\documentstyle{article}
\title{\LARGE Integrable Mappings for Non--Commutative Objects}
\author{A.~N.~Leznov\thanks{Institute for High Energy Physics,142284 Protvino,
Moscow Region, Russia}~~and~~E.~A.~Yuzbashyan\thanks{Joint Institute for
Nuclear Research, 141980 Dubna, Moscow Region, Russia}} \date{}
\newcommand{\lef}[2]{\stackrel{\leftarrow#2}{#1}}
\newcommand{\rig}[2]{\stackrel{#2\rightarrow}{#1}}
\newcommand{\al}{\alpha}
\newcommand{\be}{\beta}
\newcommand{\th}{\theta}

\begin{document}
\maketitle

\begin{abstract}
\noindent The integrable mappings formalism is generalized on non--commutative case. Arising hierarchies of integrable systems are
invariant with respect to this "quantum" discrete transformations without any
assumption about commutative properties of unknown operators they
include.  Partially, in the scope of this construction are the equations for
Heisenberg operators of quantum (integrable) systems.


\end{abstract}

\section{Introduction}
In papers \cite{1} was shown that the theory of
integrable systems (under the assumption of commutativety of all involved
functions) can be reformulated in a form, where the key role plays
the group of integrable mappings and its theory of representation.
It arose the question, what will happen with this construction,
if we will consider equations of motion for Heisenberg
operators, or in other words, when unknown functions of integrable
systems changed on non--commutative variables?

The goal of the present paper is to give the partial answer to this question.

Each quantum system with the same success can be described in many
different (in form) representations. The most known and used are
Schr\"odinger and Heisenberg pictures. The first deal with the wave
functions (the state vectors in a Hilbert space), the second with
the non--commutative Heisenberg operators and equations of motion under
appropriate initial conditions (commutation relations at the fixed moment of
time).

In this paper we will show how the group
of integrable mappings conception must be changed to include
the non--commutative variables case. The equations of evolution type
(after some modifications connected with the order of a multipliers)
remain invariant with respect to the corresponding quantum discrete
transformations without any assumption about commutation rules for unknown
functions (operators). Partially they can be $s\times s$ matrix functions or
some operators acting in the arbitrary representation space.The equations of
motion for quantum Heisenberg operators are containing within this
construction.

We use the discrete transformations method as the most
adequate to solution of such kind problems \cite{1}. We restrict ourselves
by some concrete examples of integrable mappings for non--commutative objects,
in usual and supersymmetrical two--dimensional spaces and by corresponding
hierarchies of $(1+2)$--dimensional integrable systems. We now have no idea
how to enumerate all possible integrable mappings for
non--commutative objects. In this connection we can only add that in
commutative case this problem yet is very far
{}From its final solution.



\section{Non--Commutative Darboux--Toda Substitution in Two--Dimensional
space}
  \subsection{Definitions}

 Let us denote $u,v$ the pair of operators defined in some representation
space and depending on $x,y$ coordinates of two--dimension space. We assume,
that partial derivatives up to some sufficient large order and inverse
operators $u^{-1}, v^{-1}$ are exist. Only associativety
is assumed for multiplication, nowhere we will assume any commutation
relations.

We will consider the following mapping:
  \begin{equation}
  \begin{array}{cc}
\lef{u}{}= v^{-1}&\lef{v}{}= [vu-(v_x v^{-1})_y]v\equiv v[uv- (v^{-1} v_y)_x],

  \end{array}
  \label{10}
  \end{equation}

{\emergencystretch=5pt
 where $\lef{u}{}; \lef{v}{}$
  denotes final, transformed operators. In the case, when $u,v$ are some
$s\times s$ matrices (\ref{10}) was considered in \cite{10}. In classical
case, when $u,v$ are usual commutative functions (\ref{10}) is the well-known
Darboux--Toda substitution.


  Substitution (\ref{10}) is invertible, i.e. the initial operators can be
  expressed in the terms of final ones. Denoting $\rig{u}{}; \rig{v}{}$ the
result of inverse transformation we have:
\begin{equation} \begin{array}{cc}
  \rig{v}{}= u^{-1}&\rig{u}{}= [uv-(u_y u^{-1})_x]u\equiv u[vu- (u^{-1}u_y)_x]
  \end{array} \label{11}
  \end{equation}

 Operator $ f(u,v)$ after application of the s--times direct
transformation we will denote  $ \lef{f}{s}\equiv f(\lef{u}{s},\lef{v}{s})$
and after s--times inverse transformation as
$\rig{f}{s}~\equiv~f~(\rig{u}{s}~,~\rig{v}{s}~)$,
with the agreement $\lef{f}{-m}\equiv\rig{f}{m}$, $m\ge0$.

 }
\noindent If
\begin{equation}
\begin{array}{c}
u_t=F_1(u,v,u_x,v_x,u_y,v_y,...)\\
v_t=F_2(u,v,u_x,v_x,u_y,v_y,...)
\end{array}\label{611}
\end{equation}
  is given evolution type system then
  the condition of its invariance with respect to the transformation
 (\ref{10}) (it means that in terms of $\lef{u}{},\lef{v}{}$
equations (\ref{611}) will be exactly the same as they are in terms of $u,v$)
can be derived by differentiation of (\ref{10}) by some parameter and has
  the following form:
 \begin{equation}
\begin{array}{rl} \lef{F}{}_1=
\lef{u}{}_t=&-v^{-1}v_t v^{-1}=-v^{-1}F_2 v^{-1}\\ \lef{F}{}_2=
\lef{v}{}_t=&([vu-(v_x v^{-1})_y]v)_t=[F_2 u+v F_1-(F_{2x}v^{-1})_y+\\
&+(v_xv^{-1}F_2v^{-1})_y]v+[vu-(v_xv^{-1})_y]F_2
\end{array}   \label{13}
\end{equation}
This is the functional symmetry equation for substitution (\ref{10}).
Unknown operators here are $F_1, F_2$. If some operators $F_1(u,v), F_2(u,v)$
are solution of (\ref{13}) then the corresponding system (\ref{611}) will be
invariant with respect to (\ref{10}). (\ref{13}) is a
linear system, i.e. if ${F_1}', {F_2}'$ and ${F_1}'', {F_2}''$ are
solutions then $F_1= a{F_1}'+b{F_1}'', F_2=a{F_2}'+b{F_2}''$,
where $a, b$ are arbitrary numerical parameters, is also solution.
Every symmetry equation possesses trivial solution $F_1=au_x+bu_y,
F_2=av_x+bv_y$.
  Substitution is called
integrable if its symmetry equation have at least one non--trivial
solution.


\subsection{Solution of the Symmetry Equation}

The method we will use here to find solutions of (\ref{13}) is analogues
to the method we used in \cite{14} in the case of commutative functions.
But it is not exactly the same as many
transformations can not be done because of non--commutativety of variables
under consideration.

First of all let us take $F_2=\al_0 v$, $F_1=u\be_0$. We obtain:
\begin{equation}
\begin{array}{l}
\be_0=-\rig{\al}{}_0\\ {\al_0}_{xy}=(\al_0-\lef{\al}{}_0)\lef{T_0}{\:}+
T_0(\al_0-\rig{\al_0}{\!})+\th {\al_0}_y,
-{\al_0}_y\th \label{2}
\end{array}
\end{equation}
where $T_0=vu$, $\th=v_x v^{-1}$.
This system possesses obvious partial solution
${\al_0}^{(0)}=-{\be_0}^{(0)}=1$, which gives the first term of  hierarchy:
$F_1=-u, F_2=v$.
Two following equations for $ T_0 $ and $ \th $, which are the direct
corollary of (\ref{10}), are important for father calculations:

\begin{equation}
{T_0}_x=\th T_0-T_0 \rig{\th}{}\qquad
\th_y=T_0-\lef{T_0}{\:} \label{3}
\end{equation}
In fact, (\ref{3}) is the substitution (\ref{10}) rewritten in terms of
$T_0$ and $\th$.
Now let us take ${\al_0}_y=\al_1\lef{T_0}{\:}+T_0\be_1$. One can treat
this expression as analog to the decomposition of some vector by basic
vectors. {}From (\ref{2}), expressing $\lef{T_0}{\:}_x $ and $T_{0x}$ with the
help of (\ref{3}) and equating to zero coefficients before $\lef{T}{}_0$ and
after  $T_0$ (which is some additional assumption), we have:
\begin{equation}
\begin{array}{l}
{\al_1}_x=\al_0-\lef{\al}{}_0+\th\al_1-\al_1\lef{\theta}{}\\

{\be_1}_x=\al_0-\rig{\al_0}{\!}+\rig{\th}{}\be_1-\be_1\th
\end{array}\label{1000}
\end{equation}
The second relation obviously can be rewritten as:
$$
{\lef{\be}{}_1}_x=-(\al_0-\lef{\al}{}_0)+\th\lef{\be}{}_1-
\lef{\be}{}_1\lef{\th}{}
$$

{}From what it follows that system (\ref{1000}) possesses partial solution of
the form $\lef{\be}{}_1=-\al_1$. After taking  $y$--derivative of
equation for $\al_1$, this partial solution gives the  following system:

\begin{equation}
\begin{array}{l}
\be_1=-\rig{\al_1}{\!}\\
{\al_1}_{xy}=(\al_1-\lef{\al}{}_1)\lef{T_0}{2}+T_0(\al_1-\rig{\al_1}{\!})+
\th{\al_1}_y -{\al_1}_y
\lef{\th}{}\\
\end{array} \label{4}
\end{equation}
This system is analogues to the (\ref{2}). It also has
partial solution $\al_1=-\be_1=1$, which leads to
${\al_0}^{(1)}=\int(\lef{T_0}{\:}-T_0)dy$, and gives the next solution of the
symmetry equation (\ref{13}).  Taking now
${\al_1}_y=\al_2\lef{T_0}{2}+T_0\be_2$ we are able to continue by the same
scheme. The system for $\al_2, \be_2$  has the same structure as
previous systems. Its partial solution $\al_2=-\be_2$ allows to find the
third term of hierarchy $$ {\al_0}^{(2)}=\int dy\,\Bigl[\Bigl(\int
dy\,\Bigl(\lef{T_0}{2}-T_0\Bigr)\Bigr) \lef{T_0}{\:}-T_0\int
dy\,\Bigl(\lef{T_0}{\:}-\rig{T_0}{\!}\Bigr)\Bigr] $$
By induction it can be proved that in general case equations for
$\al_n, \be_n $ have the form:

\begin{equation}
\begin{array}{l}
\be_n=-\rig{\al_n}{\!}\\
{\al_n}_{xy}=(\al_n-\lef{\al}{}_n)\lef{T_0}{(n+1)}+T_0(\al_n-\rig{\al_n}{\!})
+\th{\al_n}_y-{\al_n}_{y}\lef{\th}{n}\\
\end{array} \label{5}
\end{equation}
with partial solution $\al_n=-\be_n=1$.
 After this expression for ${\al_0}^{(n)}$ can be reconstructed in
the form of the sum of $2^n$ terms, which can be written in the following
symbolical form:

\begin{equation}
\begin{array}{rcl}
{\al_0}^{(n)}&=&(-1)^n \prod_{i=1}^n \left(1-L_i
exp\left[id_i+\sum_{i=k+1}^n d_k\right]\right)\times \\&&\\
&&\times\int dy (T_0 \int dy (\rig{T_0}{\!}\int dy (...\int dy
\rig{T_0}{(n-1)}...)))\label{20}
\end{array}
\end{equation}
where  $exp\, d_p$ means shifts by the unity the argument of p\--repeated
integral.

$$
\dots\int dy\lef{T_0}{p}\to\dots\int dy\lef{T_0}{(p+1)}\dots
$$
and symbol $L_r$\---transposition of terms in the r\--th brackets
$$
(A_1(\dots(A_r[\dots])\dots)) \to (A_1(\dots([\dots]A_r)\dots))
$$
with the following multiplication rules:
$$
L_i exp[...]_1 L_j exp[...]_2=L_i L_j exp\left[ [...]_1+[...]_2\right]
$$
Comparing (\ref{20}) with \cite{14} we see that here for non-commutativety we
are forced to introduce the new operators $L_i$ which are discount the order 
of the multipliers.

\subsection{Examples}
\subsubsection*{n=0}
$$
v_t=v\qquad u_t=-u
$$
\subsubsection*{n=1}
$$
v_t=v_x\qquad u_t=u_x
$$
\subsubsection*{n=2}
$$
v_t=v_{xx}-2\int (vu)_x dy\times v \quad u_t=-u_{xx}+2u\int (vu)_x dy
$$
This is the Davey--Stewartson system, described in \cite{12}
\subsubsection*{n=3}
$$
v_t=v_{xxx}-3\int (vu)_x dy\times v_x-3\int (v_x u)_x dy\times v-
$$
$$
-3\int \left[vu\int (vu)_x dy-\int (vu)_x dy\times vu\right]dy\times v
$$
\vspace{1em}
$$
u_t=-u_{xxx}-3u_x \int (vu)_x dy-3u\int (v_x u)_x dy-
$$
$$
-3u\int \left[vu\int dy (vu)_x-\left( \int dy (vu)_x \right) vu\right]dy
$$
In commutative case this is the Veselov--Novikov system.

\section{Non--Commutative Darboux--Toda Transformation in Two--Dimensional
Super Space}

\subsection{Definitions}
Here we will analyze the situation, when non-commutative operators under
consideration in addition to usual space and time coordinates $x, y, t$
are depend upon Grassman variables $\th_+, \th_-$. We will consider the
following mapping:
  \begin{equation}
  \begin{array}{cc}
\lef{u}{}= v^{-1}&\lef{v}{}=-[D_-(D_+v\times v^{-1})+vu]v\equiv
v[D_+(v^{-1}D_-v)-uv],
  \end{array}
  \label{31}
  \end{equation}
where
$$
D_+=\frac{\partial}{\partial\th_+}+\th_+\frac{\partial}{\partial x}\quad
D_-=\frac{\partial}{\partial\th_-}+\th_-\frac{\partial}{\partial y}\quad
D^2_+=\frac{\partial}{\partial x}\quad D^2_-=\frac{\partial}{\partial y}
$$
Other notations are the same as in the
previous section. Substitution (\ref{31}) is invertible. Inverse
transformation has the form:
  \begin{equation}
  \begin{array}{cc}
\rig{v}{}= u^{-1}&\rig{u}{}=-[D_+(D_-u\times u^{-1})+uv]u\equiv
u[D_-(u^{-1}D_+u)-vu],
  \end{array}
  \label{33}
  \end{equation}
The symmetry equation for (\ref{31}) is the following:
 \begin{equation}
\begin{array}{rcl}
\lef{F_1}{\:}&=
& -v^{-1}F_2 v^{-1}\\ \lef{F_2}{\,}&=&
F_2[D_+(v^{-1}D_-v)-uv]+v[D_+(-v^{-1}F_2v^{-1}D_-v)+\\&&+D_+(v^{-1}D_-F_2)-F_1v-
uF_2]
\end{array}   \label{35}
\end{equation}
\subsection{Solution of the Symmetry Equation}
Here we will get the hierarchy of solutions of the symmetry equation (\ref{35}).
For this we will use the same general method as in the previous section.
But there is an interesting and in some sense important difference. As we will
see bellow, partial solutions of (\ref{35}) can be found only at even steps,
when unknown operators are Bosonic--like variables, whereas at odd steps they
are Fermionic--like.

 After substitution in (\ref{35}) $F_1=u\be_0, F_2=\al_0 v$ we have:
\begin{equation}
\begin{array}{l}
\be_0=-\rig{\al_0}{\!}\\ D_+D_-\al_0=(\lef{\al}{}_0-\al_0)\lef{T_0}{\:}+
T_0(\al_0-\rig{\al_0}{\!})+\th D_-\al_0+D_-\al_0\th, \label{62}
\end{array}
\end{equation}
where $T_0=vu, \th=D_+v\times v^{-1}$. This system has partial solution
$\al_0=-\be_0=1$, which correspond to: $F_1=-u,F_2=v$.
Transformation (\ref{31}) can be rewritten in terms of $T_0, \th$  as:
\begin{equation}
D_+T_0=\th T_0-T_0 \rig{\th}{}\qquad
D_-\th=-T_0-\lef{T_0}{\:} \label{43}
\end{equation}
Taking now
$D_-\al_0=\al_1\lef{T_0}{\:}+T_0\be_1$
, for $\al_1, \be_1$ we have:
$$
D_+\al_1=\lef{\al}{}_0-\al_0+\th\al_1+\al_1\lef{\th}{}
$$
$$
D_+\be_1=\al_0-\rig{\al}{}_0+\rig{\th}{}\be_1+\be_1\th
$$
For $\lef{\be}{}_1=\al_1$ the second relation directly follows from the first
one. For this case, acting on the equation for $\al_1$ with $D_-$ operator, we
have:
\begin{equation}
\begin{array}{l}
\be_1=\rig{\al_1}{\!}\\
-D_+D_-\al_1=(\al_1+\lef{\al}{}_1)\lef{T_0}{2}-T_0(\al_1+\rig{\al_1}{\!})+
D_-\al_1\lef{\th}{}-\th D_-\al_1\\
\end{array} \label{44}
\end{equation}
This is the typical system for odd steps. Comparing it with (\ref{4}) we
notice that the difference between those systems is that (\ref{44}) have not
numerical partial solutions ($\al_1, \be_1$ are Fermionic--like operators).
However it is possible to continue reduction using decomposition
$D_-\al_1=\al_2\lef{T_0}{2}+T_0\be_2$. We have:
$$
-D_+\al_2=\lef{\al}{}_1+\al_1-\th\al_2+\al_2\lef{\th}{2}
$$
$$
-D_+\be_2=-(\al_1+\rig{\al_1}{\!})-\rig{\th}{}\be_2+\be_2\lef{\th}{}
$$
Taking $\lef{\be}{}_2=-\al_2$, after usual simple
calculations we will find:
\begin{equation}
\begin{array}{l}
\be_2=-\rig{\al_2}{\!}\\
D_+D_-\al_2=(\lef{\al}{}_2-\al_2)\lef{T_0}{3}+T_0(\al_2-\rig{\al_2}{\!})+
D_-\al_1\lef{\th}{2}+\th D_-\al_2\\
\end{array} \label{54}
\end{equation}
The partial solution of this system is: $\al_2=-\be_2=1$; it correspond
to the trivial system $F_1=au_x+bv_y, F_2=av_x+bv_y$.
All systems received on even steps will be similar to (\ref{54}). Partial
solution of each next system of that kind gives non--trivial, nonlinear
evolution type system invariant with respect to the transformation (\ref{31})
(see $k=2$ example).
By induction easily can be proved that for arbitrary $n=2k+1$ we will have:
$$
D_-\al_{n-1}=\al_n\lef{T_0}{\:n}+T_0\rig{\al}{}_n
$$
$$
D_-\al_n=\al_{n+1}\lef{T_0}{\:n+1}-T_0\rig{\al}{}_{n+1}
$$
\begin{equation}
\begin{array}{rr}
D_+D_-\al_{n-1}=(\lef{\al}{}_{n-1}-\al_{n-1})\lef{T_0}{n}-T_0(\al_{n-1}-
\rig{\al}{}_{n-1})+\\+D_-\al_{n-1}\lef{\th}{n-1}+\th D_-\al_{n-1}
\end{array}
\label{144}
\end{equation}
$$
-D_+D_-\al_n=(\lef{\al}{}_n+\al_n)\lef{T_0}{n+1}-
T_0(\al_n+\rig{\al}{}_{n})+
D_-\al_n\lef{\th}{n}\\-\th D_-\al_n \label{145}
$$

After this using $\al_{2k}=1$ partial solution of the system (\ref{144}) it is
possible to construct the $k$--th term of hierarchy. One can prove using
induction that the final result can be represented as:
\begin{equation}
\begin{array}{rcl}
{\al_0}^{(k)}&=&(-1)^k \prod_{i=1}^{2k} \left(1-(-1)^iL_i
exp\left[id_i+\sum_{i=k+1}^{2k} d_k\right]\right)\times \\&&\\
&&\times D_-^{-1}(T_0 D_-^{-1}(\rig{T_0}{\!}D_-^{-1} (...D_-{-1}
\rig{T_0}{(n-1)}...)))\label{320}
\end{array}
\end{equation}
 The meaning of notations here is the same as in formula (\ref{20}).

\subsection{Examples}
\subsubsection*{k=0}
$$
v_t=v\qquad u_t=-u
$$
\subsubsection*{k=1}
$$
v_t=v_x\qquad u_t=u_x
$$
\subsubsection*{k=2}
$$
v_t=v_{xx}-2D_-^{-1}(vu)_x D_+v-2D_-^{-1}(vD_+ u)\times v+
$$
$$
+2D_-^{-1} \left[vuD_-^{-1}(vu)_x +D_-^{-1}(vu)_x \times vu\right]
$$
\vspace{1em}
$$
u_t=-u_{xx}+D_+uD_-^{-1}(vu)_x -2uD_-^{-1}(D_+vu)_x-
$$
$$
-2uD_-^{-1} \left[vuD_-^{-1}(vu)_x +D_-^{-1}(vu)_x \times vu\right]
$$

\section{Conclusion}
The main concrete result of the paper is the
explicit form of quantum integrable systems (\ref{20}), (\ref{320}) in the
mentioned above sense.



 It is interesting that the scheme of our calculations is similar to the
computer program algorithm--there are many identical operations with
possibility to interrupt them at arbitrary step. Obviously, in this scheme is
coded the structure of
the group of integrable mappings, more exactly one of the possible
connections between the integrable system by itself and its
symmetry equation. If it will be possible to
translate it on the group--theoretical language, then we will be near to
understand the integrable substitutions role and near to the classification
theorem for them.



It is well known that quantum integrable systems are closely connected with
so--called "quantum" algebras \cite{16}. Moreover this object of mathematics in
essential part was discovered and developed under the investigations of
the integrable systems in quantum region.

So it arise more wide, deep and interesting problem--to find the
connection between the approach of this paper and sufficiently developed
formalism of quantum algebras. The equations for Heisenberg
operators, as it was mentioned in the introduction, are only one of the
possible representations of the quantum picture. We hope that further
investigations will find some bridge connecting quantum integrable mappings
of the present paper with quantum algebras of the traditional approach.
But now we are not ready and able to go so far and hope to return to this
problem in the future publications.

\section{Acknowledgments}
 The authors wish to thank the Russian Foundation for Fundamental Researches
for partial support trough the Grant 95--01--00249

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\bibitem{10}
Leznov A.N., Yuzbashyan E.A.//{\sl Lett. in Math. Phys. 35:345-349, 1995}

\bibitem{12}
Leznov A.N., Yuzbashyan E.A.--Preprint MPI--96-37, Bonn 1996.

\bibitem{14}
Leznov A.N., Yuzbashyan E.A.--Preprint IHEP--95-28, Protvino 1995

\bibitem{16}
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L.D. Fadeev, N.Yu. Reshetikhin, L.A. Takhtajan Algebra\&Analiz 1(89)178
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\end{thebibliography}

\end{document}



