mathcal{K}_{n}=kK(n)$. Let $\epsilon_{i}$ correspond to the trivial path
point $i$, $i\in\\{1,2\\}.$ In the following, we give a brief construction of
the covering $T(n)$ of Kronecker quiver $K(n)$ [Daniel, 7.1]. Let
$Q=(Q_{0},Q_{1},s,t)$ be a quiver. We define formal inverse $(Q_{1})^{-1}$ on
$Q_{1}$, where

  1. $(a)$

$(Q_{1})^{-1}:=\\{\alpha^{-1}\mid\alpha\in Q_{1}\\}$;

  2. $(b)$

$s(\alpha^{-1}):=t(\alpha)$ and $t(\alpha^{-1}):=s(\alpha)$ for any
$\alpha^{-1}\in(Q_{1})^{-1}$;

  3. $(c)$

we say that $\omega$ is a walk of $Q_{1}$, provided there exists
$\omega=\alpha^{\varepsilon_{n}}_{n}\cdots\alpha^{\varepsilon_{1}}_{1}$, where
$\alpha_{i}\in Q_{1}$, $\varepsilon\in\\{1,-1\\}$ and
$s(\alpha^{\varepsilon_{i+1}}_{i+1})=t(\alpha^{\varepsilon_{i}}_{i})$ for all
$i<n$;

  4. $(d)$

$s(\omega):=s(\alpha^{\varepsilon_{1}}_{1})$ and
$t(\omega):=t(\alpha^{\varepsilon_{n}}_{n})$.

Let $W:=\\{\text{walks of $K(n)$}\\}$. We introduce an equivalence relation
$\sim$ on $W$, which is generated by
$\gamma^{-1}_{i}\gamma_{i}\sim\epsilon_{1}$ and
$\gamma_{i}\gamma^{-1}_{i}\sim\epsilon_{2}$. We define an involution
$(-)^{-1}$ on $W$, where

$(-)^{-1}$: $W\rightarrow
W;(\alpha^{\varepsilon_{n}}_{n}\cdots\alpha^{\varepsilon_{1}}_{1})^{-1}\mapsto\alpha^{-\varepsilon_{1}}_{1}\cdots\alpha^{-\varepsilon_{n}}_{n}$.

Let $\pi(K(n))$ be the fundamental group of $K(n)$ in the point 1, i.e.
$\pi(K(n))=\\{[\alpha]\mid s(\alpha)=t(\alpha)=1\\}$, where $[\alpha]$ is the
equivalence classe of unoriented path $\alpha$. The multiplication is the
concatenation of paths. We define [$\omega$]-1:=[$\omega^{-1}$] and the
identity element is [$\epsilon_{1}$]. Then the quiver $T(n)$ is given by the
following data:

  1. $(a)$

$(T(n))_{0}$ is the set of equivalence classes of walks starting in 1.

  2. $(b)$

When $\omega^{{}^{\prime}}\sim\gamma_{i}\omega$ for some
$i\in\\{1,\cdots,r\\}$, we say that there exists an arrow from [$\omega$] to
[$\omega^{{}^{\prime}}$].

Let $\pi:T(n)\rightarrow K(n);[\omega]\mapsto
t(\omega),([\omega]\rightarrow[\gamma_{i}\omega])\mapsto\gamma_{i}$. Action of
group $G=\pi(K(n))$ on $T(n)$ is the concatenation of paths:

$g.[v]=[v\omega^{-1}]$ and
$g.([u]\rightarrow[\gamma_{i}u])=([u\omega^{-1}]\rightarrow[\gamma_{i}u\omega^{-1}])$,

where $g=[\omega]\in\pi(K(n))$ and $[v],[u]\in T(n)_{0}$ with arrow
$[u]\rightarrow[\gamma_{i}u]=[v]$. Finally, we define
$T^{+}_{n}:=\pi^{-1}(\\{1\\})$, $T^{-}_{n}:=\pi^{-1}(\\{2\\})$. Let $\Omega$
be the bipartite orientation on $T(n)$. Then for $M\in$
rep${}_{k}(T(n),\Omega)$, the group $G$ acts on $M$ via:

$M^{g}:=(((M^{g})_{x})_{x\in T(n)_{0}}$, $(M^{g}(\alpha))_{\alpha\in
T(n)_{1}})$,

where $(M^{g})_{x}:=M_{g.x}$ and $M^{g}(\alpha):=M(g.\alpha)$.

Now we fix the bipartite orientation $\Omega$ for $T(n)$, and we define a
push-down functor $\pi_{\lambda}:$ rep${}_{k}(T(n),\Omega)\rightarrow$
rep${}_{k}(K(n));M\mapsto(\pi_{\lambda}(M)_{j},\pi_{\lambda}(M)(\gamma_{i}))$
factoring through $T(n)/G$, that is, $\pi_{\lambda}(M^{g})=\pi_{\lambda}(M)$,
$\forall g\in G$, where

$\pi_{\lambda}(M)_{j}:=\bigoplus_{\pi(y)=j}M_{y},$

$\pi_{\lambda}(M)(\gamma_{i}):=\bigoplus_{\pi(\beta)=\gamma_{i}}M(\beta):\pi_{\lambda}(M)_{1}\rightarrow\pi_{\lambda}(M)_{2},j\in\\{1,2\\},1\leq
i\leq n.$

Note that $\operatorname{rep}_{k}(T(n),\Omega)$ also admits AR-sequences and
the functor $\pi_{\lambda}$ is exact (cf.[Bongartz, 2.2]).

For any two sinks $x,y\in T(n)_{0}$, we can see that
$\sigma_{x}\sigma_{y}=\sigma_{y}\sigma_{x}$ by the definition. Then we can
define a reflection functor
$\sigma:\operatorname{rep}_{k}(T(n),\Omega)\rightarrow\operatorname{rep}_{k}(T(n),\sigma\Omega)$
as the composition of the $\sigma_{x}$, for all sinks $x$. The functor
$\sigma$ is independent of the order used in the composition, so that $\sigma$
is well defined. We sometimes call such functor $\sigma$ the shift functor. By
[Claus, 2.6], the composition $\sigma^{2}$ is the translation $\tau$.

We call modules in $\operatorname{rep}_{k}(T(n),\Omega)$ the graded Kronecker
modules (or graded modules, or simply modules). We often use
mod$(T(n),\Omega)$ to denote those modules. Without specific emphasis, we
always mean quiver $(T(n),\Omega)$ when we talk about the tree $T(n)$. An
indecomposable module $M\in\operatorname{mod}(T(n),\Omega)$ is said to be
regular if $\sigma^{t}(M)\neq 0$ for any $t\in\mathbb{Z}$. Let
$\sigma^{\prime}=\mathcal{S}^{+}_{2}$ and
$\sigma^{\prime-}=\mathcal{S}^{-}_{1}$ for $K(n)$.

######  Lemma 1.

Let $M\in\operatorname{mod}(T(n),\Omega)$ be an indecomposable module. Then
$\pi_{\lambda}(\sigma M)=\sigma^{\prime}(\pi_{\lambda}(M))$ and
$\pi_{\lambda}(\sigma^{-}M)=\sigma^{\prime-}(\pi_{\lambda}(M))$.

###### Proof.

We only prove $\pi_{\lambda}(\sigma M)=\sigma^{\prime}(\pi_{\lambda}(M))$, and
it is similar for the other one. If $\sigma M=0$, then $\pi_{\lambda}(M)$ is
isomorphic to a projective module, whence there is nothing to prove. Suppose
that $\sigma M\neq 0$. We need to show that the following diagram commutes.

${\text{mod}(T(n),\Omega)}$${\text{mod
}\mathcal{K}_{n}}$${\text{mod}(T(n),\sigma\Omega)}$${\text{mod
}\mathcal{K}_{n}}$$\scriptstyle{\pi_{\lambda}}$$\scriptstyle{\sigma}$$\scriptstyle{\sigma’}$$\scriptstyle{\pi_{\lambda}}$

Following the definition, we have

$\pi_{\lambda}(M)_{j}=\bigoplus_{\pi(y)=j}M_{y}$

$\pi_{\lambda}(M)(\gamma_{i})=\bigoplus_{\pi(\beta)=\gamma_{i}}M(\beta):\pi_{\lambda}(M)_{1}\rightarrow\pi_{\lambda}(M)_{2},$

for $j\in\\{1,2\\}$ and $1\leq i\leq n$. Let $N=\sigma\pi_{\lambda}(M)$, where

$N_{2}=\pi_{\lambda}(M)_{1}=\bigoplus_{\pi(y)=1}M_{y},$

and

$\begin{split}N_{1}&=(\sigma\pi_{\lambda}(M))_{1}\\\ &=\text{ker}\
(\bigoplus_{\gamma_{j}}\pi_{\lambda}(M)_{s(\gamma_{j})}\rightarrow\pi_{\lambda}(M)_{2})\\\
&=\text{ker}\
(\bigoplus_{\gamma_{j}}\bigoplus_{\pi(\beta)=\gamma_{j}}M_{s(\beta)}\rightarrow\bigoplus_{\pi(z)=2}M_{z}).\end{split}$

Each $N(\gamma_{i})$ is the composition of the inclusion of $N_{1}$ into
$\bigoplus_{\gamma_{j}}\pi_{\lambda}(M)_{s(\gamma_{j})}$ with the projection
onto the direct summand $\pi_{\lambda}(M)_{s(\gamma_{i})}$ [Assem1, 7.5.5.].
On the other hand, let $N^{\prime}=\sigma^{\prime}(M)$ and
$N^{\prime\prime}=\pi_{\lambda}(N^{\prime})$. Note that,

$N^{\prime\prime}_{2}=\bigoplus_{\pi(y^{\prime})=2}N^{\prime}_{y^{\prime}}=\bigoplus_{\pi(y)=1}M_{y}=N_{2},$

and

$\begin{split}N^{\prime\prime}_{1}&=\bigoplus_{\pi(y^{\prime})=1}N^{\prime}_{y^{\prime}}\\\
&=\bigoplus_{\pi(y^{\prime})=1}{\sigma(M)}_{y^{\prime}}\\\
&=\bigoplus_{\pi(z)=2}\text{ker}\ (\bigoplus_{\beta:s(\beta)\rightarrow
z}M_{s(\beta)}\rightarrow M_{z})\\\ &=\text{ker}\
(\bigoplus_{\gamma_{j}}\bigoplus_{\pi(\beta)=\gamma_{j}}M_{s(\beta)}\rightarrow\bigoplus_{\pi(z)=2}M_{z})\\\
&=\text{ker}\
(\bigoplus_{\gamma_{j}}\pi_{\lambda}(M)_{s(\gamma_{j})}\rightarrow\pi_{\lambda}(M)_{2})\\\
&=N_{1}.\end{split}$

Each $N^{\prime\prime}(\gamma_{i})$ is the composition of inclusion of
$N^{\prime\prime}_{1}=N_{1}$ into
$\bigoplus_{\gamma_{j}}\pi_{\lambda}(M)_{s(\gamma_{j})}$ with the projection
onto the direct summand $\pi_{\lambda}(M)_{s(\gamma_{i})}$. Then we know that

$N({\gamma_{i}})=N^{\prime\prime}({\gamma_{i}})$, $1\leq i\leq n$.

Hence $N=N^{\prime\prime}$.

∎

######  Lemma 2.

Functors $\sigma$ and $\sigma^{-}$ send regular indecomposable modules to
regular indecomposable modules.

###### Proof.

Let $M\in\operatorname{mod}(T(n),\Omega)$ be a regular indecomposable module.
According to [Claus4, 6.3], we can see that
$\sigma^{\prime}(\pi_{\lambda}(M))$ is regular indecomposable. However, Lemma
1 tells us that $\pi_{\lambda}(\sigma M)=\sigma^{\prime}(\pi_{\lambda}(M))$,
whence $\sigma M$ is regular indecomposable. For the functor $\sigma^{-}$, the
proof is similar. ∎

(2.5) Index and Balls Let $\varnothing\neq S\subseteq T(n)_{0}$ be a set of
vertices, and let $M\in\operatorname{mod}(T(n),\Omega)$.

  1. •

The unique minimal tree in $T(n)$ containing $S$ is denoted by $T(S)$.

  2. •

A vertex $x\in T(S)_{0}$ is said to be a leaf of $T(S)$ if
$\mid\mathcal{N}(x)\cap T(S)_{0}\mid\leq 1$.

  3. •

The set $\operatorname{supp}(M):=\\{y\in T(n)_{0}\mid M_{y}\neq(0)\\}$ is
called the support of $M$, and the vertex $x$ is said to be a leaf $M$ if $x$
is a leaf of $T(M):=T(\text{supp}(M))$.

Let $M\in\operatorname{mod}(T(n),\Omega)$ be an indecomposable module. Since
$\dim_{k}M<\infty$, $T(M)=T(\operatorname{supp}(M))$ is a finite tree. Using
the fact that all diameter paths of a finite tree have the same center
(cf.[Claus]), we can define

$d(M):=$ the length of diameter paths of $M$,

$C(M):=$ the center of diameter paths of $M$,

$r(M):=$ the radius of diameter paths of $M$.

We call them the diameter, center, and radius of $M$, respectively [Claus,
2.4]. Actually, we have $d(M)=d(T(M))$, $C(M)=C(T(M))$ and $r(M)=r(T(M))$. An
indecomposable module $M$ is said to be a sink (source) module if a diameter
path (and thus all) of $T(M)$ starts and ends in sinks (sources). Otherwise we
call module $M$ a flow module. For any $x,c,c_{1},c_{2}\in T(n)_{0}$ and
$r\in\mathbb{N}$ [Claus, 2.3], let $B_{r}(c)=\\{x\in T(n)_{0}\mid d(x,c)\leq
r\\}$, $B_{r}(c_{1},c_{2})=\\{x\in T(n)_{0}\mid d(x,c_{i})\leq r\\}$,
$i\in\\{1,2\\}$. We call such sets $B_{r}(c)$ and $B_{r}(c_{1},c_{2})$ the
balls with radius $r$ and with center $c$ or $\\{c_{1},c_{2}\\}$,
respectively.

We now recall some important results in [Claus].

######  Theorem 2.1.

[Claus, Theorem 1] Let $M$ be a regular indecomposable module of
mod$(T(n),\Omega)$. Then the shift orbit of $M$ contains a unique sink module
$M_{0}$ with smallest possible radius, saying with radius $r_{0}=r_{0}(M)$.
Let $M_{i}=\sigma^{i}M_{0}$ for all $i\in\mathbb{Z}$. Then there is an integer
$0\leq b\leq r_{0}(M)$ and a path $(a_{0},\cdots,a_{b})$ in $T(n)$ with the
following properties:

  * (1)

For $i\geq 0$, the module $M_{-i}$ is a sink module with radius $r_{0}+i$ and
center $a_{0}$.

  * (2)

For $1\leq i\leq b(M)$, the module $M_{i}$ is a flow module with radius
$r_{0}-1$ and center $\\{a_{i-1},a_{i}\\}$.

  * (3)

For $i\geq 0$, the module $M_{b+1+i}$ is a source module with radius $r_{0}+i$
and center $a_{b}$.

The integer $r_{0}(M)$ is positive. If $r_{0}(M)$ is even, then $a_{0}$ is a
sink, otherwise a source.

Let $M\in\operatorname{mod}(T(n),\Omega)$ be a regular indecomposable module.
According to Theorem 2.1, all the modules $M_{i}$ with $i\leq 0$ are sink
modules. Moreover, we define its index $\iota(M):=t$ when
$M=\sigma^{t}M_{0},t\in\mathbb{Z}$.

######  Theorem 2.2.

[Claus, Theorem 3] If $M$, $M^{\prime}\in$ mod$(T(n),\Omega)$ are regular
indecomposable modules and $M\rightarrow M^{\prime}$ is an irreducible map,
then $\iota(M^{\prime})=\iota(M)-1$.

Note that the regular Auslander-Reiten components of the category
$\operatorname{mod}(T(n),\Omega)$ are of the form $\mathbb{Z}A_{\infty}.$ We
now introduce an operator $\eta$ on the set of isomorphism classes of regular
indecomposable modules. If $\mathcal{D}$ is a regular component of
$\operatorname{mod}(T(n),\Omega)$ and $0\rightarrow X\rightarrow Y\oplus
Y^{\prime}\rightarrow Z\rightarrow 0$ is an Auslander-Reiten sequence in
$\mathcal{D}$ with $Y,Y^{\prime}$ indecomposable and $r(Y)<r(Y^{\prime})$,
then we define $\eta(Y)=Y^{\prime}$.

######  Theorem 2.3.

[Claus, Theorem 4] Let $Y\in$ mod$(T(n),\Omega)$ be an indecomposable regular
module. Then $Y$ is a sink module (or a flow module, or a source module), if
and only if $\eta Y$ is a sink module (or a flow module, or a source module,
respectively). Also, $\eta Y$ has the same center as $Y$ and $r(\eta
Y)=r(Y)+2$.

######  Proposition 2.4.

[Claus, 3.1] Let $M\in$ mod$(T(n),\Omega)$ be a sink module with center $c$ ,
then

$d(M)-2\leqslant d(\sigma M)\leqslant d(M)$.

There are following three possibilities:

  1. (1)

$d(\sigma M)=d(M)-2$ if and only if $\sigma M$ is a sink module. In this case,
the center of $\sigma M$ is $c$.

  2. (2)

$d(\sigma M)=d(M)-1$ if and only if $\sigma M$ is a flow module. The center of
$\sigma M$ is of the form $\\{c,c^{{}^{\prime}}\\}$ with a path
$(x_{0},x_{1},\cdots,x_{r})$, where $x_{0}=c,x_{1}=c^{{}^{\prime}}$ such that
$x_{r}$ is a source for $\sigma\Omega$.

  3. (3)

$d(\sigma M)=d(M)$ if and only if $\sigma M$ is a source module. In this case,
the center of $\sigma M$ is $c$ again.

######  Proposition 2.5.

[Claus, 3.2] Let $M\in$ mod$(T(n),\Omega)$ be a flow module . We assume that
$(a_{0},\cdots,a_{d})$ with $d=2r+1$ is a diameter path of $M$ with $a_{0}$ a
sink and $a_{d}$ a source. Then

$d(M)\leqslant d(\sigma M)\leqslant d(M)+1.$

There are the following two possibilities:

  1. (1)

$d(M)=d(\sigma M)$, thus $\sigma M$ is a flow module. Let $a_{d+1}$ be a
neighbour of $a_{d}$ different from $a_{d-1}$. Then $(a_{1},\cdots,a_{d+1})$
is a diameter path for $\sigma M$. In particular, the center of $\sigma M$ is
$\\{a_{r+1},a_{r+2}\\}$ (and we note that $a_{1}$ is a sink for $\sigma\Omega$
and $a_{d+1}$ is a source for $\sigma\Omega$).

  2. (2)

$d(\sigma M)=d(M)+1$, then $\sigma M$ is a source module and the center of
$\sigma M$ is $a_{r+1}.$

######  Proposition 2.6.

[Claus, 3.3] Let $M\in$ mod$(T(n),\Omega)$ be a source module with center $c$
and radius $r$. Then $\sigma M$ is a source module with center $c$ and radius
$r+1.$

Let $M\in\operatorname{mod}(T(n),\Omega)$ be an indecomposable module. Suppose
that $c$, or $\\{c_{1},c_{2}\\}$, and $r$ are the center and radius of $M$,
respectively. We define

$B(M):=B_{r}(c)$, or $B(M):=B_{r}(c_{1},c_{2})$.

Clearly, sets $T(M)$ and $B(M)$ have the same radius and same center. On the
other hand, we can introduce a new set $B_{+1}(M)=\\{x\in(T(n),\Omega)_{0}\mid
d(x,B(M))\leq 1\\}$. Then $B_{+1}(M)$ is the ball having the same center with
$B(M)$, and its radius $r_{+1}=r+1$, where $r$ is the radius of $B(M)$. In
particular, these balls $B(M)$, $B_{+1}(M)$ can be naturally seen as
subquivers of $(T(n),\Omega)$, whence we sometimes call them subquivers
$B(M)$, $B_{+1}(M)$. Such subquivers are finite.

From now on, we always assume that $M$ is regular indecomposable when we talk
about a module $M\in$ mod$(T(n),\Omega)$ in this section. Note that $d(M)\geq
1$. Let $1,\cdots,n_{M}$ be all sink points in $B_{+1}(M)$, and let
$1^{\prime},\cdots,n^{\prime}_{M}$ be all source points in $B_{+1}(M)$,
respectively, $n_{M},n^{\prime}_{M}\in\mathbb{N}$.

######  Lemma 3.

Let $M\in\operatorname{mod}(T(n),\Omega)$ be a regular indecomposable module.
Then

$\sigma(M)=\mathcal{S}^{+}_{1}\cdots\mathcal{S}^{+}_{n_{M}}(M)$.

###### Proof.

Let $x\in T(n)_{0}\setminus B_{+1}(M)$ be a sink point. Then $d(x,B(M))\geq
2$, whence we have $y\nin$ supp$(M)$ for any arrow $\alpha:y\rightarrow x$.
Then $\mathcal{N}(x)\cap T(M)_{0}=\varnothing$, whence
$\mathcal{S}^{+}_{x}(M)=M$. On the other hand, reflection functors
$\sigma_{x_{1}}$ and $\sigma_{x_{2}}$ commute for any two different sink
points $x_{1},x_{2}\in T(n)_{0}$. Hence we have
$\sigma(M)=\mathcal{S}^{+}_{1}\cdots\mathcal{S}^{+}_{n_{M}}(M)$. ∎

Recall that the duality
$D:\operatorname{mod}\mathcal{K}_{n}\rightarrow\operatorname{mod}\mathcal{K}_{n};(M_{i},M(\gamma_{j}))\mapsto(M^{*}_{i},M^{*}(\gamma_{j}))$,
where $M^{*}_{i}$ is the $k$-dual of $M_{3-i}$, and $M^{*}(\gamma_{j})$ is the
$k$-dual of $M(\gamma_{j})$. Then there exists a duality
$D_{T(n)}:\operatorname{mod}(T(n),\Omega)\rightarrow\operatorname{mod}(T(n),\Omega)$,
and we have $\pi_{\lambda}\circ D_{T(n)}(M)\cong D\circ\pi_{\lambda}(M)$ for
all $M\in$ mod$(T(n),\Omega)$ [Daniel, Section 7.2]. Using the duality
$D_{T(n)}$, we can see that functor $D_{T(n)}^{2}$ is equivalent to the
identity functor on the category mod$(T(n),\Omega)$.

Since Lemma 3 tells us that we can take regular indecomposable module $M$ as a
representation of subquiver $B_{+1}(M)$, we have
$\sigma^{-}(D_{T(n)}M)=\mathcal{S}^{-}_{1}\cdots\mathcal{S}^{-}_{n_{M}}(D_{T(n)}M)$
by duality. Then we have the following result.

######  Corollary 4.

Let $M\in\operatorname{mod}(T(n),\Omega)$ be a regular indecomposable module.
Then

$\sigma^{-}(M)=\mathcal{S}^{-}_{1^{\prime}}\cdots\mathcal{S}^{-}_{n^{\prime}_{M}}(M)$.

###### Proof.

This is a dual version of Lemma 3. ∎

######  Lemma 5.

Let $M\in\operatorname{mod}(T(n),\Omega)$ be a regular indecomposable module.
Then

$\sigma^{-}\sigma(M)=\mathcal{S}^{-}_{n_{M}}\cdots\mathcal{S}^{-}_{1}\mathcal{S}^{+}_{1}\cdots\mathcal{S}^{+}_{n_{M}}(M)\cong
M$.

###### Proof.

Let $x\in(T(n),\Omega)_{0}\setminus B_{+1}(M)$ be a sink point with
$y\in\mathcal{N}(x)$, that is, $x$ is a source point for $\sigma\Omega$. Then
$y$ is a sink point for $\sigma\Omega$. Moreover, the point $y$ either
satisfies $\\{y\\}\cap B_{+1}(M)=\varnothing$ or $y$ is a leaf of $B_{+1}(M)$
by the choice of $x$. Then we have $\\{y\\}$ $\cap$ supp$(M)=\varnothing$.
Hence $(\sigma M)_{y}=M_{y}=0$. It indicates that $y\nin$ supp$(\sigma M)$. We
have

$\mathcal{N}(x)$ $\cap$ supp$(\sigma M)=\varnothing$.

Since $\sigma M$ is indecomposable by Lemma 2, we have $x\nin$ supp$(\sigma
M)$. It is obvious that $x\nin$ supp$(\sigma M)$ when $x$ is source point for
$\Omega$. Then we have supp$(\sigma M)\subseteq B_{+1}(M)$ as sets of points,
and $\mathcal{S}^{-}_{x}(\sigma M)=\sigma M$ for any
$x\in(T(n),\Omega)_{0}\setminus B_{+1}(M)$ being a sink point. Finally, we can
see that $\sigma^{-}$ is the composition of those source points that contained
in $B_{+1}(M)$ for $\sigma\Omega$, and they are exactly the points
$1,\cdots,n_{M}$, whence we get our claim.

∎

Note that we can also get $B(\sigma M)\subseteq B_{+1}(M)$ as unoriented
graphs from supp$(\sigma M)\subseteq B_{+1}(M)$ in the proof of Lemma 5.
Similarly, we can get the following result.

######  Corollary 6.

Let $M\in\operatorname{mod}(T(n),\Omega)$ be a regular indecomposable module.
Then

$\sigma\sigma^{-}(M)=\mathcal{S}^{+}_{n^{\prime}_{M}}\cdots\mathcal{S}^{+}_{1^{\prime}}\mathcal{S}^{-}_{1^{\prime}}\cdots\mathcal{S}^{-}_{n^{\prime}_{M}}(M)\cong
M$.

###### Proof.

This is a dual version of Lemma 5. That is, we can change $\sigma M$ into
$\sigma^{-}M$ in the proof of Lemma 5 and consider $x\in T(n)_{0}\setminus
B_{+1}(M)$ being a source point for $\Omega$, then $x$ is a sink point for
$\sigma\Omega$. Comparing supp$(\sigma^{-}M)$ with $B_{+1}(M)$, we can finally
get $\mathcal{N}(x)$ $\cap$ supp$(\sigma^{-}M)=\varnothing$ and $x\nin$
supp$(\sigma^{-}M)$. Then supp$(\sigma^{-}M)\subseteq B_{+1}(M)$ and
$\mathcal{S}^{+}_{x}(\sigma^{-}M)=\sigma^{-}M$. Hence $\sigma$ is the
composition of those sink points that contained in $B_{+1}(M)$ for
$\sigma\Omega$, and they are exactly the points
$1^{\prime},\cdots,n^{\prime}_{M}$, whence we get this.

∎

We also have $B(\sigma^{-}M)\subseteq B_{+1}(M)$ from the proof of Corollary
6. According to Lemma 5 and Corollary 6, we can take

$M,\sigma M,\sigma^{-}M,\sigma^{-}\sigma(M)$ and $\sigma\sigma^{-}(M)$

as representations of subquivers $(B_{+1}(M),\Omega)$ and
$(B_{+1}(M),\sigma\Omega)$, respectively.

######  Lemma 7.

There is a natural isomorphism between the functor $\sigma^{-}\circ\sigma$ and
the identity functor.

###### Proof.

Let $1_{T(n)}$ denote the identity functor on the full subcategory of
$\operatorname{mod}(T(n),\Omega)$ consisting of regular modules. By Lemma 2,
we only need to talk about regular indecomposable modules.

Let $M\in$ mod$(T(n),\Omega)$ be a regular indecomposable module. We now
consider $M$ as a representation of subquiver $B_{+1}(M)$. According to Lemma
5, there exists an isomorphism $\theta_{M}:M\rightarrow\sigma^{-}\sigma(M)$.
Let $N\in\operatorname{mod}(T(n),\Omega)$ be regular indecomposable, and let
$f:M\rightarrow N$ be a morphism. Let $B_{+1}(M,N)$ be the minimal ball
containing $B_{+1}(M)$ and $B_{+1}(N)$. Then $B_{+1}(M,N)\subseteq T(n)_{0}$
is finite. It allows us to take $M,N$ as representations of subquiver
$B_{+1}(M,N)$. Let $F=\sigma^{-}\circ\sigma,$ and let $1,\cdots,n_{M,N}$ be
the sink points of subquiver $B_{+1}(M,N)$. Consider the following diagram

${M}$${F(M)}$${N}$${F(N)}$$\scriptstyle{\theta_{M}}$$\scriptstyle{f}$$\scriptstyle{F(f)}$$\scriptstyle{\theta_{N}}$

Note that $B_{+1}(M),B_{+1}(N)\subseteq B_{+1}(M,N)$. By the proof of Lemma 5,
it is not hard to see that

$F(X)\cong\mathcal{S}^{-}_{n_{M,N}}\cdots\mathcal{S}^{-}_{1}\mathcal{S}^{+}_{1}\cdots\mathcal{S}^{+}_{n_{M,N}}(X)\cong
X,$

where $X\in\\{M,N\\}$. Moreover, we have

$\operatorname{Hom}_{k}(M,N)\cong\operatorname{Hom}_{k}(F(M),F(N))$.

Then there exists such $F(f)\in\operatorname{Hom}_{k}(F(M),F(N))$ so that the
above diagram commutes. Hence there exists a natural isomorphism
$\theta:1_{T(n)}\rightarrow F$. ∎

######  Corollary 8.

There is a natural isomorphism between the functor $\sigma\circ\sigma^{-}$ and
the identity functor.

###### Proof.

By Lemma 2, we only need to talk about regular indecomposable modules. Using
Corollary 6, we can finally get a dual proof with that of the Lemma 7.

∎

######  Lemma 9.

The functors $\sigma$ and $\sigma^{-}$ are quasi-inverses and induce
equivalences of the $k$-linear full subcategories of
$\operatorname{mod}(T(n),\Omega)$ and $\operatorname{mod}(T(n),\sigma\Omega)$
consisting of regular modules.

###### Proof.

Combining Lemma 7 and Corollary 8, the functors $\sigma\circ\sigma^{-}$ and
$\sigma^{-}\circ\sigma$ are equivalent to the identity functor on regular
modules of mod$(T(n),\Omega)$, that is, $\sigma^{-}$, $\sigma$ are the inverse
functors of $\sigma$, $\sigma^{-}$, respectively. Hence we have the claim.

∎

######  Lemma 10.

Let $M\in\operatorname{mod}(T(n),\Omega)$ be a regular indecomposable module.
Then we have $D_{T(n)}\circ\sigma(M)\cong\sigma^{-1}\circ D_{T(n)}(M)$ and
$D_{T(n)}\circ\sigma^{-}(M)\cong\sigma\circ D_{T(n)}(M)$.

###### Proof.

As a consequence of Lemma 9, we have these isomorphisms. ∎

##  3\. middle terms of AR-sequences of T(n)

In this section, we discuss the middle terms of AR-sequences in the regular
components of $\operatorname{mod}(T(n),\Omega)$. Now let
$M\in\operatorname{mod}(T(n),\Omega)$ be a regular indecomposable module.
Combining with Theorem 2.1, and sticking to its notation and conventions in
[Claus], we define

  1. $(a)$

$\pi(M):=(a_{0},\cdots,a_{b})$ the center path of $M$,

  2. $(b)$

$p(M):=a_{0}$, $q(M):=a_{b}$.

Note that $p(M)$ is the center of all sink modules in the $\sigma$-orbit of
$M$ and $q(M)$ is the center of all its source modules, respectively. If,
furthermore, $b(M)$ is the number of flow modules in the $\sigma$-orbit of
$M$, then $b(M)=d(p(M),q(M))$. Recall that $i=\iota(M)$. By Theorem 2.1, we
can get the following corollary.

######  Corollary 3.1.

Let $M\in\operatorname{mod}(T(n),\Omega)$ be a regular indecomposable module.
Then

  1. $(a)$

$M$ is a sink module if and only if $i\leq 0$,

  2. $(b)$

$M$ is a flow module if and only if $0<i\leq b(M)$,

  3. $(c)$

$M$ is a source module if and only if $i>b(M)$.

Let $\mathcal{D}$ be a regular component of $\operatorname{mod}(T(n),\Omega)$,
and let $M,M^{\prime}\in$ $\mathcal{D}$ be two regular indecomposable modules.
Suppose that the $\sigma$-orbit of $M$ (or $M^{\prime}$) contains $b$ (or
$b^{\prime}$) flow modules. According to [Claus, Section 7], we have
$p(M)=p(M^{\prime})$, $q(M)=q(M^{\prime})$ and $b=b^{\prime}$. This implies
that for the $\sigma$-orbits of $M$ and $M^{\prime}$, the centers of all the
sink modules or source modules are the same. We also have
$r_{0}(M)-ql(M)=r_{0}(M^{\prime})-ql(M^{\prime})$, where $r_{0}(M)=r(M_{0})$.
Then we define

$p(\mathcal{D})=p(M)$,

$q(\mathcal{D})=q(M)$,

$b(\mathcal{D})=d(p(M),q(M))$,

$r(\mathcal{D})=r_{0}(M)-ql(M)$.

We have $p(M)=p(\sigma^{t}M)=p(M^{\prime})=p(\sigma^{t^{\prime}}M^{\prime})$
and $q(M)=q(\sigma^{t}M)=q(M^{\prime})=q(\sigma^{t^{\prime}}M^{\prime})$ for
any $t,t^{\prime}\in\mathbb{Z}$. That is,
$\pi(\sigma^{t}M)=\pi(\sigma^{t^{\prime}}M^{\prime})$ and
$b(M)=b(\sigma^{t}M)=b(M^{\prime})=b(\sigma^{t^{\prime}}M^{\prime})=b$.

We say that a module $Y\in$ mod$(T(n),\Omega)$ is a sink (source, or flow)
module if all its indecomposable direct summands are sink (source, or flow)
modules with the same center. Suppose that $Y=\oplus_{i\in I}Y^{i}$, where
$Y^{i}$ is an indecomposable sink (source, or flow) module for all $i$, and
$C(Y^{i})=C(Y^{j}),i,j\in I$. Then we define $r(Y)=\max\\{r(Y^{i})\mid i\in
I\\},C(Y)=C(Y^{i})$. Let $0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow
0$ be an AR-sequence in a regular component $\mathcal{D}$. According to
[Claus1, Lemma 2.2], there exist at most two indecomposable direct summands in
the middle term $Y$. Now suppose that $Y=Y^{1}\oplus Y^{2}$, where
$Y^{1},Y^{2}\in\operatorname{mod}(T(n),\Omega)$ are indecomposable and
$ql(Y^{1})<ql(Y^{2})$. Then we have $b(Y^{1})=b(Y^{2})$ and
$\iota(Y^{1})=\iota(Y^{2})$ [Theorem 2.2]. In addition, $C(Y^{1})=C(Y^{2})$
and $Y^{1},Y^{2}$ are of the same type modules [Theorem 2.3]. Hence we get
$C(Y)=C(Y^{2})$, $r(Y)=r(Y^{2})$. We define

$\iota(Y):=\iota(Y^{2})$ and $b(Y):=b(Y^{1})=b(Y^{2})$.

In the following, we focus on such middle term $Y$ and let $Y=Y^{2}$ when $Y$
is indecomposable.

######  Lemma 3.2.

Let $X,Y,Z$ $\in\operatorname{mod}(T(n),\Omega)$ be regular modules. Assume
that $0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0$ is an AR-
sequence. Then $\iota(Y)=\iota(\sigma Z)$. Moreover, $Y$ is a sink (flow, or
source) module if and only if $\sigma Z$ is a sink (flow, or source) module.

###### Proof.

We already know that $\iota(Y^{1})=\iota(Y^{2})$ by Theorem 2.3. We just need
to prove $\iota(Y^{2})=\iota(\sigma Z)$. By Theorem 2.2, we get
$\iota(Z)=\iota(Y^{2})-1$. Then $\iota(\sigma Z)=\iota(Y^{2})$. This also
tells us that $Y$ and $\sigma Z$ keep the same type [Corollary 3.1].

∎

######  Theorem 3.3.

Let $\mathcal{D}$ be a regular component of $\operatorname{mod}(T(n),\Omega)$.
Suppose that $0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0$ is an AR-
sequence in $\mathcal{D}$. Then $C(Y)=C(\sigma Z)$ and $r(Y)=r(\sigma Z)+1$.

###### Proof.

We firstly assume that $\sigma Z$ is a sink module, then $Y$ and $Z$ would be
sink modules using Lemma 3.2 and Proposition 2.4$(1)$. Then
$C(Y)=p(Y)=p(Z)=C(Z)=C(\sigma Z)$. Module $\sigma^{2}Z=X$ would be a source
module whenever modules $Y$ and $\sigma Z$ are both source modules by
Proposition 2.6, whence we still have $C(Y)=q(Y)=q(Z)=C(\sigma^{2}Z)=C(\sigma
Z)$ in this case.

By [Claus, Proposition 1.$(b)$], the radii of all its flow modules are the
same in the $\sigma$-orbit of a regular indecomposable module. Moreover,
centers of those flow modules just follow the path from $p(M)$ to $q(M)$, that
is, $\pi(M)$ [Proposition 2.5$(1)$]. However, $\pi(Y)=\pi(Z)=\pi(\sigma Z)$
and $b(Y)=b(Z)=b(\sigma Z)$. Hence we must have $C(Y)=C(\sigma Z)$ when both
of them are flow modules.

For the second assertion, we use induction on the quasi-length of $Z$. Suppose
that $ql(Z)=1$. This implies that $Z$ is quasi-simple and $Y=Y^{2}$. In
addition, $Z$ is the quasi-top of $Y$. Suppose that $\sigma Z$ is a sink
module. Then $Y$ is a sink module [Lemma 3.2]. By [Claus, Lemma 3], we get
$B(Y)=B(Z)$, whence $Z$ is a sink and $r(Z)=r(\sigma Z)+1$ [Proposition
2.4$(1)$]. Then $r(Y)=r(Z)=r(\sigma Z)+1$. Suppose that $\sigma Z$ is a source
module. Then $Y$ is a source module and $X=\sigma^{2}Z$ is the quasi-socle of
$Y$. Then $B(Y)=B(\sigma^{2}Z)$ [Claus, Lemma 3∗]. On the other hand, there
exists $r(\sigma^{2}Z)=r(\sigma Z)+1$, whence $r(Y)=r(\sigma Z)+1$ in this
case using Proposition 2.6. Suppose that $\sigma Z$ is a flow module. We get
$Y$ being a flow module [Lemma 3.2]. But we have seen that all flow modules
have the same radius in the $\sigma$-orbit of a regular indecomposable module.
Moreover, $r(\sigma Z)=r(Z_{0})-1$ and $r(Y)=r(Y_{0})-1$. Note that
$r_{0}(Y)-ql(Y)=r_{0}(Z)-ql(Z)$, $ql(Y_{0})=ql(Y)$ and $ql(Z)=ql(Z_{0})$
[Lemma 9]. Since $ql(Y)=ql(Z)+1$, we have
$r_{0}(Y)=r(Y_{0})=r_{0}(Z)+1=r(Z_{0})+1$. Finally, we get $r(Y)=r(\sigma
Z)+1$.

Suppose that $ql(Z)=2$. Then there exists a module
$Z^{\prime\prime}\in\mathcal{D}$ such that

$\circ$$\cdots$
$\circ$$\circ$$\circ$$\vdots$$\cdots$$\cdots$$\circ$$\circ$$X$$\circ$$Y^{1}$$Y^{2}$$Z$$\circ$$Z^{\prime\prime}$$\circ$$\circ$$\circ$$\vdots$$\circ$$\circ$
$\cdots$

We have $ql(Y^{1})=ql(Z^{\prime\prime})=1$ and $ql(Z)=2$. The sequences
$0\rightarrow Y^{1}\rightarrow Z\rightarrow Z^{\prime\prime}\rightarrow 0$ and
$0\rightarrow\sigma Y^{1}\rightarrow\sigma Z\rightarrow\sigma
Z^{\prime\prime}\rightarrow 0$ are AR-sequences [Lemma 9]. There are the
equalities

$r(Z)=r(\sigma Z^{\prime\prime})+1$, and $r(\sigma
Z)=r(\sigma^{2}Z^{\prime\prime})+1$.

Hence we have

$r(Y)=r(Y^{2})=r(Y^{1})+2=r(\sigma^{2}Z^{\prime\prime})+2=r(\sigma Z)+1$

by Theorem 2.3. Suppose that $ql(Z)>2$. Similarly, there exist modules
$Z^{\prime},W\in\mathcal{D}$ such that

$\cdots$ $\circ$$\cdots$
$\circ$$\circ$$\vdots$$\circ$$\circ$$\circ$$\vdots$$\circ$$\vdots$$\circ$$X$$Y^{1}$$Y^{2}$$\circ$$Z^{\prime}$$Z$$W$$\vdots$$\circ$$\circ$$\circ$
$\cdots$$\circ$ $\cdots$

In this case, we have $r(Y^{1})=r(\sigma Z^{\prime})+1$ by induction. However,
$r(Y^{2})=r(Y^{1})+2$ [Theorem 2.3]. We get $r(Y)=r(\sigma
Z^{\prime})+3=r(\sigma Z)+1$ by $r(Z)=r(\sigma Z^{\prime})+2$. ∎

######  Proposition 3.4.

Let $\mathcal{D}$ be a regular component of $\operatorname{mod}(T(n),\Omega)$.
Then $r(\mathcal{D})=r(\sigma\mathcal{D})$.

###### Proof.

Let $M\in\mathcal{D}$ be a regular indecomposable module. By the definition,
we have $r(\mathcal{D})=r_{0}(M)-ql(M)$. However, $r_{0}(M)=r(M_{0})$ and
$ql(M)=ql(\sigma M)$ [Lemma 9]. Hence $r(\sigma\mathcal{D})=r_{0}(\sigma
M)-ql(\sigma M)=r(M_{0})-ql(\sigma M)=r(\mathcal{D})$.

∎

######  Proposition 3.5.

Let $X,Y,Z\in\operatorname{mod}(T(n),\Omega)$ be regular modules. Suppose that

$0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0$

is an AR-sequence with $T(X)_{0}\cap T(Z)_{0}=\varnothing$. Then $Y$ is a flow
module.

###### Proof.

By Lemma 3.2, it is enough to prove that $\sigma Z$ is a flow module.

  1. $(a)$

Suppose that $\sigma Z$ is a sink module with $C(\sigma Z)=x_{0}$, where
$x_{0}\in T(\sigma Z)_{0}$. According to Proposition 2.4$(1)$, $Z$ is a sink
module and $C(Z)=x_{0}$. For $\sigma^{2}Z=X$, we have $C(X)=C(\sigma Z)=x_{0}$
when $X$ is a source module [Proposition 2.4$(3)$ ], and $C(X)\cap C(Z)=x_{0}$
when $X$ is a flow module [Proposition 2.4$(2)$ ]. Hence we always have
$C(Z)\cap C(X)=C(\sigma Z)\cap C(\sigma^{2}Z)\neq\varnothing$ in this case.
This contradicts our assumption $T(X)_{0}\cap T(Z)_{0}=\varnothing$.

  2. $(b)$

Suppose that $\sigma Z$ is a source module with $C(\sigma Z)=y_{0}$, $y_{0}\in
T(\sigma Z)_{0}$. Then $\sigma^{2}Z=X$ is a source module and $C(X)=C(\sigma
Z)$ [Proposition 2.6]. Module $Z$ has 3 possibilites: sink, flow or source. By
Proposition 2.4, 2.5, 2.6, we know that $C(Z)=C(\sigma Z)=y_{0}$ when $Z$ is a
sink or source module. When $Z$ is a flow module, we have $C(Z)\cap C(\sigma
Z)=y_{0}$, where $C(Z)=\\{y^{\prime}_{0},y_{0}\\},y^{\prime}_{0}\in T(n)_{0}$.
Since $T(X)_{0}\cap T(Z)_{0}=\varnothing$ and $C(X)\cap C(Z)=C(\sigma Z)\cap
C(Z)\neq\varnothing$, this case also cannot happen.

Together with $(a)$ and $(b)$, we can see that $\sigma Z$ is a flow module. ∎

  * Example.

Let $Z\in$ mod$(T(3),\Omega)$ be a sink module with the following support,

$a_{0}$$\circ$$\circ$$\circ$$a_{3}$$\circ$$\circ$$a_{1}$$\circ$$\circ$

where dim${}_{k}Z_{a}=1,a\in T(Z)_{0}$ and
$Z(\gamma_{i})=\lambda_{i},\lambda_{i}\in k\setminus\\{0\\},i\in\\{1,3\\}$.
Suppose that $0\rightarrow X\rightarrow Y\rightarrow Z\rightarrow 0$ is an AR-
sequence. Then support of $X=\sigma^{2}Z$ would look like the following.

$\circ$$b_{2}$$b_{1}$$b_{3}$$\circ$$\circ$$\circ$$\circ$$\circ$$\circ$

Hence $X$ is a source module. Moreover, we have $T(X)_{0}\cap
T(Z)_{0}=\varnothing$. Let $(a_{1},a_{0},a_{3})$ be a diameter path of $T(Z)$.
By direct computation, module $\sigma
Z:k\xrightarrow{\gamma_{2}}k,\gamma_{2}\in k\setminus\\{0\\}$ is a flow
module. Then $C(\sigma Z)=\\{a_{0},b_{2}\\}$ and $r(\sigma Z)=0$. According to
Lemma 3.2 or Proposition 3.5, $Y$ is a flow module. By Theorem 3.3, we have
$r(Y)=r(\sigma Z)+1=1$ and $C(Y)=C(\sigma Z)=\\{a_{0},b_{2}\\}$.

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## References