\subsection{Quantum cluster algebra}

The solutions of $Q$-system for $A_r$ have been shown to form particular
clusters in a suitable cluster algebra \cite{Ke07}. On the other hand,
cluster algebras are known to have quantum analogues, 
the so-called quantum cluster algebras \cite{BZ}. In particular, there exists a quantum deformation of the $Q$-system relations.

The cluster algebra for the $A_r$ $Q$-system is a rank $2r$ cluster algebra
with trivial coefficients, and with initial cluster $\bx_0$ and exchange matrix $B_0$:
$$ \bx_0=(R_{1,0},R_{2,0},...,R_{r,0},R_{1,1},R_{2,1},...,R_{r,1})\qquad  B_0=\begin{pmatrix}
0 & -C^t \\ C & 0\end{pmatrix} $$
where $C$ is the Cartan matrix for $A_r$.
The subset of (forward) mutations we have considered so far is such that each mutation amounts
to one application of the $Q$-system equation, allowing to update the cluster
via a substitution of the form: 
$R_{i,n-1}\to R_{i,n+1}=(R_{i,n}^2+R_{i+1,n}R_{i-1,n})/R_{i,n-1}$ (mutation $\mu_i$
if $n$ is odd, $\mu_{r+i}$ if $n$ is even). Recall also that the compound mutation
$\mu=\mu_{2r}\mu_{2r-1}\cdots \mu_1$ acts on $\bx_0$ as a global
translation of all the $n$ indices by $+2$ (i.e. $R_{i,n}\to R_{i,n+2}$ for all $i$), preserving $B_0$.
The ``half-translation" of all indices by $+1$ is realized by acting only with 
$\mu_{\rm even}=\mu_r\mu_{r-1}\cdots \mu_1$
or $\mu_{\rm odd}=\mu_{2r}\mu_{2r-1}\cdots \mu_{r+1}$ in alternance,
which reverse the sign of the exchange matrix.

A quantum cluster algebra with trivial coefficients corresponding to this involves 
the {\it same} exchange matrix, subject to the same evolution under mutation, but
the cluster variables are now non-commuting variables
$\bR_{i,n}\in \cA$ whose commutation relations within each cluster are dictated 
by the exchange matrix as follows.
We have for any pair $(x_i,x_j)$ of variables within the same cluster the following $q$-commutation relation:
\begin{equation}\label{xqcomm} 
x_ix_j=q^{\Lambda_{i,j}} x_jx_i
\end{equation}
where $\Lambda$ is a suitably normalized $2r\times 2r$ matrix proportional to the inverse of the 
exchange matrix. Note that in a different cluster the commutation relations 
are different, as they are entirely dictated by the exchange matrix, 
and may involve other cluster variables.

For the initial cluster, in order for $\Lambda$
to have only integer entries, we choose 
$\Lambda=(r+1)B_0^{-1}=\begin{pmatrix} 0 &\lambda \\-\lambda & 0\end{pmatrix}$, where 
$\lambda=(r+1)C^{-1}$. We obtain the following $q$-commutation relations between variables
of the fundamental cluster:

\begin{lemma}
For $1\leq i,j \leq r$, we have:
\begin{eqnarray}
\bR_{i,0}\bR_{j,1}&=&q^{\lambda_{i,j}} \bR_{j,1}\bR_{i,0}
\label{qcomm1} \\
\bR_{i,n}\bR_{j,n}&=&\bR_{j,n}\bR_{i,n} \qquad  (n=0,1)\ \label{qcomm2} 
\end{eqnarray}
where
\begin{equation}\label{lambdamat}
\lambda_{i,j}={\rm Min}(i,j)\Big( r+1- {\rm Max}(i,j)\Big) \qquad (1\leq i,j \leq r)
\end{equation}
\end{lemma}
\begin{proof}
By explicitly inverting the Cartan matrix $C$, with entries  $2\delta_{i,j}-\delta_{|i-j|,1}$.
\end{proof}

Note that the relations \eqref{qcomm1} and \eqref{qcomm2} are preserved under the above-mentioned 
half-translation of all $n$ indices by $+1$, as $B_0\to -B_0$, and therefore $\Lambda\to -\Lambda$
for $\mu_{\rm even}$ and vice versa for $\mu_{\rm odd}$.
Hence the relations \eqref{qcomm1}-\eqref{qcomm2} extend to
\begin{eqnarray}
\bR_{i,n}\bR_{j,n+1}&=&q^{\lambda_{i,j}} \bR_{j,n+1}\bR_{i,n}
\label{qcom1} \\
\bR_{i,n}\bR_{j,n}&=&\bR_{j,n}\bR_{i,n} \ \label{qcom2} 
\end{eqnarray}
for all $n\in \Z$.
In particular, the initial sequence $\bR_n=\bR_{1,n}$, $n\in\Z$, has the $q$-commutation relations:
\begin{equation}
\bR_n \bR_{n+1} =q^r \, \bR_{n+1}\bR_n
\end{equation}

In the following we consider the {\it same} subset of cluster mutations as in the commuting $A_r$
$Q$-system cluster algebra, leading to a subset of cluster
nodes indexed by Motzkin paths $\bm\in \cM$. Each of these clusters $\bx_\bm$
is now made of $2r$ non-commuting
variables $\{\bR_{i,m_i},\bR_{i,m_i+1} \}_{i=1}^r$, ordered as in the commuting case.
As before, these will play the role of possible initial conditions for the quantum $Q$-system below.


\subsection{Quantum $Q$-system}

The particular form of the quantum $Q$-system we choose is borrowed from 
the $T$-system case \eqref{NCQ}, 
and compatible up to multiplicative redefinition of the variables with the mutations
of the quantum cluster algebra.

\begin{defn}
We define the quantum $Q$-system for $A_r$ as:
\begin{equation}\label{Qquant}
\bR_{i,k+1}(\bR_{i,k})^{-1}\bR_{i,k-1}=\bR_{i,k}+\bR_{i+1,k}(\bR_{i,k})^{-1}\bR_{i-1,k}
\qquad (i\geq 1;\, k\in\Z)
\end{equation}
with the boundary conditions
\begin{equation}\label{boundQquant}
\bR_{0,k}={\bf 1}\qquad  \bR_{r+1,k}={\bf 1} \qquad (k\in \Z)
\end{equation}
\end{defn}

Note that the latter imply a relation 
\begin{equation}\label{truncR}
\bR_{r+2,k}=0 \qquad (k\in \Z)
\end{equation}
as $\lambda_{r+1,r+1}=0$. We also have $\bR_{-1,n}=0$ as $\lambda_{0,0}=0$.

Using the 
$q$-commutations \eqref{qcom1}-\eqref{qcom2}, we may rewrite the system \eqref{Qquant}
in a simpler form:

\begin{lemma}
The $A_r$ quantum $Q$-system is equivalent to:
\begin{equation}\label{quantQ}
q^{\lambda_{i,i}} \bR_{i,k+1}\bR_{i,k-1}=(\bR_{i,k})^2+\bR_{i+1,k}\bR_{i-1,k}
\qquad (i\geq 1;\, k\in\Z)
\end{equation}
with $\lambda_{i,j}$ as in \eqref{lambdamat}, and
with the same boundary condition \eqref{boundQquant}.
\end{lemma}

\begin{remark}
The relation \eqref{quantQ} is equivalent to the relevant quantum 
cluster mutations of \cite{BZ} up to a scalar renormalization
of the cluster variables. Indeed, defining the variables
 $\bX_{i,n}=q^{{r+1\over 4}\lambda_{i,i}} \bR_{i,n}$, we find:
\begin{equation}\label{quantQsys}
\bX_{i,k+1}=q^{\lambda_{i,i}}\, (\bX_{i,k-1})^{-1}(\bX_{i,k})^2
+q^{{1\over 2}(\lambda_{i,i-1}+\lambda_{i,i+1})}\, (\bX_{i,k-1})^{-1}\bX_{i-1,k}\bX_{i+1,k}
\end{equation}
in agreement with the definitions of \cite{BZ}. Note that boundary conditions remain the same: 
$\bX_{0,n}=\bX_{r+1,n}={\bf 1}$ for all $n\in \Z$.
\end{remark}

\subsection{Quasi-Wronskian formulation}

The quasi-Wronskians \eqref{wronskdefnc} of the variables $\bR_n=\bR_{1,n}$ 
may be expressed as simple monomials
in terms of the solutions of the quantum $Q$-system \eqref{Qquant}-\eqref{boundQquant}.
We have:

\begin{thm}\label{quantdelt}
\begin{equation}\label{deltaquant}
\Delta_{i,n}=q^{-{1\over 2}(i-1)(2r+2-i)}\, \bR_{i,n}\bR_{i-1,n-1}^{-1} \qquad (i>1,n\in \Z)
\end{equation}
\end{thm}
\begin{proof}
By induction on $i$. Define 
$\Gamma_{i,n}=q^{-{1\over 2}(i-1)(2r+2-i)}\, \bR_{i,n}\bR_{i-1,n-1}^{-1}$, the r.h.s. of \eqref{deltaquant}
and $a_i={1\over 2}(i-1)(2r+2-i)$.
For $i=1$, we have $\Delta_{1,n}=\bR_{1,n}=\Gamma_{1,n}$
from the boundary condition \eqref{boundQquant}. 
Assume \eqref{deltaquant} holds for all $1\leq j \leq i$.
We use the Hirota equation \eqref{nchiro} to express
\begin{equation}\label{hire}
\Delta_{i+1,n}-\Gamma_{i+1,n}=\Gamma_{i,n+1}-\Gamma_{i+1,n}-
\Gamma_{i,n}\Gamma_{i,n-1}^{-1}(\Gamma_{i,n-1}-\Gamma_{i-1,n})\Gamma_{i-1,n}^{-1}\Gamma_{i,n}
\end{equation}
Moreover, defining $B_{i,n}=(\bR_{i,n})^2+\bR_{i+1,n}\bR_{i-1,n}-q^{\lambda_{i,i}}\bR_{i,n+1}\bR_{i,n-1}$
($=0$ due to the quantum Q-system equation \eqref{quantQ}), we find that
$$ \Gamma_{i-1,n}-\Gamma_{i,n-1}=q^{-a_i} \Big((\bR_{i-1,n-1})^2-B_{i-1,n-1}\Big)
\bR_{i-2,n-1}^{-1}\bR_{i-1,n-2}^{-1}$$
due to the $q$-commutation relations \eqref{qcom1}-\eqref{qcom2}
and the identity $\lambda_{i-1,i-1}-\lambda_{i-2,i-1}-a_{i}=-a_{i-1}$.
Substituting this into \eqref{hire}, we get
\begin{eqnarray*}
\Delta_{i+1,n}-\Gamma_{i+1,n}&=&q^{-a_{i+1}}\bR_{i,n}^2
\bR_{i-1,n}^{-1}\bR_{i,n-1}^{-1}\\
&&-q^{-a_i} \Gamma_{i,n}\Gamma_{i,n-1}^{-1}(\bR_{i-1,n-1})^2
\bR_{i-2,n-1}^{-1}\bR_{i-1,n-2}^{-1}
\Gamma_{i-1,n}^{-1}\Gamma_{i,n}\\
&=&0
\end{eqnarray*}
by use of the $q$-commutation relations \eqref{qcom1}-\eqref{qcom2}.
\end{proof}

\begin{remark}\label{quantumwronsk}
Theorem \ref{quantdelt} allows to interpret $\bR_{i,n}$ as a ``quantum Wronskian", namely 
a ``quantum determinant" of the Wronskian matrix used in Definition \ref{ncwdef}.
Indeed, we may write
$$\bR_{i,n}=q^{{i(i-1)\over 6}(3r+2-i)}\, \Delta_{i,n}\Delta_{i-1,n-1}\cdots \Delta_{1,n-i+1}
\qquad (i\geq 1)$$
which is a product of principal quasi-minors of the Wronskian matrix, like in the standard definitions of
quantum determinants \cite{EtingofRetakh} (note however that here the quasi-minors do not commute).
In the cases $i=1,2,3$, using
the $q$-commutations and the quantum $Q$-system, we find
the following expressions for the quantum Wronskian:
\begin{eqnarray*}
\bR_{1,n}&=&\bR_n\\
\bR_{2,n}&=&q^r \bR_{n+1}\bR_{n-1}-\bR_{n}^2\\
\bR_{3,n}&=& q^{3r-1}\bR_{n+2}\bR_n\bR_{n-2}
-q^{2r-1}(\bR_{n+2}\bR_{n-1}^2+\bR_{n+1}^2\bR_{n-2})\\
&&+q^{r-1}(q^{r+1}+1)\bR_{n+1}\bR_n\bR_{n-1}-\bR_n^3
\end{eqnarray*}
\end{remark}


Applying Theorem \ref{quantdelt} for $i=r+2$, and using the equation \eqref{truncR}, we get:

\begin{cor}
The solution of the quantum $Q$-system \eqref{Qquant}-\eqref{boundQquant} satisfies the finite truncation
condition $\Delta_{r+2,n}=0$ for all $n\in \Z$.
\end{cor}

We may therefore apply the results of Section \ref{section3}. Assuming $\bR_{i,n}$ is a solution
of \eqref{Qquant}-\eqref{boundQquant}, we consider the usual subset of clusters $\bx_\bm$,
also viewed as initial data for the quantum $Q$-system.
Recall that the commutation relations between the quantum cluster variables within 
each cluster $\bx_\bm$ are determined by the exchange matrix $B_\bm$.
Using the explicit expressions for the
exchange matrix $B_\bm$ of Ref. \cite{DFK3}, it is an easy exercise to compute the $q$-commutation
relations between the cluster variables within any given cluster $\bx_\bm$. We find that:

\begin{lemma}\label{qcomgen}
\begin{equation}\label{genqcomm}
\bR_{i,n}\bR_{j,n+p}=q^{p\lambda_{i,j}}\, \bR_{j,n+p}\bR_{i,n}
\end{equation}
for all $n,p,i,j$ such that the variables belong to the same cluster $\bx_\bm$. 
\end{lemma}

\begin{remark}
Lemma \ref{qcomgen} does not cover for instance the case $j=i$, $p=2$,
as $\bR_{i,n}$ and $\bR_{i,n+2}$ never belong to the same cluster $\bx_\bm$. Using
the quantum $Q$-system and the valid $q$-commutations, we find for instance
$$ \bR_{i,n}\bR_{i,n+2}=q^{2\lambda_{i,i}}\,
\left( q^{-r-1}\, \bR_{i,n+2}\bR_{i,n}+( q^{-r-1}-1)\bR_{i,n+1}^2\right)$$
However, we may apply Lemma \ref{qcomgen} to compute the commutation of $\Delta_{i,n}$
and $\Delta_{j,m}$, using Theorem \ref{quantdelt}, when the
four variables $\bR_{i,n},\bR_{i-1,n-1},\bR_{j,m},\bR_{j-1,m-1}$ belong to the same
cluster $\bx_\bm$:
$$ \Delta_{i,n}\Delta_{j,m}=q^{(n-m)(\lambda_{i,j-1}+\lambda_{j,i-1})+r(j-i)}\, 
\Delta_{j,m} \Delta_{i,n}$$
\end{remark}


Using the $q$-commutation relations \eqref{genqcomm}
we easily get the following:

\begin{lemma}\label{uselem}
The quantities $C_{i,n}$ and $D_{i,n}$ \eqref{defCD} are expressed as:
$$C_{i,n}=q^{i(i-1)\over 2} \, \bR_{i,n}\bR_{i,n-1}^{-1} \qquad D_{i,n}= \bR_{i+1,n}\bR_{i,n}^{-1}
\bR_{i,n-1}^{-1}\bR_{i-1,n-1}$$


\end{lemma}
In particular, the value of
the conserved quantity \eqref{conserveC} follows from Lemma \ref{uselem} and the boundary condition
\eqref{boundQquant}:
$$ C_{r+1} = q^{r(r+1)\over 2}. $$
Note that in this case, this conserved quantity is central.

\begin{thm}\label{mainquanthm}
Let $\bR_{i,n}$ be a solution of the quantum $Q$-system \eqref{Qquant}-\eqref{boundQquant} 
and $\bR_n=\bR_{1,n}$.
The quantity $\bR_{n+m_1}\bR_{m_1}^{-1}$ for $n\geq 0$,
when expressed in terms of initial data $\bx_\bm$, is the partition function for non-commutative
weighted paths on the graph $\Gamma_\bm$ from the root to itself with $n$ steps towards the root
and with skeleton weights:
\begin{eqnarray}
\qquad \by_{2i-1}(\bm)&=&
q^{i-1} \, \bR_{i,m_i+1}\bR_{i,m_i}^{-1} 
 \bR_{i-1,m_{i-1}}\bR_{i-1,m_{i-1}+1}^{-1}
\label{oddyq} \\
\by_{2i}(\bm)&=&\left\{  \begin{matrix} 
\bR_{i+1,m_{i+1}+1}\bR_{i+1,m_{i+1}}^{-1}
\bR_{i+1,m_{i}}\bR_{i+1,m_{i}+1}^{-1}
& {\rm if} \ m_i=m_{i+1}+1\\
{\bf 1} & {\rm otherwise} 
\end{matrix}\right\}\nonumber \\
&&\times\, 
 \bR_{i+1,m_i+1}\bR_{i,m_i+1}^{-1}
 \bR_{i,m_i}^{-1}\bR_{i-1,m_i} \nonumber \\
&& \times \left\{ \begin{matrix} 
\bR_{i-1,m_{i}+1}\bR_{i-1,m_{i}}^{-1}
\bR_{i-1,m_{i-1}}\bR_{i-1,m_{i-1}+1}^{-1}
& {\rm if} \ m_i=m_{i-1}-1\\
{\bf 1} & {\rm otherwise} 
\end{matrix}\right\}\label{evenyq} 
\end{eqnarray}
that are all non-commutative Laurent monomials of the initial data, times some possible integer powers of $q$.
As such it is a Laurent polynomial of the initial data with
coefficients in $\Z_+[q,q^{-1}]$.
\end{thm}
\begin{proof}
We apply Theorem \ref{finlut} with the weights
\eqref{ncweightcalc}. We then rewrite all the skeleton 
weights using Lemma \ref{uselem}.
Finally, the redundant weights
\eqref{redunc} are also monomials of the initial data, with a possible integer power of $q$ in factor.
Reordering and rearranging the terms may only produce coefficients in $\Z_+[q,q^{-1}]$.
\end{proof}


\begin{example}
In the case of non-decreasing Motzkin paths (see Example \ref{incmotz} above),
we have the following expressions for the skeleton weights:
\begin{eqnarray*}
\by_{2i-1}(\bm)&=&\left\{ \begin{matrix}
\bR_{i,m_i+1}\bR_{i-1,m_{i-1}+1}^{-1}
\bR_{i,m_i}^{-1} \bR_{i-1,m_{i-1}}
& {\rm if} \ m_i=m_{i-1}\\
\bR_{i,m_i+1}
\bR_{i-1,m_{i-1}+1}^{-1}\bR_{i-1,m_{i-1}}\bR_{i,m_i}^{-1} 
& {\rm if} \ m_i=m_{i-1}+1
 \end{matrix}\right.\\
\by_{2i}(\bm)&=& \bR_{i+1,m_i+1}\bR_{i,m_i+1}^{-1}
 \bR_{i,m_i}^{-1}\bR_{i-1,m_i}
 \end{eqnarray*}
where we have used the $q$ commutations \eqref{genqcomm}
and the identity $\lambda_{i-1,i}-\lambda_{i-1,i-1}+i-1=0$ 
to simplify the monomials.
For the fundamental initial condition $\bx_0=\bx_{\bm_0}$, this gives:
\begin{eqnarray*}
\by_{2i-1}(\bm_0)&=&
\bR_{i,1}\bR_{i-1,1}^{-1}\bR_{i,0}^{-1}\bR_{i-1,0}\\
\by_{2i}(\bm_0)&=& \bR_{i+1,1}\bR_{i,1}^{-1}\bR_{i,0}^{-1}\bR_{i-1,0}
\end{eqnarray*}
For the Stieltjes point $\bx_{\bm_1}$, we have:
\begin{eqnarray*}
\by_{2i-1}(\bm_1)&=&\bR_{i,i}\bR_{i-1,i-1}^{-1}\bR_{i-1,i-2}\bR_{i,i-1}^{-1}\\
\by_{2i}(\bm_1)&=&\bR_{i+1,i}\bR_{i,i}^{-1}\bR_{i,i-1}^{-1}\bR_{i-1,i-1}
\end{eqnarray*}
\end{example}

\begin{example}
Let us consider the case $r=2$ of Example \ref{atwoex}. Setting $\bR_n=\bR_{1,n}$ and $\bS_n=\bR_{2,n}$,
we have the following commutation relations:
\begin{equation}\label{coma2}
\begin{matrix}\bR_n\bR_{n+1}=q^2 \, \bR_{n+1}\bR_n  &  \bS_n\bS_{n+1}=q^2 \, \bS_{n+1}\bS_n &
\bR_n\bS_{n}=\bS_{n}\bR_n \\
\bS_n\bR_{n+1}=q \, \bR_{n+1}\bS_n & \bR_n\bS_{n+1}=q \, \bS_{n+1}\bR_n & \ \ \ (n\in \Z)\end{matrix}
\end{equation}
The $A_2$  quantum $Q$-system reads:
\begin{eqnarray*}
q^2\bR_{n+1}\bR_{n-1}&=& (\bR_n)^2 +\bS_n\\
q^2\bS_{n+1}\bS_{n-1}&=& (\bS_n)^2 +\bR_n
\end{eqnarray*}
and the weights at the various Motzkin paths may be expressed as:
$$ \begin{matrix}
\by_1(0,0)&=&\bR_1\bR_0^{-1}\\
\by_2(0,0)&=&\bS_1\bR_1^{-1}\bR_0^{-1}\\
\by_3(0,0)&=&\bS_1\bR_1^{-1}\bS_0^{-1}\bR_0\\
\by_4(0,0)&=&\bS_1^{-1}\bS_0^{-1}\bR_0\\
\by_5(0,0)&=&\bS_1^{-1}\bS_0
\end{matrix}
\qquad
\begin{matrix}
\by_1(0,1)&=&\bR_1\bR_0^{-1}\\
\by_2(0,1)&=&\bS_1\bR_1^{-1}\bR_0^{-1}\\
\by_3(0,1)&=&\bS_2\bR_1^{-1}\bR_0\bS_1^{-1}\\
\by_4(0,1)&=&\bS_2^{-1}\bS_1^{-1}\bR_1\\
\by_5(0,1)&=&\bS_2^{-1}\bS_1
\end{matrix}
\qquad
\begin{matrix}
\by_1(1,0)&=&\bR_2\bR_1^{-1}\\
\by_2(1,0)&=&\bS_0^{-1}(\bS_1)^2\bR_1^{-1}\bR_2^{-1}\\
\by_3(1,0)&=&\bS_1\bR_1\bS_0^{-1}\bR_2^{-1}\\
\by_4(1,0)&=&\bS_0^{-1}\bS_1^{-1}(\bR_1)^2\bR_2^{-1}\\
\by_5(1,0)&=&\bS_1^{-1}\bS_0
\end{matrix}
$$
where we have used \eqref{coma2} to eliminate the $q$-dependent prefactors.
The redundant weight of the last case $\bm=(1,0)$ is 
$\bw_{3,1}=t\by_4\by_3^{-1}\by_2=tq^{-1}\bR_2^{-1}\bS_0^{-1}$.
Note that the conservation law $\by_5(\bm)\by_3(\bm)\by_1(\bm)=C_3=q^3$ is satisfied for the three 
Motzkin paths.
\end{example}

\begin{remark} One can show that the conserved quantities $P_m$ of
Remark \ref{consrem} commute with each other in this case. This suggests an
analogy with the definition of classical integrability, where the
integrals of the motion are in involution. In fact, in the $A_1$ quantum $Q$-system case,
the Hamiltonian structure and integrability were shown in \cite{FV}. The case of the $A_1$ $T$-system
was treated in \cite{FKV}.
\end{remark}


\begin{remark}
  The quasi-Wronskians $\Delta_{i,n}$ are ``bar-invariant" quantities.
  Let us define an anti-automorphism (called usually the bar
  involution), which we denote by $*$, by $(ab)^*=b^* a^*$ for all
  $a,b\in\cA$, ${\bf 1}^*={\bf 1}$ and $q^*=q^{-1}$, while
  $(\bR_n)^*=\bR_n$ for all $n\in\Z$.  Then, as a trivial consequence
  of the Hirota equation \eqref{nchiro} and of the initial condition
  $\Delta_{1,n}=\bR_n$, we have that
$$(\Delta_{i,n})^*=\Delta_{i,n}$$ 
\end{remark}
{"config": "arxiv", "file": "1006.4774/section5.tex"}