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%{���ࠫ� 2002}\\
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%{nlin.SI/0206044}
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\begin{center}
{\sl Theoretical and Mathematical Physics\footnote{Translated from 
Teoreticheskaya i Matematicheskaya Fizika
{\bf 132} (2002) 222.}\\
{\bf 132} (2002) 1080, nlin.SI/0206044}
\end{center}

\begin{center}
{\LARGE {\bf N=(1$|$1) supersymmetric }}\\[0.3cm]
{\LARGE {\bf dispersionless}}\\[0.3cm]
{\LARGE {\bf Toda lattice hierarchy}}\\[0.5cm]

{}~

{\large V.G. Kadyshevsky$^{(a)}$ and A.S. Sorin$^{(b)}$}

{}~\\
\quad \\


{{Bogoliubov Laboratory of Theoretical Physics,}}\\
{{Joint Institute for Nuclear Research,}}\\
{\em 141980 Dubna, Moscow Region, Russia}~\quad\\
\em {~$~^{(a)}$ Email: kadyshev@jinr.ru}\\
\em {~$~^{(b)}$ Email: sorin@thsun1.jinr.ru}\\

\end{center}

%\vfill

{}~

\centerline{{\bf Abstract}}
\noindent
Generalizing the graded commutator in superalgebras, we propose a new 
bracket
operation on the space of graded operators with an involution. We study
properties of this operation and show that the Lax representation of the
two-dimensional $N{=}(1|1)$ supersymmetric Toda lattice hierarchy can be
realized via the generalized bracket operation; this is important in
constructing the semiclassical (continuum) limit of this 
hierarchy. We construct the continuum limit of the $N{=}(1|1)$ Toda 
lattice hierarchy, the dispersionless $N{=}(1|1)$ Toda hierarchy. In this 
limit, we obtain the Lax representation, with the generalized graded 
bracket becoming the corresponding Poisson bracket on the graded phase 
superspace. We find bosonic symmetries of the dispersionless $N{=}(1|1)$ 
supersymmetric Toda equation.   
\vskip 2mm

\begin{center} 
\noindent{\sl Dedicated to the $75$th birthday of Academician \\ Anatolii
Alekseevich Logunov} 
\end{center} 

%{}~


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\section{Introduction}


In several recent decades, quantum field theory, having incorporated
efficient mathematical methods, has become a theory satisfying the most
rigorous mathematical requirements~\cite{1}. After the formulation of
supersymmetric quantum field theories, the main attention of investigators
was attracted to numerous problems whose solution is interesting in both
mathematical physics and important physical applications.

In this paper, we consider an integrable $N{=}(1|1)$ supersymmetric
generalization of the two-dimensional bosonic Toda lattice hierarchy
(2DTL hierarchy)~\cite{2} proposed in~\cite{3},~\cite{4}. It is given by an
infinite system of evolution equations (flows) for an infinite set of bosonic
and fermionic lattice fields evolving in two bosonic and two fermionic
infinite ``towers" of times; as a subsystem, it involves an $N{=}(1|1)$
supersymmetric integrable generalization of the 2DTL equation, which is
called the $N{=}(1|1)$ 2DTL in what follows.

Two new infinite series of fermionic flows of the $N{=}(1|1)$ 2DTL hierarchy
were constructed in~\cite{5}--\cite{7}. This hierarchy was shown to actually
have a higher symmetry, namely, the $N=(2|2)$ supersymmetry. Together with
the previously known bosonic flows of the $N{=}(1|1)$ 2DTL hierarchy, these
flows are symmetries of the $N{=}(1|1)$ 2DTL equation. The continuum limit of
that equation with respect to the lattice constant~\cite{8} is a
three-dimensional nonlinear equation, called the continuum or dispersionless
$N{=}(1|1)$ 2DTL equation. The solution of the corresponding Cauchy problem
was also considered in~\cite{8}.

Although the $N{=}(1|1)$ 2DTL hierarchy and the dispersionless $N{=}(1|1)$
2DTL equation have been known for a relatively long time, the problem of
constructing the continuum (semiclassical) limit with respect to 
the lattice
constant (which plays the role of the Planck constant here) for all the
$N{=}(1|1)$ 2DTL hierarchy flows is still to be solved and is quite
complicated.\footnote{We note that similar problems for
differential--difference equations were considered in~\cite{9}--\cite{16}.}
Apart from the purely academic significance of this problem, its solution is
also interesting in relation to a number of important physical and
mathematical applications. In particular, these include the semiclassical
limit of the bosonic preimage of the $N{=}(1|1)$ 2DTL hierarchy, the
dispersionless 2DTL hierarchy~\cite{17}, which unifies the 2DTL hierarchy
flows arising in the leading semiclassical approximation constructed
in~\cite{18} (see also~\cite{19}). Possible applications of the
dispersionless 2DTL hierarchy are

1. construction of a number of self-dual vacuum metrics and
Einstein--Weyl metrics,

2. twistor theory,

3. two-dimensional conformal and topological field theory, and

4. two-dimensional string theory

(also see~\cite{20}--\cite{31} and the references therein).

In view of a deep relation between the 2DTL and $N{=}(1|1)$ 2DTL hierarchies,
it is natural to hypothesize that the dispersionless $N{=}(1|1)$
supersymmetric 2DTL hierarchy must also admit similar applications in
supersymmetric generalizations of the theories listed above. This motivates
our construction of the dispersionless $N{=}(1|1)$ 2DTL hierarchy here.

We emphasize that the existing general algorithm for constructing
semiclassical asymptotic expansions, which was applied to the 2DTL hierarchy
in~\cite{18}, encounters a number of difficulties, both formal and informal,
in its direct extension to the $N{=}(1|1)$ 2DTL hierarchy. It is well-known
that in the semiclassical limit, all the operators are mapped into their
symbols defined on the corresponding phase space and the (anti)commutators
involved in the standard Lax representations are replaced with the
corresponding Poisson brackets. But the Lax representation of the $N{=}(1|1)$
2DTL hierarchy proposed in~\cite{3} is not of the (anti)commutator form, and
it is therefore unclear how the above qualitative recipe for the transition
to the semiclassical regime can be applied literally.

The analysis is also complicated by another principally important,
qualitatively new feature of the $N{=}(1|1)$ 2DTL hierarchy compared with its
bosonic preimage, namely, the operators entering the Lax representation are
defined on a space with two $Z_2$-gradings but have only one diagonal
$Z_2$-grading. This is why the symbols of these operators in the general case
cannot be expected to commute, even in the semiclassical limit.

We also mention possible complications related to the fact that the fermionic
and bosonic fields of the $N{=}(1|1)$ 2DTL hierarchy can have noncoincident
semiclassical asymptotic behaviors, which must then be specified
self-consistently. A similar situation occurs for the fermionic and bosonic
times of this hierarchy, but these can be easily made consistent using
additional dimension arguments.

Very recently, a certain progress has been observed toward constructing the
semiclassical limit of the $N{=}(1|1)$ 2DTL hierarchy. In~\cite{7}, it was
found that the Lax representation for this hierarchy can be brought to the
commutator form (Eq.~(39) in~\cite{7}) by introducing a number of new
auxiliary fermionic constants. Although the commutator representation thus
obtained seems artificial at first glance, it actually involves a new
important generalized graded bracket operation, which can be defined in
sufficiently general terms to allow a broad spectrum of applications. We
introduce and use it in solving the problem addressed here.

It turns out that all the basic relations defining the $N{=}(1|1)$ 2DTL
hierarchy can be expressed in terms of precisely this bracket operation; this
does not require introducing any auxiliary objects like the fermionic
constants mentioned above. In the semiclassical limit, moreover, precisely
this bracket operation is replaced with the corresponding Poisson superbracket
on the phase superspace. This therefore produces the Lax representation of
the dispersionless $N{=}(1|1)$ 2DTL hierarchy.

The structure of this paper is as follows. In Sec.~2, we introduce the
generalized graded bracket operation on the space of graded operators with an
involution that generalizes the graded commutator for superalgebras; we
describe its properties and give the corresponding generalized Jacobi
identities. We then obtain the Lax representation of the $N{=}(1|1)$ 2DTL
hierarchy and all the basic defining relations in terms of the generalized
graded bracket. We give an explicit expression for the $N{=}(1|1)$ 2DTL
hierarchy flows and for bosonic symmetries of the $N{=}(1|1)$ 2DTL equation,
which are consequently used in Sec.~3 to obtain the respective dispersionless
analogues.

In Sec.~3, we also find the semiclassical limit of the $N{=}(1|1)$ 2DTL
hierarchy and postulate the corresponding asymptotic behavior of the
fermionic and bosonic fields parameterizing the Lax operators. Using these
data, we then evaluate the asymptotic behavior of all the composite operators
entering the Lax representation and the corresponding field evolution
equations. We next obtain regular leading terms in the semiclassical
expansion of these evolution equations, which are by definition the flows of
the dispersionless $N{=}(1|1)$ 2DTL hierarchy. This a posteriori demonstrates
self-consistency of the postulates underlying all our calculations. The next
step is to model the Poisson superbracket on the phase superspace obtained by
extending the phase space of the dispersionless 2DTL hierarchy by one
Grassmann coordinate. Replacing the Lax operators with their symbols and
replacing the generalized graded bracket with the above Poisson superbracket
in the Lax representation of the $N{=}(1|1)$ 2DTL hierarchy and in all its
defining relations, we show by a direct calculation that the operator
representation thus obtained correctly reproduces the flows of the
dispersionless $N{=}(1|1)$ 2DTL hierarchy that we constructed previously and
is therefore the sought Lax representation of the above hierarchy.

In Sec.~4, we briefly summarize the main results obtained in this paper.

\section{$N{=}(1|1)$ 2DTL hierarchy}

In this section, we introduce a new graded bracket operation and use it to
propose a new form of the Lax representation for the $N{=}(1|1)$ 2DTL
hierarchy.   

\subsection{Generalized graded brackets}

We consider the space of operators ${\bO}_k$ with the grading 
$d_{{\bO}_k}$  ($d_{{\bO}_k}\in {\bZ}$),  
\begin{eqnarray}
d_{{\bO}_1{\bO}_2}= d_{{\bO}_1}+d_{{\bO}_2}, \label{grad}
\end{eqnarray}
and the involution $*$, 
\begin{eqnarray}
{\bO}^{*(2)}_k = {\bO}_k, \label{inv}
\end{eqnarray}
where  ${{\bO}_k}^{*(m)}$  denotes the $m$-fold action of the involution 
$*$ on the operator ${\bO}_k$. On this space, we can define the 
generalized graded bracket operation $[\,\cdot\,,\,\cdot\,\}$, 
\begin{eqnarray}
\Bigl[ {\bO}_1, {\bO}_2 \Bigr\}:={\bO}_1{\bO}_2
 - (-1)^{d_{{\bO}_1}d_{{\bO}_2}}~{{\bO}_2}^{*(d_{{\bO}_1})}~
{{\bO}_1}^{*(d_{{\bO}_2})}
  \label{brack}
\end{eqnarray}
with the easily verified properties of

{\sl a.  symmetry}               
\begin{eqnarray}
\Bigl[ {\bO}_1, {\bO}_2 \Bigr\}=(-1)^{d_{{\bO}_1}d_{{\bO}_2}+1}
\Bigl[{{\bO}_2}^{*(d_{{\bO}_1})},{{\bO}_1}^{*(d_{{\bO}_2})}
\Bigr\}, \label{prop1}
\end{eqnarray}

{\sl b. derivation}    
\begin{eqnarray}
\Bigl[ {\bO}_1, {\bO}_2{\bO}_3\Bigr\}=
\Bigl[{\bO}_1,{\bO}_2\Bigr\}{\bO}_3 +
(-1)^{d_{{\bO}_1}d_{{\bO}_2}}{{\bO}_2^{*(d_{{\bO}_1})}}
\Bigl[{{\bO}_1}^{*(d_{{\bO}_2})},{\bO}_3 \Bigr\}, \label{prop2}
\end{eqnarray}

{\sl c. Jacobi identity} 
\begin{eqnarray}
&&(-1)^{d_{{\bO}_1}d_{{\bO}_3}}\Bigl[\Bigl[
{{\bO}_1}^{*(d_{{\bO}_3)}}, {\bO}_2
\Bigr\},{{\bO}_3}^{*(d_{{\bO}_1)}}\Bigr\} \nonumber\\
&+&(-1)^{d_{{\bO}_2}d_{{\bO}_1}} \Bigl[\Bigl[
{{\bO}_2}^{*(d_{{\bO}_1)}},
{\bO}_3\Bigr\},{{\bO}_1}^{*(d_{{\bO}_2)}}\Bigr\} \nonumber\\&+&
(-1)^{d_{{\bO}_3}d_{{\bO}_2}} \Bigl[\Bigl[
{{\bO}_3}^{*(d_{{\bO}_2)}},
{\bO}_1\Bigr\},{{\bO}_2}^{*(d_{{\bO}_3)}}\Bigr\}=0. \label{jacobi}
\end{eqnarray}
Relations (\ref{prop1}--\ref{jacobi}) generalize the corresponding 
properties of the graded commutator in Lie superalgebras. We emphasize 
that in the particular case where the involution $*$ acts as the
identity transformation, bracket \p{brack} reproduces the graded Lie
superalgebra commutator. In the general case of the involution action, 
this bracket is a nontrivial generalization of that commutator.            

\subsection{Lax representation and flows}

We begin this section by detailing the space of operators, their grading, and
involution which are relevant in context of the $N{=}(1|1)$ 2DTL hierarchy. 
These operators can be represented in the general form 
\begin{eqnarray}
{\bO}_m=\sum^{\infty}_{k=-\infty}
f^{(m)}_{k,j}e^{(k-m){\partial}}, \quad m\in {\bZ}, \label{optoda}
\end{eqnarray}
parameterized by the functions $f^{(m)}_{2k,j}$ ($f^{(m)}_{2k+1,j}$) 
that are $Z_2$-graded bosonic (fermionic) lattice fields ($j \in {\bZ}$),
\begin{eqnarray}
d^{~'}_{f^{(m)}_{k,j}}=|k| \quad \mbox{mod} ~~ 2; \label{grad1}
\end{eqnarray}
the operator $e^{l{\partial}}$ ($l \in {\bZ}$) acting on these fields as 
the discrete lattice shift                                                               
\begin{eqnarray}
e^{l{\partial}} f^{(m)}_{k,j} \equiv f^{(m)}_{k,j+l}
e^{l{\partial}} \label{rule1}
\end{eqnarray}
has another $Z_2$-grading given by 
\begin{eqnarray}
d_{e^{l{\partial}}}=|l| \quad \mbox{mod}  ~~ 2. \label{grad2}
\end{eqnarray}
Operators \p{optoda} allow specifying only one diagonal 
$Z_2$-grading 
\begin{eqnarray}
d_{{\bO}_m}=d^{~'}_{f^{(m)}_{k,j}}+d_{e^{(k-m){\partial}}} =|m|
\quad \mbox{mod}  ~~ 2
 \label{gradop}
\end{eqnarray}
and the involution 
\begin{eqnarray}
{\bO}^{*}_m=\sum^{\infty}_{k=-\infty} (-1)^k
f^{(m)}_{k,j}e^{(k-m){\partial}}. \label{invtoda}
\end{eqnarray}
In what follows, we also need the projections ${({\bO}_m)}_{\pm}$
of the operators ${\bO}_m$ \p{optoda} defined as 
\begin{eqnarray}
({\bO}_m)_{+}=\sum^{\infty}_{k=m}
f^{(m)}_{k,j}e^{(k-m){\partial}}, \quad
{({\bO}_m)}_{-}=\sum^{m-1}_{k=-\infty}
f^{(m)}_{k,j}e^{(k-m){\partial}}.
 \label{optoda1}
\end{eqnarray}

The Lax operators $L^{\pm}$ of the $N{=}(1|1)$ 2DTL hierarchy belong to 
the space of operators \p{optoda} \cite{3,7}
\begin{eqnarray}
L^{+}=\sum^{\infty}_{k=0} u_{k,j}e^{(1-k){\partial}},\quad
u_{0,j}= 1, \quad L^{-}=\sum^{\infty}_{k=0}
v_{k,j}e^{(k-1){\partial}}, \quad v_{0,j}\neq 0 \label{laxs1}
\end{eqnarray}
and have the grading $d_{L^{\pm}}=1$.

We now have all the ingredients necessary to express the Lax 
representation of the $N{=}(1|1)$ 2DTL hierarchy in terms of bracket 
operation \p{brack}, thereby bringing it to a very simple form, 
\begin{eqnarray}
D^{\pm}_n L^{{\alpha}}&=& \mp {\alpha} (-1)^n
\Bigr[{(((L^{\pm})^{n}_{*})}_{-{\alpha} })^{*},
L^{{\alpha}}\Bigl\}, \quad {\alpha} =+,-, \quad n \in {\bN},
\label{laxreprs1}
\end{eqnarray}
where $D^{\pm}_{2n}$ ($D^{\pm}_{2n+1}$) are bosonic (fermionic) evolution
derivatives.                                         

For the composite operators $(L^{\pm})^{n}_{*}$ entering this 
representation, we can also obtain very simple expressions in terms of the 
Lax operators and bracket operation \p{brack},
\begin{eqnarray}
(L^{{\alpha}})^{2n}_{*}:= \Bigl(~\frac{1}{2}~\Bigr[
(L^{{\alpha}})^{*}, (L^{{\alpha}})\Bigl\}~\Bigr)^n, \quad
(L^{{\alpha}})^{2m+1}_{*}:=L^{{\alpha}}~(L^{{\alpha}})^{2n}_{*}.
\label{rule2}
\end{eqnarray}
Similarly to the Lax operators $L^{\pm}$, the operators 
$(L^{\pm})^{n}_{*}$ belong to the space of operators \p{optoda} 
and can be represented as 
\begin{eqnarray}
(L^{+})^{m}_{*}:=\sum^{\infty}_{k=0}
u^{(m)}_{k,j}e^{(m-k){\partial}}, \quad u^{(m)}_{0,j}=1, \quad
(L^{-})^{m}_{*}:=\sum^{\infty}_{k=0}
v^{(m)}_{k,j}e^{(k-m){\partial}}, \label{laxs2}
\end{eqnarray}
where  $u^{(m)}_{k,j}$ and $v^{(m)}_{k,j}$ (with $u^{(1)}_{k,j}\equiv
u_{k,j},~v^{(1)}_{k,j}\equiv v_{k,j}$) are functionals of the original 
fields $\{u_{k,j},~v_{k,j}\}$. It must be noted here that in Lax
representation \p{laxreprs1}, the  $Z_2$-grading of the operator�
$(L^{\pm})^{n}_{*}$, which has the form $d_{(L^{\pm})^{2n}_{*}}=0$ and 
$d_{(L^{\pm})^{2n+1}_{*}}=1$, agrees with another $Z_2$-grading 
$d^{~'}_{D^{\pm}_{2n}}=0$ and $d^{~'}_{D^{\pm}_{2n+1}}=1$ 
that corresponds to the statistics of the evolution derivatives 
$D^{\pm}_{n}$.

Using bracket properties (\ref{prop1}--\ref{jacobi}) and relations
(\ref{rule2}) as definitions of $(L^{\pm})^{n}_{*}$, we can easily obtain 
the useful identities   
\begin{eqnarray}
&&\Bigr[ (L^{{\alpha}})^{2n}_{*},
(L^{{\alpha}})^{2m}_{*}\Bigl\}=0, \nonumber\\
&&\Bigr[ ((L^{{\alpha}})^{2n}_{*})^{*},
(L^{{\alpha}})^{2m+1}_{*}\Bigl\}=0, \quad \Bigr[
(L^{{\alpha}})^{2n+1}_{*},
(L^{{\alpha}})^{2m}_{*}\Bigl\}=0, \nonumber\\
&&\Bigr[ ((L^{{\alpha}})^{2n+1}_{*})^{*},
(L^{{\alpha}})^{2m+1}_{*}\Bigl\}=2(L^{{\alpha}})^{2(n+m+1)}_{*}.
\label{ident}
\end{eqnarray}
Next, using Eqs. (\ref{prop1}--\ref{jacobi}) 
and (\ref{laxreprs1}--\ref{rule2}), we can derive equations of motion for 
the composite operators $(L^{\pm})^{n}_{*}$,
\begin{eqnarray}
D^{\pm}_n (L^{{\alpha}})^{m}_{*}&=& \mp {\alpha} (-1)^{nm} \Bigr[
{(((L^{\pm})^{n}_{*})}_{-{\alpha} })^{*(m)},
(L^{{\alpha}})^{m}_{*}\Bigl\}, \label{laxreprs2}
\end{eqnarray}
and the evolution equations for the functionals 
$\{u^{(m)}_{k,j},~v^{(m)}_{k,j}\}$ in \p{laxs2} implied by Eqs. 
\p{laxreprs2} found above, 
\begin{eqnarray}
D^{+}_n u^{(2m)}_{k,j}&=&\sum^{n}_{p=0}
(u^{(n)}_{p,j}u^{(2m)}_{k-p+n,j-p+n} \nonumber\\
&-&(-1)^{(p+n)(k-p+n)}u^{(n)}_{p,j-k+p-n+2m}u^{(2m)}_{k-p+n,j}),
\nonumber\\ D^{+}_{2n} u^{(2m+1)}_{k,j}&=&\sum^{2n}_{p=0}
((-1)^{p}u^{(2n)}_{p,j}u^{(2m+1)}_{k-p+2n,j-p+2n} \nonumber\\
&-&(-1)^{p(k-p)}u^{(2n)}_{p,j-k+p-2n+2m+1}u^{(2m+1)}_{k-p+2n,j}),
\nonumber\\ D^{+}_{2n+1} u^{(2m+1)}_{k,j}&=&\sum^{k}_{p=1}
((-1)^{p+1}u^{(2n+1)}_{p+2n+1,j}u^{(2m+1)}_{k-p,j-p} \nonumber\\
&+&(-1)^{p(k-p)}u^{(2n+1)}_{p+2n+1,j-k+p+2m+1}u^{(2m+1)}_{k-p,j}),
\label{equationss1}
\end{eqnarray}
\begin{eqnarray}
D^{-}_n u^{(m)}_{k,j}&=&\sum^{n-1}_{p=0}
((-1)^{(p+n)m}v^{(n)}_{p,j}u^{(m)}_{k+p-n,j+p-n} \nonumber\\
&-&(-1)^{(p+n)(k+p-n)}v^{(n)}_{p,j-k-p+n+m}u^{(m)}_{k+p-n,j}),
\label{equationss2}
\end{eqnarray}
\begin{eqnarray}
D^{+}_n v^{(m)}_{k,j}&=&\sum^{n}_{p=0}
((-1)^{(p+n)m}u^{(n)}_{p,j}v^{(m)}_{k+p-n,j-p+n} \nonumber\\
&-&(-1)^{(p+n)(k+p-n)}u^{(n)}_{p,j+k+p-n-m}v^{(m)}_{k+p-n,j}),
\label{equationss3}
\end{eqnarray}
\begin{eqnarray}
D^{-}_n v^{(2m)}_{k,j}&=&\sum^{n-1}_{p=0}
(v^{(n)}_{p,j}v^{(2m)}_{k-p+n,j+p-n} \nonumber\\
&-&(-1)^{(p+n)(k-p+n)}v^{(n)}_{p,j+k-p+n-2m}v^{(2m)}_{k-p+n,j}),
\nonumber\\ D^{-}_{2n} v^{(2m+1)}_{k,j}&=&\sum^{2n-1}_{p=0}
((-1)^{p}v^{(2n)}_{p,j}v^{(2m+1)}_{k-p+2n,j+p-2n} \nonumber\\
&-&(-1)^{p(k-p)}v^{(2n)}_{p,j+k-p+2n-2m-1}v^{(2m+1)}_{k-p+2n,j}),
\nonumber\\ D^{-}_{2n+1} v^{(2m+1)}_{k,j}&=&\sum^{k}_{p=0}
((-1)^{p+1}v^{(2n+1)}_{p+2n+1,j}v^{(2m+1)}_{k-p,j+p} \nonumber\\
&+&(-1)^{p(k-p)}v^{(2n+1)}_{p+2n+1,j+k-p-2m-1}v^{(2m+1)}_{k-p,j})
\label{equationss4b}
\end{eqnarray}
(in the right-hand side of these equations, all the fields  
$\{u^{(m)}_{k,j},~v^{(m)}_{k,j}\}$ with $k < 0$  must be set equal to 
zero).

Lax representation \p{laxreprs1} generates a non-Abelian algebra of flows 
of the $N{=}(1|1)$ 2DTL hierarchy, 
\begin{eqnarray}
[D^{+}_{n}~,~D^{-}_{l}\}=[D^{\pm}_{n}~,~D^{\pm}_{2l}]=0,\quad
\{D^{\pm}_{2n+1}~,~D^{\pm}_{2l+1}\}=2D^{\pm}_{2(n+l+1)},
\label{algebras1}
\end{eqnarray}
which can be realized as    
\begin{eqnarray}
D^{\pm}_{2n} ={\partial}^{\pm}_{2n}, \quad D^{\pm}_{2n+1}
={\partial}^{\pm}_{2n+1}+
\sum^{\infty}_{l=1}t^{\pm}_{2l-1}{\partial}^{\pm}_{2(k+l)},\quad
\partial^{\pm}_n:={\frac{\partial}{\partial t^{\pm}_n}},
\label{covder}
\end{eqnarray}
where $t^{\pm}_{2n}$ ($t^{\pm}_{2n+1}$) 
are bosonic (fermionic) evolution times.
The dispersionless $N{=}(1|1)$ 2DTL hierarchy flows are based on the
continuum limit of flows (\ref{equationss1}--\ref{equationss4b}) 
constructed in Sec.~3.1. 

\subsection{Bosonic symmetries of the $N{=}(1|1)$ 2DTL equation}

The supersymmetric $N{=}(1|1)$ 2DTL equation 
\begin{eqnarray}
D^{+}_1D^{-}_1 \ln v_{0,j}= v_{0,j+1} - v_{0,j-1} \label{todabs}
\end{eqnarray}
belongs to system of equations (\ref{equationss1}--\ref{equationss4b}). 
It can be obtained from Eq. \p{equationss2} with $\{n=m=k=1\}$,
\begin{eqnarray}
D^{-}_1 u_{1,j} = - v_{0,j} - v_{0,j+1}, \label{todabs1}
\end{eqnarray}
and from Eq. \p{equationss3} with $\{n=m=1,~ k=0\}$,
\begin{eqnarray}
D^{+}_1 v_{0,j}= v_{0,j}(u_{1,j} - u_{1,j-1}), \label{todabs2}
\end{eqnarray}
after eliminating the field $u_{1,j}$. Its bosonic symmetries 
$D^{\pm}_{2n} v_{0,j}$, with
\begin{eqnarray}
\{D^{+}_1,D^{\pm}_{2n}\}= \{D^{-}_1,D^{\pm}_{2n}\}=0,
\label{algebra2new}
\end{eqnarray}
were described in~\cite{5}--\cite{7} in terms of the iteration procedure  
\begin{eqnarray}
D^{\pm}_{2n}
v_{0,j}&=&v_{0,j}(u^{(2n)\pm}_{2n,j}-u^{(2n)\pm}_{2n,j-1}), \quad
u^{(2n)\pm}_{0,j}=1, \nonumber\\ \pm D^{\mp}_1
u^{(2n)\pm}_{k,j}&=&v_{0,j}u^{(2n)\pm}_{k-1,j-1} +
(-1)^{k}v_{0,j-k+2n+1}u^{(2n)\pm}_{k-1,j}, \label{flowss+-bos}
\end{eqnarray}
where the functions $u^{(n)\pm}_{k,j}$ are related to the original 
functionals $\{u^{(n)}_{k,j}, v^{(n)}_{k,j}\}$ as
\begin{eqnarray}
u^{(n)+}_{k,j}= u^{(n)}_{k,j}, \quad u^{(n)-}_{k,j}=
\frac{v^{(n)}_{k,-j-1}}{{\sum}^{n-k}_{m=1} v_{0,k+m-n-j-1}}
\label{relation}
\end{eqnarray}
(for detail, see \cite{7}).

The continuum limit of $N{=}(1|1)$ 2DTL equation \p{todabs} and of its
symmetries $D^{\pm}_{2n} v_{0,j}$ \p{flowss+-bos} is derived in Sec.~3.2.

\section{The dispersionless $N{=}(1|1)$ 2DTL hierarchy}

In this section, we construct the continuum (semiclassical) limit of the
$N{=}(1|1)$ 2DTL hierarchy with respect to the lattice constant, which 
gives the dispersionless $N{=}(1|1)$ 2DTL hierarchy, and construct the
corresponding Lax representation.             

\subsection{Semiclassical limit}

Flows (\ref{equationss1}--\ref{equationss4b}) of the $N{=}(1|1)$ 2DTL 
hierarchy do not involve an explicit dependence on dimensional constants, 
and the lattice with the dimensionless coordinate $j$ ($j\in {\bZ})$ 
has the unit spacing constant. To study the continuum limit, we explicitly 
introduce the lattice spacing length. This parameter is denoted by $\hbar$ 
because it plays the role of the Planck constant in what follows. Instead 
of $j$, there then arises the combination                            
\begin{eqnarray}
{\hbar} j \equiv s, \label{coord1}
\end{eqnarray}
and all the lattice fields acquire a dependence on the parameter $\hbar$. 
The continuum (semiclassical) limit can then be defined as   
\begin{eqnarray}
{\hbar}\rightarrow 0, \quad s = {\lim}_{{\hbar}\rightarrow 0, ~j
>> 1}({\hbar} j), \label{displimit1}
\end{eqnarray}
with $s$ playing the role of the continuum ``lattice" coordinate.

For the flows in Eqs. (\ref{equationss1}--\ref{equationss4b})
to be nontrivial and regular in the limit in Eq. \p{displimit1},
we must additionally perform several scaling transformations of the 
dependent and independent variables in the system. We postulate the rules 
for transition to the new evolution times and fields of the hierarchy 
given by                                            
\begin{eqnarray}
t^{\pm}_{2n+1} \rightarrow \frac{1}{\sqrt{\hbar}} t^{\pm}_{2n+1},
~~ t^{\pm}_{2n} \rightarrow \frac{1}{{\hbar}} t^{\pm}_{2n}
~\Leftrightarrow ~ D^{\pm}_{2n+1}  \rightarrow \sqrt{\hbar}
D^{\pm}_{2n+1}, ~~ D^{\pm}_{2n}  \rightarrow {\hbar}
D^{\pm}_{2n},~~~~~~ \label{displimit2}
\end{eqnarray}
\begin{eqnarray}
&&u_{2k,j} \quad ~\rightarrow ~~~~u_{2k}({\hbar} j), \quad
\quad ~~~v_{2k,j} ~\quad \rightarrow ~~~~v_{2k}({\hbar} j),\nonumber\\
&&u_{2k+1,j}~\rightarrow ~ \frac{1}{\sqrt{\hbar}}~u_{2k+1}({\hbar}
j), \quad ~~v_{2k+1,j}~~\rightarrow ~
\frac{1}{\sqrt{\hbar}}~v_{2k+1}({\hbar} j), \label{displimit3}
\end{eqnarray}
and assume they are nonsingular at $\hbar=0$. 

We can now establish two important properties of semiclassical limit 
(\ref{displimit1}--\ref{displimit3}), which have a key importance in 
what follows and are verified by direct calculations.   

The first property is that the new composite fields defined in accordance
with the rules    
\begin{eqnarray}
&&u^{(m)}_{2k,j} \quad ~\rightarrow ~~~~u^{(m)}_{2k}({\hbar} j),
\quad \quad ~v^{(m)}_{2k,j} \quad \rightarrow
~~~~~v^{(m)}_{2k}({\hbar} j), \nonumber\\
&&u^{(2m+1)}_{2k+1,j}\rightarrow ~\frac{1}{\sqrt{\hbar}}~
u^{(2m+1)}_{2k+1}({\hbar}j), \quad v^{(2m+1)}_{2k+1,j}\rightarrow
~ \frac{1}{\sqrt{\hbar}}~v^{(2m+1)}_{2k+1}({\hbar} j),\nonumber\\
&&u^{(2m)}_{2k+1,j}~\rightarrow ~\sqrt{\hbar}~
u^{(2m)}_{2k+1}({\hbar} j), \quad \quad
v^{(2m)}_{2k+1,j}~\rightarrow ~\sqrt{\hbar}~
v^{(2m)}_{2k+1}({\hbar}j), \label{displimit4}
\end{eqnarray}
are regular in the semiclassical limit.   

From rules (\ref{displimit1}--\ref{displimit4}) and the obvious 
identities  
\begin{eqnarray}
(L^{{\alpha}})^{2(m+1)}_{*}:= (L^{{\alpha}})^{2}_{*}
(L^{{\alpha}})^{2m}_{*}, \quad (L^{{\alpha}})^{(2m+1)}_{*}:=
L^{{\alpha}} (L^{{\alpha}})^{2m}_{*}, \label{ident1}
\end{eqnarray}
which follow from Eqs. \p{rule2}, we can, for example, obtain the important
recursive relations for the leading terms of the functionals  
$u^{(m)}_{k}\equiv u^{(m)}_{k}(s)$,
\begin{eqnarray}
&& u^{(2(l+1))}_{2k}
=\sum^{k}_{n=0}u^{(2)}_{2n}u^{(2l)}_{2(k-n)},\quad
u^{(2(l+1))}_{2k+1}=\sum^{2k+1}_{n=0}u^{(2)}_{n}u^{(2l)}_{2k-n+1},
\nonumber\\
&& u^{(2l+1)}_{2k} ~=\sum^{2k}_{n=0}u_{n}u^{(2l)}_{2k-n},\quad
\quad ~~u^{(2l+1)}_{2k+1}=\sum^{k}_{n=0}u_{2n+1}u^{(2l)}_{2(k-n)},
\label{rec1}
\end{eqnarray}
\begin{eqnarray}
&& u^{(2)}_{2k} =\sum^{k}_{n=0}u_{2n}u_{2(k-n)} +2\sum^{k-1}_{n=0}
(k-n-1)u_{2(k-n)-1}{\partial}_su_{2n+1}, \nonumber\\
&& u^{(2)}_{2k+1} =\sum^{k}_{n=0}\Big[(1-2n)u_{2n}{\partial}_s
u_{2(k-n)+1} +2(k-n)u_{2(k-n)+1} {\partial}_su_{2n}\Big],
\label{rec2}
\end{eqnarray}
where ${\partial}_s:= \frac{{\partial}}{{\partial s}}$.

The second property is that in semiclassical limit 
(\ref{displimit1}--\ref{displimit4}) flows 
(\ref{equationss1}--\ref{equationss4b}) of the $N{=}(1|1)$ 2DTL hierarchy 
are nontrivial and regular. Explicit expressions for their leading terms 
are as follows.                                        

For $\{m=2l, ~k=2r\}$ or $\{m=2l+1, ~k=2r+1\}$, we have 
\begin{eqnarray}
%�� \quad \{m=2l, ~&k&=2r\}, \quad ��� \quad \{m=2l+1, ~k=2r+1\} 
%\nonumber\\
D^{-}_{2n+1} u^{(m)}_{k}&=&\sum^{n-1}_{p=0}
\Bigl[2(p-n)v^{(2n+1)}_{2p+1}{\partial}_s u^{(m)}_{k+2(p-n)}\nonumber\\
&+&(k-m+2(p-n))({\partial}_sv^{(2n+1)}_{2p+1})u^{(m)}_{k+2(p-n)}\Bigr]
\nonumber\\
&+&2(-1)^k\sum^{n}_{p=0}v^{(2n+1)}_{2p}u^{(m)}_{k+2(p-n)-1},
\nonumber\\
D^{-}_{2n} u^{(m)}_{k}&=&\sum^{n-1}_{p=0}
\Bigl[2(p-n)v^{(2n)}_{2p}{\partial}_s u^{(m)}_{k+2(p-n)}\nonumber\\
&+&(k-m+2(p-n))({\partial}_sv^{(2n)}_{2p})u^{(m)}_{k+2(p-n)}
\nonumber\\
&+&2(-1)^kv^{(2n)}_{2p+1}u^{(m)}_{k+2(p-n)+1}\Bigr], \nonumber\\
D^{+}_{2n} u^{(m)}_{k}&=&\sum^{n}_{p=0}
\Bigl[2(n-p)u^{(2n)}_{2p}{\partial}_s u^{(m)}_{k+2(n-p)}\nonumber\\
&+&(k-m+2(n-p))({\partial}_su^{(2n)}_{2p})u^{(m)}_{k+2(n-p)}\Bigr]
\nonumber\\&+& 2(-1)^k \sum^{n-1}_{p=0}
u^{(2n)}_{2p+1}u^{(m)}_{k+2(n-p)-1}, \nonumber\\
D^{+}_{2n+1} v^{(m)}_{k}&=&\sum^{n}_{p=0}
\Bigl[2(n-p)u^{(2n+1)}_{2p+1}{\partial}_s v^{(m)}_{k+2(p-n)}\nonumber\\
&-&(k-m+2(p-n))({\partial}_su^{(2n+1)}_{2p+1})v^{(m)}_{k+2(p-n)} \nonumber\\
&+& 2(-1)^k u^{(2n+1)}_{2p}v^{(m)}_{k-1+2(p-n)}\Bigr], \nonumber\\
D^{+}_{2n} v^{(m)}_{k}&=&\sum^{n}_{p=0}
\Bigl[2(n-p)u^{(2n)}_{2p}{\partial}_s v^{(m)}_{k+2(p-n)}\nonumber\\
&-&(k-m+2(p-n))({\partial}_su^{(2n)}_{2p})v^{(m)}_{k+2(p-n)}\Bigr]
\nonumber\\&+& 2(-1)^k \sum^{n-1}_{p=0}
u^{(2n)}_{2p+1}v^{(m)}_{k+1+2(p-n)}, \nonumber\\
D^{-}_{2n} v^{(m)}_{k}&=&\sum^{n-1}_{p=0}
\Bigl[2(p-n)v^{(2n)}_{2p}{\partial}_s v^{(m)}_{k+2(n-p)}\nonumber\\
&-&(k-m+2(n-p))({\partial}_sv^{(2n)}_{2p})v^{(m)}_{k+2(n-p)}\nonumber\\
&+&2(-1)^kv^{(2n)}_{2p+1}v^{(m)}_{k+2(n-p)-1}\Bigr].
\label{equ1+}
\end{eqnarray}
For $\{m=2l, ~k =2r+1\}$ or $\{m=2l+1, ~k=2r\}$, we have 
\begin{eqnarray}
%�� \quad \{m=2l, ~&k&=2r+1\}, \quad ��� \quad \{m=2l+1, ~k=2r\} 
%\nonumber\\
D^{-}_n u^{(m)}_{k}&=&\sum^{n-1}_{p=0}
(-1)^{m(p+n)}\Bigl[(p-n)v^{(n)}_{p}{\partial}_s u^{(m)}_{k+p-n}\nonumber\\
&+&(k+p-n-m)({\partial}_sv^{(n)}_{p})u^{(m)}_{k+p-n}\Bigr],
\nonumber\\ D^{+}_{2n} u^{(m)}_{k}&=&\sum^{2n}_{p=0}
(-1)^{mp}\Bigl[(2n-p)u^{(2n)}_{p}{\partial}_s u^{(m)}_{k-p+2n}\nonumber\\
&+&(k-p+2n-m)({\partial}_su^{(2n)}_{p})u^{(m)}_{k-p+2n}\Bigr],
\nonumber\\ D^{+}_{n} v^{(m)}_{k}&=&\sum^{n}_{p=0}
(-1)^{m(p+n)}\Bigl[(n-p)u^{(n)}_{p}{\partial}_s v^{(m)}_{k+p-n}\nonumber\\
&-&(k+p-n-m)({\partial}_su^{(n)}_{p})v^{(m)}_{k+p-n}\Bigr],
\nonumber\\
D^{-}_{2n} v^{(m)}_{k}&=&\sum^{2n-1}_{p=0}
(-1)^{mp}\Bigl[(p-2n)v^{(2n)}_{p}{\partial}_s v^{(m)}_{k-p+2n}\nonumber\\
&-&(k-p+2n-m)({\partial}_sv^{(2n)}_{p})v^{(m)}_{k-p+2n}\Bigr],
\label{equ2+}
\end{eqnarray}
and also   
\begin{eqnarray}
D^{+}_{2n+1} u^{(2m)}_{2k}&=&2\sum^{n}_{p=0}
\Bigl[(n-p)u^{(2n+1)}_{2p+1}{\partial}_s u^{(2m)}_{2(k-p+n)}\nonumber\\
&+&(k-p+n-m)({\partial}_su^{(2n+1)}_{2p+1})u^{(2m)}_{2(k-p+n)}
\nonumber\\ &+& u^{(2n+1)}_{2p}u^{(2m)}_{2(k-p+n)+1}\Bigr],
\nonumber\\ D^{+}_{2n+1} u^{(2m+1)}_{2k+1}&=&2\sum^{k}_{p=1}
\Bigl[pu^{(2n+1)}_{2(p+n)+1}{\partial}_s u^{(2m+1)}_{2(k-p)+1}\nonumber\\
&+&(m+p-k)({\partial}_su^{(2n+1)}_{2(p+n)+1})u^{(2m+1)}_{2(k-p)+1}\Bigr]
\nonumber\\ &+& 2\sum^{k}_{p=0}
u^{(2n+1)}_{2(p+n+1)}u^{(2m+1)}_{2(k-p)},\nonumber\\
D^{+}_{2n+1} u^{(2m+1)}_{2k}&=&\sum^{2k}_{p=1}
(-1)^p\Bigl[pu^{(2n+1)}_{p+2n+1}{\partial}_s u^{(2m+1)}_{2k-p}\nonumber\\
&+&(2(m-k)+p+1)({\partial}_su^{(2n+1)}_{p+2n+1})u^{(2m+1)}_{2k-p}\Bigr],
\nonumber\\ D^{+}_{2n+1} u^{(2m)}_{2k+1}&=&\sum^{2n+1}_{p=0}
\Bigl[(2n-p+1)u^{(2n+1)}_{p}{\partial}_s u^{(2m)}_{2(k+n+1)-p}\nonumber\\
&+&(2(k+n-m+1)-p)({\partial}_su^{(2n+1)}_{p})u^{(2m)}_{2(k+n+1)-p}\Bigr],
\nonumber\\D^{-}_{2n+1} v^{(2m)}_{2k}&=&2\sum^{n-1}_{p=0}
\Bigl[(p-n)v^{(2n+1)}_{2p+1}{\partial}_s v^{(2m)}_{2(k-p+n)}\nonumber\\
&-&(k-p+n-m)({\partial}_sv^{(2n+1)}_{2p+1})v^{(2m)}_{2(k-p+n)}\Bigr]
\nonumber\\ &+&2\sum^{n}_{p=0}
v^{(2n+1)}_{2p}v^{(2m)}_{2(k-p+n)+1}, \nonumber\\D^{-}_{2n+1}
v^{(2m+1)}_{2k+1}&=&2\sum^{k}_{p=0}
\Bigl[-pv^{(2n+1)}_{2(p+n)+1}{\partial}_s v^{(2m+1)}_{2(k-p)+1}\nonumber\\
&+&(k-p-m){\partial}_sv^{(2n+1)}_{2(p+n)+1})v^{(2m+1)}_{2(k-p)+1}
\nonumber\\ &+& v^{(2n+1)}_{2(p+n+1)}v^{(2m+1)}_{2(k-p)}\Bigr],
\nonumber\\D^{-}_{2n+1} v^{(2m+1)}_{2k}&=&\sum^{k}_{p=0}
(-1)^p\Bigl[-pv^{(2n+1)}_{p+2n+1}{\partial}_s v^{(2m+1)}_{2k-p}\nonumber\\
&-&(2(m-k)+p+1)({\partial}_sv^{(2n+1)}_{p+2n+1})v^{(2m+1)}_{2k-p}\Bigr],
\nonumber\\D^{-}_{2n+1} v^{(2m)}_{2k+1}&=&\sum^{2n}_{p=0}
\Bigl[-(2n-p+1)v^{(2n+1)}_{p}{\partial}_s v^{(2m)}_{2(k+n+1)-p}\nonumber\\
&-&(2(k+n-m+1)-p)({\partial}_sv^{(2n+1)}_{p})v^{(2m)}_{2(k+n+1)-p}\Bigr],
\label{eqvv6-}
\end{eqnarray}
where all the fields $\{u^{(m)}_{k},~v^{(m)}_{k}\}$ with  $k < 0$
must be set equal to zero in the right-hand sides. Flows
(\ref{equ1+}--\ref{eqvv6-}) thus derived are said to constitute the 
dispersionless $N{=}(1|1)$ 2DTL hierarchy. The corresponding Lax 
representation is constructed in Sec.~3.3.            

\subsection{The dispersionless $N{=}(1|1)$ 2DTL equation and its bosonic
symmetries}

The dispersionless limits of $N{=}(1|1)$ 2DTL equation \p{todabs} and of 
its bosonic symmetries (\ref{flowss+-bos}) can be easily obtained using 
Eqs. (\ref{relation}--\ref{displimit4}). We respectively obtain 
\begin{eqnarray}
D^{+}_1D^{-}_1 \ln v_{0}= 2 {\partial}_s v_{0}
\label{disptoda}
\end{eqnarray}
and 
\begin{eqnarray}
D^{\pm}_{2n} v_{0}&=&v_{0}{\partial}_s
u^{(2n)\pm}_{2n}, \quad u^{(2n)\pm}_{0,j}=1, \nonumber\\
\mp D^{\mp}_1 u^{(2n)\pm}_{2k+1}&=&v_{0}{\partial}_s
u^{(2n)\pm}_{2k} + 2(n-k)({\partial}_sv_{0})u^{(2n)\pm}_{2k},
\nonumber\\ \pm D^{\mp}_1
u^{(2n)\pm}_{2k}&=&2v_{0}u^{(2n)\pm}_{2k-1}.
\label{disp-flowss+-bos}
\end{eqnarray}

Eliminating $u^{(2n)\pm}_{2k+1}$ from system of equations 
\p{disp-flowss+-bos}, we finally obtain a recursive system of equations 
for the generation of bosonic symmetries of dispersionless $N{=}(1|1)$ 
2DTL equation \p{disptoda}
\begin{eqnarray}
D^{\pm}_{2n} v_{0}&=&v_{0}{\partial}_s
u^{(2n)\pm}_{2n}, \quad u^{(2n)\pm}_{0,j}=1, \nonumber\\
-D^{\mp}_1 u^{(2n)\pm}_{2k}&=&2v_{0} {(D^{\mp}_1)}^{-1} \Bigl[
v_{0}{\partial}_s u^{(2n)\pm}_{2(k-1)} +
2(n-k+1)({\partial}_sv_{0})u^{(2n)\pm}_{2(k-1)}\Bigr].
\label{disp-flowss+-bos1}
\end{eqnarray}
We note that symmetries of the bosonic dispersionless 2DTL equation were
found in~\cite{32} (also see~\cite{26}) by solving the corresponding
sufficiently complicated symmetry equation.                           

\subsection{Lax representation}

In Sec.~3.1, we constructed flows (\ref{equ1+}--\ref{eqvv6-}) of the
dispersionless $N{=}(1|1)$ 2DTL hierarchy. We now find the corresponding 
Lax representation. Because Lax representation \p{laxreprs1} 
of the $N{=}(1|1)$ 2DTL hierarchy can be expressed in terms of generalized 
graded bracket \p{brack}, it is natural to expect that to derive its 
semiclassical limit, the corresponding dispersionless $N{=}(1|1)$ 2DTL 
hierarchy, it would suffice to replace all occurrences of this bracket 
with a certain Poisson superbracket and the operators with their symbols 
defined on the corresponding phase space.

Our immediate problem is to model the phase space and the Poisson    
superbracket starting from certain properties with which they must be
endowed. Recalling that the Lax operators of the $N{=}(1|1)$ 2DTL 
hierarchy, being defined on the space of operators graded by two different
$Z_2$-gradings (\ref{grad1}) and (\ref{grad2}), have only one diagonal
$Z_2$-grading \p{gradop}, we can assume that the phase space inherits these
properties. With the $Z_2$-grading $d_{e^{{\partial}}}=1, 
~d_{e^{2{\partial}}}=0$ of the lattice shift operator $e^{l{\partial}}$ 
taken into account, (\ref{grad2}), we assume that its counterpart on the 
phase space is given by two coordinates, the Grassmann coordinate $\pi$ 
(with $\pi^2=0$) and the bosonic coordinate $p$, via                                                                      
\begin{eqnarray}
e^{{\partial}}~\rightarrow ~ \frac{1}{{\sqrt\hbar}}~\pi, \quad
 e^{2{\partial}}~\rightarrow  ~p . \label{displax1}
\end{eqnarray}
Although the coordinate $\pi$ is Grassmann, it must have a property, not
typical of Grassmann coordinates, to commute with not only bosonic but 
also fermionic fields of the hierarchy, as follows from the continuum 
limit of relation \p{rule1}.

It is obvious that in addition to the coordinates $\pi$ and $p$, the phase
space must include a continuum ``lattice" coordinate $s$, Eq. \p{coord1}.
The resulting phase superspace $\{\pi, ~p,~ s\}$ of the dispersionless
$N{=}(1|1)$ 2DTL hierarchy contains the phase space $\{p,~ s\}$ of the 
dispersionless 2DTL hierarchy as a subspace. 

Having established the coordinates on the phase space, we can construct 
the Poisson superbracket for them. Recalling that the Poisson superbracket 
must agree with the superalgebra of the $Z_2$-graded operators    
$\{\sqrt{\hbar}e^{{\partial}}, ~e^{2{\partial}},~ {\hbar} j\}$, to which 
the phase superspace coordinates $\{\pi,~p,~ s\}$ correspond, we can find 
these superbrackets relatively easily. We here give the Poisson 
superbrackets between two arbitrary functions  
%$\Bbb F_{1,2}\equiv\Bbb F_{1,2}(\pi,p,s)$ 
%block letters
${\bF}_{1,2}\equiv {\bF}_{1,2}(\pi,p,s)$
on the phase space obtained as explained above,   
\begin{eqnarray}
\Bigl\{{\bF}_1,{\bF}_2\Bigr\}&=& 2p~\Bigl(\frac{\partial
{\bF}_1}{\partial p} ~\frac{\partial {\bF}_2}{\partial s}-
\frac{\partial {\bF}_1}{\partial s} ~\frac{\partial
{\bF}_2}{\partial p}+ \frac{\partial {\bF}_1}{\partial \pi}~
\frac{\partial {\bF}_2}{\partial \pi}\Bigr)\nonumber\\
&+&\pi ~\Bigl(\frac{\partial {\bF}_1}{\partial \pi}~
\frac{\partial {\bF}_2}{\partial s}- \frac{\partial
{\bF}_1}{\partial s}~ \frac{\partial {\bF}_2}{\partial \pi}\bigr).
\label{poissbrack}
\end{eqnarray}

We note that after the transition to the new basis
$\{{\widetilde s}, \widetilde p,\widetilde \pi\}$ 
in the phase space given by the formulas  
\begin{eqnarray}
{\widetilde s}&:=&\frac{s}{2}, \quad \widetilde p := \ln p, \quad
{\widetilde \pi} := \frac{\pi}{\sqrt{2p}}, \nonumber\\
s&:=&2{\widetilde s}, \quad p := e^{\widetilde p}, \quad \pi :=
\sqrt{2}{\widetilde \pi} e^{\frac{\widetilde p}{2}},
 \label{poissbrack2}
\end{eqnarray}
Poisson superbrackets \p{poissbrack} become 
\begin{eqnarray}
\Bigl\{{\bF}_1,{\bF}_2\Bigr\}&=& \frac{\partial {\bF}_1}{\partial
\widetilde p} ~\frac{\partial {\bF}_2}{\partial \widetilde s}-
\frac{\partial {\bF}_1}{\partial \widetilde s} ~\frac{\partial
{\bF}_2}{\partial \widetilde p}+ \frac{\partial {\bF}_1}{\partial
\widetilde \pi} ~\frac{\partial {\bF}_2}{\partial \widetilde \pi},
\label{poissbrack1}
\end{eqnarray}
which corresponds to the canonical orthosymplectic structure of the phase
superspace.  

We now proceed to the next stage of deriving the Lax representation of the
dispersionless $N{=}(1|1)$ 2DTL hierarchy. Heuristic formulas for the 
symbols ${\cal L}^{{\pm}}$ and $({\cal L}^{{\pm}})^{n}_{*}$ 
of the Lax operators $L^{\pm}$ \p{laxs1} and of the composite operators 
$(L^{\pm})^{n}_{*}$ (\ref{rule2}--\ref{laxs2}) are 
\begin{eqnarray}
L^{{\pm}}~ &\rightarrow & ~ \frac{1}{{\sqrt\hbar}}~{\cal
L}^{{\pm}},  \nonumber\\
{\cal L}^{+}=\sum^{\infty}_{k=0} (u_{2k+1}+u_{2k}\pi)p^{-k},
&\quad &{\cal L}^{-}=\sum^{\infty}_{k=0}
(v_{2k-1}+v_{2k}\pi)p^{k-1}
\label{displaxs1}
\end{eqnarray}
and
\begin{eqnarray}
&&(L^{{\pm}})^{2m}_{*}~ \rightarrow ~({\cal L}^{{\pm}})^{2m}_{*},
\quad (L^{{\pm}})^{2m+1}_{*}~ \rightarrow ~
\frac{1}{{\sqrt\hbar}}~({\cal L}^{{\pm}})^{2m+1}_{*}, \nonumber\\
&&\quad \quad \quad ({\cal L}^{+})^{2m}_{*}~~~:=~
\sum^{\infty}_{k=0} (u^{(2m)}_{2k}+u^{(2m)}_{2k-1}\pi)p^{m-k}, \nonumber\\
&&\quad \quad \quad ({\cal L}^{-})^{2m}_{*}~~~:=~
\sum^{\infty}_{k=0} (v^{(2m)}_{2k}+v^{(2m)}_{2k+1}\pi)p^{k-m}, \nonumber\\
&&\quad \quad \quad ({\cal L}^{+})^{2m+1}_{*}:=~
\sum^{\infty}_{k=0}
(u^{(2m+1)}_{2k+1}+u^{(2m+1)}_{2k}\pi)p^{m-k},\nonumber\\
&&\quad \quad \quad ({\cal L}^{-})^{2m+1}_{*}:=~
\sum^{\infty}_{k=0}
(v^{(2m+1)}_{2k-1}+v^{(2m+1)}_{2k}\pi)p^{k-m-1}, \label{diplaxs2}
\end{eqnarray}
respectively. By definition, all the fields  
$\{u^{(m)}_{k},~v^{(m)}_{k}\}$ with $k<0$  must be set equal to zero. In 
obtaining these expressions we have used substitutions \p{displimit4} and 
\p{displax1}. We note that the above symbols are not commutative in 
general,                        
\begin{eqnarray}
({\cal L}^{{\alpha}})^{k}_{*} ({\cal L}^{{\beta}})^{m}_{*}=
(-1)^{km}(({\cal L}^{{\beta}})^{m}_{*})^{*(k)} (({\cal
L}^{{\alpha}})^{k}_{*})^{*(m)}, \quad {\alpha},{\beta} =+,-,
\label{dispprop3}
\end{eqnarray}
which is related to the atypical property of the Grassmann coordinate 
$\pi$ noted above (see the remark after Eq. \p{displax1}).

In Lax representations (\ref{laxreprs1}) and \p{laxreprs2} for the 
$N{=}(1|1)$ 2DTL hierarchy and in defining relations (\ref{rule2}), 
we finally replace the Lax operators with their symbols
(\ref{displaxs1}--\ref{diplaxs2}) and the generalized
graded bracket (\ref{brack}) with Poisson superbracket \p{poissbrack} 
in accordance with                           
\begin{eqnarray}
\Bigl[~ . ~, ~. ~\Bigr\} ~\rightarrow  ~
{\hbar}~\Bigl\{~.~,~.~\Bigr\} \label{dispbrac}
\end{eqnarray}
and then perform substitution \p{displimit2} and take limit 
\p{displimit1}. This gives the expressions
\begin{eqnarray}
D^{\pm}_n {\cal L}^{{\alpha}}&=& \mp {\alpha} (-1)^n \Bigr\{
{((({\cal L}^{\pm})^{n}_{*})}_{-{\alpha} })^{*}, {\cal
L}^{{\alpha}}\Bigl\}, \quad n \in {\bN}, \quad {\alpha} =+,-,
\label{displaxreprs1}
\end{eqnarray}
\begin{eqnarray}
D^{\pm}_n ({\cal L}^{{\alpha}})^{m}_{*}&=& \mp {\alpha} (-1)^{nm}
\Bigr\{ {((({\cal L}^{\pm})^{n}_{*})}_{-{\alpha} })^{*(m)}, ({\cal
L}^{{\alpha}})^{m}_{*}\Bigl\},\quad n,m \in {\bN},
\label{displaxreprs2}
\end{eqnarray}
\begin{eqnarray}
({\cal L}^{{\alpha}})^{2m}_{*}:= \Bigl(~\frac{1}{2}~\Bigl\{({\cal
L}^{{\alpha}})^{*},{\cal L}^{{\alpha}}\Bigr\}~\Bigr)^{m},\quad
({\cal L}^{{\alpha}})^{2m+1}_{*}:={\cal L}^{{\alpha}}~({\cal
L}^{{\alpha}})^{2m}_{*} \label{disprule2}
\end{eqnarray}
for the dispersionless $N{=}(1|1)$ 2DTL hierarchy. Direct calculations
confirm that obtained relations (\ref{displaxreprs2}--\ref{disprule2}) 
correctly reproduce dispersionless flows (\ref{rec1}--\ref{eqvv6-}) and 
are therefore the sought Lax representation for the dispersionless 
$N{=}(1|1)$ 2DTL hierarchy. 

%zdes' okonchil vchera

\section{ Conclusions}

We have proposed new bracket operation (\ref{brack}) with the properties 
in Eqs. (\ref{prop1}--\ref{jacobi}) on the space of graded operators with 
an involution; this bracket generalizes the graded commutator in  
superalgebras. We then obtained a new form of Lax representation
(\ref{laxreprs1}--\ref{rule2}) and \p{laxreprs2} 
for the two-dimensional $N{=}(1|1)$ supersymmetric Toda lattice
hierarchy in terms of the generalized graded bracket. We next used this
representation to construct semiclassical limit 
(\ref{displimit1}--\ref{displimit4}) of this hierarchy\,---\,the 
dispersionless $N{=}(1|1)$ Toda hierarchy in 
Eqs. (\ref{equ1+}--\ref{eqvv6-})\,---\,and its Lax representation
(\ref{displaxs1}--\ref{diplaxs2}), (\ref{displaxreprs1}--\ref{disprule2}) 
on the graded phase superspace with Poisson bracket \p{poissbrack}. 
Finally, we established bosonic symmetries \p{disp-flowss+-bos1}
of dispersionless $N{=}(1|1)$ supersymmetric Toda equation \p{disptoda}.

{}~

{\sl One of us (V.~K.) first became acquainted with A.~A.~Logunov 
while still a second-year student in the Physics Department of Moscow 
State University (1956) and over the decades since then have always 
felt his firm support in matters both within and outside science. All our 
best wishes, dear Anatolii Alekseevich!}   

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%{���ࠫ� 2002}\\
%{JINR E2-???-2002}\\
%{nlin.SI/0206044}
%\end{flushright}

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%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{center}
{\sl Theoretical and Mathematical Physics\footnote{Translated from
Teoreticheskaya i Matematicheskaya Fizika
{\bf 132} (2002) 222.}\\
{\bf 132} (2002) 1080, nlin.SI/0206044}
\end{center}  


\begin{center}
{\LARGE {\bf N=(1$|$1) supersymmetric }}\\[0.3cm]
{\LARGE {\bf dispersionless}}\\[0.3cm]
{\LARGE {\bf Toda lattice hierarchy}}\\[0.5cm]

{}~

{\large V.G. Kadyshevsky$^{(a)}$ and A.S. Sorin$^{(b)}$}

{}~\\
\quad \\


{{Bogoliubov Laboratory of Theoretical Physics,}}\\
{{Joint Institute for Nuclear Research,}}\\
{\em 141980 Dubna, Moscow Region, Russia}~\quad\\
\em {~$~^{(a)}$ Email: kadyshev@jinr.ru}\\
\em {~$~^{(b)}$ Email: sorin@thsun1.jinr.ru}\\

\end{center}

%\vfill

{}~

\centerline{{\bf Abstract}}
\noindent
Generalizing the graded commutator in superalgebras, we propose a new
bracket operation on the space of graded operators with an involution. We
study properties of this operation and show that the Lax representation of
the two-dimensional $N{=}(1|1)$ supersymmetric Toda lattice hierarchy can
be realized via the generalized bracket operation; this is important in
constructing the semiclassical (continuum) limit of this
hierarchy. We construct the continuum limit of the $N{=}(1|1)$ Toda
lattice hierarchy, the dispersionless $N{=}(1|1)$ Toda hierarchy. In this
limit, we obtain the Lax representation, with the generalized graded
bracket becoming the corresponding Poisson bracket on the graded phase
superspace. We find bosonic symmetries of the dispersionless $N{=}(1|1)$
supersymmetric Toda equation.   
\vskip 2mm

\newpage

%\vfill

%{\it {\Large �}���頥��� 75--���� ��������� {\Large �}��⮫�� 
%{\Large  }���ᥥ���
%{\Large �}��㭮��}

%{}~

\begin{center}
{\LARGE {\bf N=(1$|$1) �㯥�ᨬ����筠� }}\\[0.3cm]
{\LARGE {\bf �����ᯥ�ᨮ����}}\\[0.3cm]
{\LARGE {\bf ���筠� ������ ����}}\\[0.5cm]

{\large {\LARGE �}.{\LARGE �}. {\LARGE �}���襢᪨�$^{(a)}$ � �.{\LARGE �}.
{\LARGE �}�ਭ$^{(b)}$}{}~\\
\quad \\

{{{\Large �}������� {\Large �}�����᪮� {\Large �}����� ��.
{\Large �}.{\Large �}. {\Large �}������,}}\\
{{��ꥤ������ {\Large �}������ {\Large �}����� {\Large �}�᫥�������,}}\\
{\em {141980 �㡭�, {\Large �}�᪮�᪠� �������, {\Large �}����}}~\quad\\
{\em {~$~^{(a)}$ e-mail: kadyshev@jinr.ru}}\\
{\em {~$~^{(b)}$ e-mail: sorin@thsun1.jinr.ru}}

\end{center}

{}~

\centerline{{\bf �������}}
\noindent
{\Large �}।������ ᪮��筠� ������ �� ����࠭�⢥ �ࠤ�஢�����
�����஢ � ������樥�, �������� �ࠤ�஢���� ��������
�㯥ࠫ����. {\Large �}�������, �� �।�⠢����� {\Large �}��� ��� ��㬥୮�
$N=(1|1)$ �㯥�ᨬ����筮� ���筮� ����娨 ���� ����� ����
ॠ�������� ��� ����饭��� ᪮��筠� ������, �� ����� ���
����஥��� �����������᪮�� (�����뢭���) �।��� �⮩ ����娨.
{\Large �}���஥� �����뢭� �।�� $N=(1|1)$ ���筮� ����娨
���� --- �����ᯥ�ᨮ���� $N=(1|1)$ ������ ����, � ����祭�
��� �।�⠢����� {\Large �}���, ��� ����饭��� �ࠤ�஢����� ᪮���
���室�� � ᮮ⢥�������� ᪮��� {\Large �}��ᮭ� �� �ࠤ�஢�����
䠧���� �㯥����࠭�⢥. {\Large �}������ ������� ᨬ���ਨ
�����ᯥ�ᨮ����� $N=(1|1)$ �㯥�ᨬ����筮�� �ࠢ����� ����.

{}~


{\it {\Large �}���頥��� 75--���� ��������� {\Large �}��⮫�� {\Large �}���ᥥ���
{\Large �}��㭮��}



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\section{{\Large �}�������}

$\quad ~${\Large �} ��᫥���� ��᪮�쪮 ����⨫�⨩ ����⮢�� ⥮�� ����
({\Large ��}), �ਭ� �� ����㦥��� ��䥪⨢�� ��⥬���᪨� ��⮤�,
�ॢ�⨫��� � ⥮��, 㤮���⢮������ ᠬ� ��ண��
��⥬���᪨� �ॡ������ \cite{blot}. {\Large �}�᫥ �������������
�㯥�ᨬ������� {\Large ��} �ᮡ�� �������� ��᫥����⥫�� �ਢ�����
������᫥��� �஡����, �����, c ����� ��஭�, �।�⠢����
����� ��� ��⥬���᪮� 䨧���, a, � ��㣮� ��஭�, ������
����� 䨧��᪨� �ਫ������.

{\Large �} �����饩 ࠡ�� �㤥� ��ᬮ�७� ��⥣��㥬�� $N=(1|1)$
�㯥�ᨬ����筮� ����饭�� ���筮� ��㬥୮� �������� ����娨
���� (2DTL ����娨) \cite{ut}, �।�������� � \cite{i,t}. {\Large �}��
�।�⠢��� ᮡ�� ��᪮����� ��⥬� �����樮���� (�� ���
������� � ��� �ନ���� ��᪮���� "����" �६��) �ࠢ�����
(��⮪��) ��� ��᪮��筮�� ����� ������ �������� � �ନ�����
����� � ᮤ�ন� ��� �����⥬� $N=(1|1)$ �㯥�ᨬ����筮�
��⥣��㥬�� ����饭�� 2DTL �ࠢ�����, ������砥��� � ���쭥�襬
���।�⢮� $N=(1|1)$ 2DTL.

{\Large �}������ � ࠡ��� \cite{ls1,ols1,vgks} �뫨 ����஥�� ��� ����
��᪮���� �ਨ �ନ����� ��⮪�� $N=(1|1)$ 2DTL ����娨 �,
��� ᫥��⢨�, �뫮 ��⠭������, �� �� ������, �
����⢨⥫쭮��, �������� ����� ��᮪�� ᨬ���ਥ�, � ������
$N=(2|2)$ c㯥�ᨬ���ਥ�. {\Large �}� ��⮪�, ᮢ���⭮ � ࠭��
������묨 ������묨 ��⮪��� $N=(1|1)$ 2DTL ����娨, �����
ᨬ����ﬨ $N=(1|1)$ 2DTL �ࠢ����� \cite{ls1,ols1,vgks}.
{\Large �}����뢭� �।�� �� 蠣� ��⪨ ��᫥����� �ࠢ����� \cite{ss}
�।�⠢��� ᮡ�� ��嬥୮� ���������� �ࠢ�����, ���뢠����
�����뢭� ��� �����ᯥ�ᨮ��� $N=(1|1)$ 2DTL �ࠢ������; �
\cite{ss} ⠪�� ��ᬠ�ਢ����� �襭�� ᮮ⢥�����饩 �����
���.

{\Large �}��� $N=(1|1)$ 2DTL ������ � �����ᯥ�ᨮ���� $N=(1|1)$ 2DTL
�ࠢ����� ������� �����筮 �����, {\it �஡���� ����஥���
�����뢭��� (�����������᪮��) �।��� �� 蠣� ��⪨,
����饣� ����� ஫� ����ﭭ�� {\Large �}�����, ��� ��� ��⮪�� $N=(1|1)$
2DTL ����娨 ���� �� �� �襭��� � �����筮 ᫮����
����祩}\footnote{{\Large �} �⮩ �裡 ����⨬, �� ��������� ����� ���
����७樠�쭮--ࠧ������ �ࠢ����� ��ᬠ�ਢ����� � 横��
ࠡ�� \cite{vgk1,vgk2,vgk3,vgk4,vgk5,vgk7,vgk8,vgk9}.}. �஬�
��� ��������᪮�� ���祭�� �⮩ �஡����, ����� � �� �襭��
�易� � �冷� ������ 䨧��᪨� � ��⥬���᪨� �ਫ������. {\Large �}���
����, � ��⭮��, � �����������᪮� �।��� ��������� �ம�ࠧ�
$N=(1|1)$ 2DTL ����娨 --- �����ᯥ�ᨮ���� 2DTL ����娨
\cite{tt1}, �।�⠢���饩 ᮡ�� ��ꥤ������ ��⮪�� 2DTL
����娨, ���������� � �������饬 �ਡ������� �����������᪮��
ࠧ�������, ����஥���� � ࠡ�� \cite{tt2} (�. ⠪�� �����
\cite{tt3}). {\Large �} ����⢥ �������樨, ����� �ਢ��� ���祭�
��������� �ਫ������ �����ᯥ�ᨮ���� 2DTL ����娨:\\
$\quad ~~~~~~~$1. ����஥��� �鸞 ᠬ��㠫��� ����㬭�� ���ਪ � ���ਪ \\
$\quad ~~~~~~~~~~$ {\Large �}���⥩��--{\Large �}����; \\
$\quad ~~~~~~~$2. ⥮ਨ ⢨��஢;\\
$\quad ~~~~~~~$3. ��㬥୮� ����ଭ�� � ⮯������᪮� ⥮ਨ
����;\\
$\quad ~~~~~~~$4. ��㬥୮� ⥮ਨ ���� \\
(�., ⠪�� \cite{bf,b,p,w,k,dss,hp,tt4,bx,ct,kkk,ak} �
�ਢ������ ⠬ ��뫪�).

{\Large �}��� � ���� ��㡮��� ��� ����� 2DTL � $N=(1|1)$ 2DTL �����ﬨ,
�।�⠢����� ����⢥��� ��������, �� � �����ᯥ�ᨮ����
$N=(1|1)$ �㯥�ᨬ����筠� 2DTL ������ ������ ���������
�ਫ������ � �㯥�ᨬ������� ����饭��� ����᫥���� ���
⥮਩. {\it {\Large �}�᫥���� �����⥫��⢮ ���� �⨬㫮� ���
����஥��� �����ᯥ�ᨮ���� $N=(1|1)$ 2DTL ����娨 � ������
ࠡ��.}

����� �� �⥫� �� �ᮡ� ����ભ���, �� �������騩 ��騩
������ ����஥��� �����������᪨� �ᨬ��⨪, �ᯮ�짮����� �
ࠡ�� \cite{tt2} ��� 2DTL ����娨, ����砥� �� ��� �ଠ����,
⠪ � ���ଠ���� �९���⢨� �� ��אַ�������� �����࠭����
�� ��砩 $N=(1|1)$ 2DTL ����娨. ���, ��饨����⭮, �� �
�����������᪮� �।��� �� ������� ���室�� � �� ᨬ����,
��������� �� ᮮ⢥�����饬 䠧���� ����࠭�⢥, �
(���)��������� � �⠭������ �।�⠢������ {\Large �}��� ����������
ᮮ⢥�����騬� ᪮����� {\Large �}��ᮭ�. {\Large �}� �� ��ᠥ��� �।���������
� ࠡ�� \cite{i} �।�⠢����� {\Large �}��� ��� $N=(1|1)$ 2DTL ����娨,
� ��� �� ����� (���)�������୮�� ����. {\Large �}��⮬� �㪢��쭮�
�ਬ������ ��襯ਢ�������� ����⢥����� �楯� ���室� �
���������ᨪ� ������� ����㤭�⥫��.

��㣮� �ਭ樯���쭮 ������, ����⢥��� ����� ��⮩ $N=(1|1)$
2DTL ����娨 �� �ࠢ����� � �� ������� �ம�ࠧ��, �᫮����饩
��ᬮ�७��, ���� �, �� �������, �室�騥 � ��
�।�⠢����� {\Large �}���, �������� �� ����࠭�⢥ � ����
$Z_2$--�ࠤ�஢����, �������, ������, ⮫쪮 ����� ��������쭮�
$Z_2$--�ࠤ�஢���. {\Large �}� �⮩ ��稭� � ��饬 ��砥 ���� �
�����������᪮� �।��� ����� ������� ������⨢���� ᨬ�����
��� �����஢.

� ᪠������� ����� ⠪�� �������� � �������� �᫮������,
�易��� � ⥬, �� �ନ���� � ������� ���� $N=(1|1)$ 2DTL
����娨 ����� ����� �� ᮢ�����騥 �����������᪨� �ᨬ��⨪�,
����� ����室��� ������ ᠬ�ᮣ��ᮢ����. �������筠� �����
����� ���� ��� �ନ����� � �������� �६�� �⮩ ����娨, ������
��᫥���� ����� ����� ᮣ��ᮢ���, �������⥫쭮 �ਢ�����
ᮮ�ࠦ���� ࠧ��୮��.

{\Large �}��ᥬ ������� ����⨫�� ��।������ �ண��� �� ��� ����஥���
�����������᪮�� �।��� $N=(1|1)$ 2DTL ����娨. ���, � ࠡ��
\cite{vgks} �뫮 �����㦥��, �� �।�⠢����� {\Large �}��� ��� �⮩
����娨 ����� ���� �ਢ����� � �������୮� �ଥ (�ࠢ�����
(39) � \cite{vgks}) ���।�⢮� �������� �鸞 �����
�ᯮ����⥫��� �ନ����� ����ﭭ�� (��⠫� �. \cite{vgks}).
{\Large �}��� ����砥��� ⠪�� ᯮᮡ�� �������୮� �।�⠢�����, ��
���� �����, ������� �����⢥���, � ����⢨⥫쭮��, � ���
������஢��� ������ ����� ����饭��� �ࠤ�஢����� ᪮��筠�
������, ����� ����� ���� ��।����� � �����筮 ����
�ନ���, �⮡� ����� �ப�� ᯥ��� �ਫ������. {\it �� ������ ��
� ��ᬮ�७�� � �����饩 ࠡ�� � �ᯮ��㥬 ��� �襭��
���砥��� ����� �஡����.}

����뢠����, �� ������ � �ନ��� ������ ᪮��筮� ����樨 �����
���� ��ࠦ��� �� �᭮��� ᮮ⭮襭��, ��।����騥 $N=(1|1)$
2DTL ������, � ��� �⮣� �� �ॡ���� �������� �������
�ᯮ����⥫��� ��ꥪ⮢, ⨯� 㯮������ ��� �ନ�����
����ﭭ��. {\Large �}���� ⮣�, � �����������᪮� �।��� ������ ��
᪮��筠� ������ ��������� ᮮ⢥�����饩 �㯥�᪮����
{\Large �}��ᮭ� �� 䠧���� �㯥����࠭�⢥. ��� ᠬ� ��������
�।�⠢����� {\Large �}��� ��� �����ᯥ�ᨮ���� $N=(1|1)$ 2DTL ����娨.

{\Large �}������ ������ ࠡ��� ⠪���. {\Large �} ��ࠣ�� 2 �� ������ ����饭���
�ࠤ�஢����� ᪮����� ������ �� ����࠭�⢥ �ࠤ�஢�����
�����஢ � ������樥�, ��������� �ࠤ�஢���� �������� ���
�㯥ࠫ����, ����뢠�� �� ᢮��⢠ � �ਢ���� ᮮ⢥�����騥
����饭�� ⮦���⢠ {\Large �}����. {\Large �}�⥬ �� ����砥�
�।�⠢����� {\Large �}���
$N=(1|1)$ 2DTL ����娨, � ⠪�� �� �᭮��� ��।����騥 ���
ᮮ⭮襭�� � �ନ��� ����饭��� �ࠤ�஢����� ᪮���. ��
�ਢ���� ����� � ��ࠦ���� ��� ��⮪�� $N=(1|1)$ 2DTL ����娨
� �������� ᨬ���਩ $N=(1|1)$ 2DTL �ࠢ�����, �����
���᫥��⢨� �ᯮ������� � ��ࠣ�� 3 ��� ����祭�� ��
�����ᯥ�ᨮ���� ��������. {\Large �} ��ࠣ�� 3 ⠪�� ��।������
�����������᪨� �।�� $N=(1|1)$ 2DTL ����娨 � ����㫨�����
ᮮ⢥�����饥 �ᨬ����᪮� ��������� �ନ����� � ��������
�����, ��ࠬ��ਧ���� ������� {\Large �}���.

{\Large �}�⥬ � ������� ��� ������ �� ����塞 �ᨬ����᪮� ���������
��� ���������� �����஢, �室��� � �।�⠢����� {\Large �}��� �
ᮮ⢥�����騥 ��� ������ �����樮��� �ࠢ�����. ����� ��
����砥� ����⨢��稢�, ॣ���� �������騥 童��
�����������᪮�� ࠧ������� ��� �����樮���� �ࠢ�����,
������, �� ��।������, ��⮪��� �����ᯥ�ᨮ���� $N=(1|1)$
2DTL ����娨. ��� ᠬ� �����ਮ� �������������
ᠬ�ᮣ��ᮢ������� ���� ����㫠⮢, ������ � �᭮�� ���
�஢������� ���᫥���.

{\Large �}�����騩 蠣 --- ������஢���� �㯥�᪮��� {\Large �}��ᮭ� �� 䠧����
�㯥����࠭�⢥, �।�⠢���饬 ᮡ�� 䠧���� ����࠭�⢮
�����ᯥ�ᨮ���� 2DTL ����娨, ���७��� ����� ���ᬠ�����
���न��⮩. {\Large �}, �������, ������� ������� {\Large �}��� �� �� ᨬ����, �
����饭��� �ࠤ�஢����� ᪮��� �� 㪠������ �㯥�᪮��� {\Large �}��ᮭ�
� �।�⠢����� {\Large �}��� ��� $N=(1|1)$ 2DTL ����娨 � �� ���
��।������ ��� ᮮ⭮襭���, �� 㡥������� ���� ���᫥����,
�� ����砥��� ⠪�� ᯮᮡ�� �����୮� �।�⠢����� �ࠢ��쭮
���ந������ ����஥��� ���� ࠭�� ��⮪� �����ᯥ�ᨮ����
$N=(1|1)$ 2DTL ����娨, �.�. ���� �᪮�� �।�⠢������
{\Large �}��� ��� ��᫥���� ����娨. {\Large �} ��ࠣ�� 4 �� ��⪮ १�㥬
�᭮��� १�����, ����祭�� � ࠡ��.

\section{N=(1$|$1) 2DTL ������}

$\quad$ {\Large �} �⮬ ��ࠣ�� �������� ����� �ࠤ�஢����� ᪮��筠�
������ � �� �⮩ �᭮�� �।�������� ����� �ଠ ���
�।�⠢����� {\Large �}��� $N=(1|1)$ 2DTL ����娨.

\subsection{{\Large �}���饭�� �ࠤ�஢���� ᪮���}

$\quad$ {\Large �}��ᬮ�ਬ ����࠭�⢮ �����஢ ${\bO}_k$ �
�ࠤ�஢��� $d_{{\bO}_k}$ ($d_{{\bO}_k}\in {\bZ}$),
\begin{eqnarray}
d_{{\bO}_1{\bO}_2}= d_{{\bO}_1}+d_{{\bO}_2}, \label{grad}
\end{eqnarray}
� ������樥� $*$,
\begin{eqnarray}
{\bO}^{*(2)}_k = {\bO}_k, \label{inv}
\end{eqnarray}
��� ����� � � ���쭥�襬 ${{\bO}_k}^{*(m)}$ ������砥�
$m$--��⭮� ����⢨� ������樨 $*$ �� ������ ${\bO}_k$. {\Large �}� �⮬
����࠭�⢥ ����� ��।�����
%᫥������
����饭��� �ࠤ�஢����� ᪮����� ������ $[..., ...
\}$:
\begin{eqnarray}
\Bigl[ {\bO}_1, {\bO}_2 \Bigr\}:={\bO}_1{\bO}_2
 - (-1)^{d_{{\bO}_1}d_{{\bO}_2}}~{{\bO}_2}^{*(d_{{\bO}_1})}~
{{\bO}_1}^{*(d_{{\bO}_2})},
  \label{brack}
\end{eqnarray}
� ᫥���騬� ����� �஢��塞묨 ᢮��⢠��:

{\it {\Large �}�������}
\begin{eqnarray}
\Bigl[ {\bO}_1, {\bO}_2 \Bigr\}=(-1)^{d_{{\bO}_1}d_{{\bO}_2}+1}
\Bigl[{{\bO}_2}^{*(d_{{\bO}_1})},{{\bO}_1}^{*(d_{{\bO}_2})}
\Bigr\}, \label{prop1}
\end{eqnarray}

{\it ����७�஢����}
\begin{eqnarray}
\Bigl[ {\bO}_1, {\bO}_2{\bO}_3\Bigr\}=
\Bigl[{\bO}_1,{\bO}_2\Bigr\}{\bO}_3 +
(-1)^{d_{{\bO}_1}d_{{\bO}_2}}{{\bO}_2^{*(d_{{\bO}_1})}}
\Bigl[{{\bO}_1}^{*(d_{{\bO}_2})},{\bO}_3 \Bigr\}, \label{prop2}
\end{eqnarray}

{\it ������⢠ {\Large �}����}
\begin{eqnarray}
&&(-1)^{d_{{\bO}_1}d_{{\bO}_3}}\Bigl[\Bigl[
{{\bO}_1}^{*(d_{{\bO}_3)}}, {\bO}_2
\Bigr\},{{\bO}_3}^{*(d_{{\bO}_1)}}\Bigr\} \nonumber\\
&+&(-1)^{d_{{\bO}_2}d_{{\bO}_1}} \Bigl[\Bigl[
{{\bO}_2}^{*(d_{{\bO}_1)}},
{\bO}_3\Bigr\},{{\bO}_1}^{*(d_{{\bO}_2)}}\Bigr\} \nonumber\\&+&
(-1)^{d_{{\bO}_3}d_{{\bO}_2}} \Bigl[\Bigl[
{{\bO}_3}^{*(d_{{\bO}_2)}},
{\bO}_1\Bigr\},{{\bO}_2}^{*(d_{{\bO}_3)}}\Bigr\}=0. \label{jacobi}
\end{eqnarray}
{\Large �}��⭮襭�� (\ref{prop1}--\ref{jacobi}) ������� ᮮ⢥�����騥
᢮��⢠ �ࠤ�஢������ �������� �㯥ࠫ���� {\Large �}�. �� �⥫� ��
����ભ���, �� � ��⭮� c��砥, ����� �������� $*$ \p{inv}
������� ��� ⮦���⢥���� �८�ࠧ������, ᪮��� \p{brack}
���ந������ �ࠤ�஢���� �������� �㯥ࠫ���� {\Large �}�. {\Large �} ��砥
�� ���ਢ���쭮�� ����⢨� ������樨 �� ᪮��� ����
���ਢ����� ����饭��� ��᫥�����.

\subsection{{\Large �}।�⠢����� {\Large �}��� � ��⮪�}

$\quad$ �� ��筥� ��� ࠧ��� � ��⠫���樨 ����࠭�⢠
�����஢, �� �ࠤ�஢�� � ������樨, ������ �⭮襭�� �
��ᬠ�ਢ����� ����� $N=(1|1)$ 2DTL ����娨.

�� ������� ����� ���� �।�⠢���� � ᫥���饬 ��饬 ����:
\begin{eqnarray}
{\bO}_m=\sum^{\infty}_{k=-\infty}
f^{(m)}_{k,j}e^{(k-m){\partial}}, \quad m\in {\bZ}, \label{optoda}
\end{eqnarray}
��� ��ࠬ��ਧ��騥 �� �㭪樨 $f^{(m)}_{2k,j}$
($f^{(m)}_{2k+1,j}$) ����� $Z_2$--�ࠤ�-\\஢���묨
%$(d^{'}_{f^{(m)}_{2k,j}}=0,~d^{'}_{f^{(m)}_{2k+1,j}}=1)$
\begin{eqnarray}
d^{~'}_{f^{(m)}_{k,j}}=|k| \quad \mbox{�� ~~ �����} ~~ 2 \label{grad1}
\end{eqnarray}
������묨 (�ନ���묨) ����묨 ���ﬨ ($j \in {\bZ}$), �
������ $e^{l{\partial}}$ ($l \in {\bZ}$) ������� �� �� ����
��� ������ ����� ᤢ��
\begin{eqnarray}
e^{l{\partial}} f^{(m)}_{k,j} \equiv f^{(m)}_{k,j+l}
e^{l{\partial}} \label{rule1}
\end{eqnarray}
� �������� ��㣮� $Z_2$--�ࠤ�஢���
\begin{eqnarray}
d_{e^{l{\partial}}}=|l| \quad \mbox{�� ~~ �����}  ~~ 2. \label{grad2}
\end{eqnarray}
������� \p{optoda} ����᪠�� ������� ⮫쪮 ����� ��������쭮�
$Z_2$--�ࠤ�஢��
\begin{eqnarray}
d_{{\bO}_m}=d^{~'}_{f^{(m)}_{k,j}}+d_{e^{(k-m){\partial}}} =|m|
\quad \mbox{�� ~~ �����}  ~~ 2
 \label{gradop}
\end{eqnarray}
� ������樨
\begin{eqnarray}
{\bO}^{*}_m=\sum^{\infty}_{k=-\infty} (-1)^k
f^{(m)}_{k,j}e^{(k-m){\partial}}. \label{invtoda}
\end{eqnarray}
{\Large �} ���쭥�襬 ��� ⠪�� ����������� �஥�樨 ${({\bO}_m)}_{\pm}$
�����஢ ${\bO}_m$ \p{optoda}, ��।��塞� ���
\begin{eqnarray}
({\bO}_m)_{+}=\sum^{\infty}_{k=m}
f^{(m)}_{k,j}e^{(k-m){\partial}}, \quad
{({\bO}_m)}_{-}=\sum^{m-1}_{k=-\infty}
f^{(m)}_{k,j}e^{(k-m){\partial}}.
 \label{optoda1}
\end{eqnarray}

������� {\Large �}��� $L^{\pm}$ $N=(1|1)$ 2DTL ����娨 �ਭ�������
����࠭��� �����஢ \p{optoda} \cite{i,vgks}
\begin{eqnarray}
L^{+}=\sum^{\infty}_{k=0} u_{k,j}e^{(1-k){\partial}},\quad
u_{0,j}= 1, \quad L^{-}=\sum^{\infty}_{k=0}
v_{k,j}e^{(k-1){\partial}}, \quad v_{0,j}\neq 0 \label{laxs1}
\end{eqnarray}
� ����� �ࠤ�஢�� $d_{L^{\pm}}=1$.

������ �� �ᯮ������ �ᥬ� ����室��묨 ����묨 ��� ⮣�, �⮡�
��ࠧ��� �।�⠢����� {\Large �}��� $N=(1|1)$ 2DTL ����娨 � �ନ���
᪮��筮� ����樨 \p{brack} � ⥬ ᠬ� �ਤ��� ��� �祭� ���⮩
���:
\begin{eqnarray}
D^{\pm}_n L^{{\alpha}}&=& \mp {\alpha} (-1)^n
\Bigr[{(((L^{\pm})^{n}_{*})}_{-{\alpha} })^{*},
L^{{\alpha}}\Bigl\}, \quad {\alpha} =+,-, \quad n \in {\bN},
\label{laxreprs1}
\end{eqnarray}
��� $D^{\pm}_{2n}$ ($D^{\pm}_{2n+1}$) -- ������� (�ନ����)
�����樮��� �ந������.

��� �室��� � �� �।�⠢����� ���������� �����஢
$(L^{\pm})^{n}_{*}$ ⠪�� ����� ���� ����祭� ���쬠 �����
��ࠦ���� � �ନ��� �����஢ {\Large �}��� � ᪮��筮� ����樨
\p{brack}
\begin{eqnarray}
(L^{{\alpha}})^{2n}_{*}:= \Bigl(~\frac{1}{2}~\Bigr[
(L^{{\alpha}})^{*}, (L^{{\alpha}})\Bigl\}~\Bigr)^n, \quad
(L^{{\alpha}})^{2m+1}_{*}:=L^{{\alpha}}~(L^{{\alpha}})^{2n}_{*}.
\label{rule2}
\end{eqnarray}
��� � ������� {\Large �}��� $L^{\pm}$, ������� $(L^{\pm})^{n}_{*}$
�ਭ������� ����࠭��� �����஢ \p{optoda} � ����� ����
�।�⠢���� � ᫥���饬 ����:
\begin{eqnarray}
(L^{+})^{m}_{*}:=\sum^{\infty}_{k=0}
u^{(m)}_{k,j}e^{(m-k){\partial}}, \quad u^{(m)}_{0,j}=1, \quad
(L^{-})^{m}_{*}:=\sum^{\infty}_{k=0}
v^{(m)}_{k,j}e^{(k-m){\partial}}, \label{laxs2}
\end{eqnarray}
��� $u^{(m)}_{k,j}$ � $v^{(m)}_{k,j}$ ($u^{(1)}_{k,j}\equiv
u_{k,j},~v^{(1)}_{k,j}\equiv v_{k,j}$) --- �㭪樮���� ��室���
����� $\{u_{k,j},~v_{k,j}\}$. {\Large �}���� ����� �⬥���, �� �
�।�⠢����� {\Large �}��� \p{laxreprs1} $Z_2$--�ࠤ�஢�� �����஢
$(L^{\pm})^{n}_{*}$ --- $d_{(L^{\pm})^{2n}_{*}}=0$ �
$d_{(L^{\pm})^{2n+1}_{*}}=1$ --- ᮣ��ᮢ��� � ��㣮�
$Z_2$--�ࠤ�஢��� $d^{~'}_{D^{\pm}_{2n}}=0$ �
$d^{~'}_{D^{\pm}_{2n+1}}=1$, ᮮ⢥�����饩 ����⨪�
�����樮���� �ந������� $D^{\pm}_{n}$.

{\Large �}ᯮ���� ᪮���� ᢮��⢠ (\ref{prop1}--\ref{jacobi}) �
ᮮ⭮襭�� (\ref{rule2}) ��� ��।������ ��� $(L^{\pm})^{n}_{*}$,
����㤭� ������� ������� ⮦���⢠
\begin{eqnarray}
&&\Bigr[ (L^{{\alpha}})^{2n}_{*},
(L^{{\alpha}})^{2m}_{*}\Bigl\}=0, \nonumber\\
&&\Bigr[ ((L^{{\alpha}})^{2n}_{*})^{*},
(L^{{\alpha}})^{2m+1}_{*}\Bigl\}=0, \quad \Bigr[
(L^{{\alpha}})^{2n+1}_{*},
(L^{{\alpha}})^{2m}_{*}\Bigl\}=0, \nonumber\\
&&\Bigr[ ((L^{{\alpha}})^{2n+1}_{*})^{*},
(L^{{\alpha}})^{2m+1}_{*}\Bigl\}=2(L^{{\alpha}})^{2(n+m+1)}_{*}.
\label{ident}
\end{eqnarray}
������, �ਬ���� (\ref{prop1}--\ref{jacobi}) �
(\ref{laxreprs1}--\ref{rule2}), ����� �뢥�� �ࠢ����� ��������
��� ���������� �����஢ $(L^{\pm})^{n}_{*}$
\begin{eqnarray}
D^{\pm}_n (L^{{\alpha}})^{m}_{*}&=& \mp {\alpha} (-1)^{nm} \Bigr[
{(((L^{\pm})^{n}_{*})}_{-{\alpha} })^{*(m)},
(L^{{\alpha}})^{m}_{*}\Bigl\}, \label{laxreprs2}
\end{eqnarray}
� ⠪�� �����樮��� �ࠢ����� ��� �㭪樮�����
$\{u^{(m)}_{k,j},~v^{(m)}_{k,j}\}$ \p{laxs2}, ��⥪��騥 ��
��������� �ࠢ����� \p{laxreprs2}:
\begin{eqnarray}
D^{+}_n u^{(2m)}_{k,j}&=&\sum^{n}_{p=0}
(u^{(n)}_{p,j}u^{(2m)}_{k-p+n,j-p+n} \nonumber\\
&-&(-1)^{(p+n)(k-p+n)}u^{(n)}_{p,j-k+p-n+2m}u^{(2m)}_{k-p+n,j}),
\nonumber\\ D^{+}_{2n} u^{(2m+1)}_{k,j}&=&\sum^{2n}_{p=0}
((-1)^{p}u^{(2n)}_{p,j}u^{(2m+1)}_{k-p+2n,j-p+2n} \nonumber\\
&-&(-1)^{p(k-p)}u^{(2n)}_{p,j-k+p-2n+2m+1}u^{(2m+1)}_{k-p+2n,j}),
\nonumber\\ D^{+}_{2n+1} u^{(2m+1)}_{k,j}&=&\sum^{k}_{p=1}
((-1)^{p+1}u^{(2n+1)}_{p+2n+1,j}u^{(2m+1)}_{k-p,j-p} \nonumber\\
&+&(-1)^{p(k-p)}u^{(2n+1)}_{p+2n+1,j-k+p+2m+1}u^{(2m+1)}_{k-p,j}),
\label{equationss1}
\end{eqnarray}
\begin{eqnarray}
D^{-}_n u^{(m)}_{k,j}&=&\sum^{n-1}_{p=0}
((-1)^{(p+n)m}v^{(n)}_{p,j}u^{(m)}_{k+p-n,j+p-n} \nonumber\\
&-&(-1)^{(p+n)(k+p-n)}v^{(n)}_{p,j-k-p+n+m}u^{(m)}_{k+p-n,j}),
\label{equationss2}
\end{eqnarray}
\begin{eqnarray}
D^{+}_n v^{(m)}_{k,j}&=&\sum^{n}_{p=0}
((-1)^{(p+n)m}u^{(n)}_{p,j}v^{(m)}_{k+p-n,j-p+n} \nonumber\\
&-&(-1)^{(p+n)(k+p-n)}u^{(n)}_{p,j+k+p-n-m}v^{(m)}_{k+p-n,j}),
\label{equationss3}
\end{eqnarray}
\begin{eqnarray}
D^{-}_n v^{(2m)}_{k,j}&=&\sum^{n-1}_{p=0}
(v^{(n)}_{p,j}v^{(2m)}_{k-p+n,j+p-n} \nonumber\\
&-&(-1)^{(p+n)(k-p+n)}v^{(n)}_{p,j+k-p+n-2m}v^{(2m)}_{k-p+n,j}),
\nonumber\\ D^{-}_{2n} v^{(2m+1)}_{k,j}&=&\sum^{2n-1}_{p=0}
((-1)^{p}v^{(2n)}_{p,j}v^{(2m+1)}_{k-p+2n,j+p-2n} \nonumber\\
&-&(-1)^{p(k-p)}v^{(2n)}_{p,j+k-p+2n-2m-1}v^{(2m+1)}_{k-p+2n,j}),
\nonumber\\ D^{-}_{2n+1} v^{(2m+1)}_{k,j}&=&\sum^{k}_{p=0}
((-1)^{p+1}v^{(2n+1)}_{p+2n+1,j}v^{(2m+1)}_{k-p,j+p} \nonumber\\
&+&(-1)^{p(k-p)}v^{(2n+1)}_{p+2n+1,j+k-p-2m-1}v^{(2m+1)}_{k-p,j})
\label{equationss4b}
\end{eqnarray}
(� �ࠢ�� ����� �� ���� $\{u^{(m)}_{k,j},~v^{(m)}_{k,j}\}$ � $k
< 0$ ������ ���� �������� ࠢ�묨 ���).

{\Large �}।�⠢����� {\Large �}��� \p{laxreprs1} �������� ��������� �������
��⮪�� $N=(1|1)$ 2DTL ����娨
\begin{eqnarray}
[D^{+}_{n}~,~D^{-}_{l}\}=[D^{\pm}_{n}~,~D^{\pm}_{2l}]=0,\quad
\{D^{\pm}_{2n+1}~,~D^{\pm}_{2l+1}\}=2D^{\pm}_{2(n+l+1)},
\label{algebras1}
\end{eqnarray}
����� ����� ���� ॠ�������� ���
\begin{eqnarray}
D^{\pm}_{2n} ={\partial}^{\pm}_{2n}, \quad D^{\pm}_{2n+1}
={\partial}^{\pm}_{2n+1}+
\sum^{\infty}_{l=1}t^{\pm}_{2l-1}{\partial}^{\pm}_{2(k+l)},\quad
\partial^{\pm}_n:={\frac{\partial}{\partial t^{\pm}_n}},
\label{covder}
\end{eqnarray}
��� $t^{\pm}_{2n}$ ($t^{\pm}_{2n+1}$) -- ������� (�ନ����)
�����樮��� �६���. {\Large �}����뢭� �।�� ��⮪��
(\ref{equationss1}--\ref{equationss4b}) �㤥� ����஥� � ��ࠣ��
3.1 � �殮� � �᭮�� ��।������ ��⮪�� �����ᯥ�ᨮ����
$N=(1|1)$ 2DTL ����娨.

\subsection{{\Large �}������ ᨬ���ਨ N=(1$|$1) 2DTL �ࠢ�����}

$\quad${\Large �}㯥�ᨬ����筮� $N=(1|1)$ 2DTL �ࠢ�����
\begin{eqnarray}
D^{+}_1D^{-}_1 \ln v_{0,j}= v_{0,j+1} - v_{0,j-1} \label{todabs}
\end{eqnarray}
�ਭ������� ��⥬� �ࠢ�����
(\ref{equationss1}--\ref{equationss4b}). ��� ��⥪��� �� �ࠢ�����
\p{equationss2} �� $\{n=m=k=1\}$
\begin{eqnarray}
D^{-}_1 u_{1,j} = - v_{0,j} - v_{0,j+1} \label{todabs1}
\end{eqnarray}
� �ࠢ����� \p{equationss3} �� $\{n=m=1, k=0\}$
\begin{eqnarray}
D^{+}_1 v_{0,j}= v_{0,j}(u_{1,j} - u_{1,j-1}) \label{todabs2}
\end{eqnarray}
��᫥ �᪫�祭�� ���� $u_{1,j}$. ��� ������� ᨬ���ਨ
$D^{\pm}_{2n} v_{0,j}$
\begin{eqnarray}
\{D^{+}_1,D^{\pm}_{2n}\}= \{D^{-}_1,D^{\pm}_{2n}\}=0
\label{algebra2new}
\end{eqnarray}
�뫨 ���ᠭ� � ࠡ��� \cite{ls1,ols1,vgks} � �ନ��� ᫥���饩
����樮���� ��楤���:
\begin{eqnarray}
D^{\pm}_{2n}
v_{0,j}&=&v_{0,j}(u^{(2n)\pm}_{2n,j}-u^{(2n)\pm}_{2n,j-1}), \quad
u^{(2n)\pm}_{0,j}=1, \nonumber\\ \pm D^{\mp}_1
u^{(2n)\pm}_{k,j}&=&v_{0,j}u^{(2n)\pm}_{k-1,j-1} +
(-1)^{k}v_{0,j-k+2n+1}u^{(2n)\pm}_{k-1,j}, \label{flowss+-bos}
\end{eqnarray}
��� �㭪樨 $u^{(n)\pm}_{k,j}$ �易�� � ��室�묨 �㭪樮������
$\{u^{(n)}_{k,j}, v^{(n)}_{k,j}\}$ ᫥���騬 ��ࠧ��:
\begin{eqnarray}
u^{(n)+}_{k,j}= u^{(n)}_{k,j}, \quad u^{(n)-}_{k,j}=
\frac{v^{(n)}_{k,-j-1}}{{\sum}^{n-k}_{m=1} v_{0,k+m-n-j-1}}
\label{relation}
\end{eqnarray}
(��⠫� �. � \cite{vgks}).

{\Large �}����뢭� �।�� $N=(1|1)$ 2DTL �ࠢ����� \p{todabs} � ���
ᨬ���਩ $D^{\pm}_{2n} v_{0,j}$ \p{flowss+-bos} �㤥� ����祭 �
��ࠣ�� 3.2.

\section{{\Large �}����ᯥ�ᨮ���� N=(1$|$1) 2DTL ������}

$\quad$ {\Large �} �⮬ ��ࠣ�� ��ந��� �����뢭� (�����������᪨�)
�� 蠣� ��⪨ �।�� $N=(1|1)$ 2DTL ����娨 ---
�����ᯥ�ᨮ���� $N=(1|1)$ 2DTL ������, � ������������
ᮮ⢥�����饥 ��� �।�⠢����� {\Large �}���.

\subsection{�����������᪨� �।��}

$\quad$ {\Large �}�⮪� (\ref{equationss1}--\ref{equationss4b}) $N=(1|1)$
2DTL ����娨, ������� � �।��饬 ��ࠣ��, �� ᮤ�ঠ� ��
����ᨬ���� �� ࠧ����� ����ﭭ��, � ��⪠ � ���ࠧ��୮�
���न��⮩ $j$ ($j\in {\bZ})$ ����� ������� 蠣. ��� ���祭��
�����뢭��� �।��� ������ � ����� 蠣� ��⪨. {\Large �}�᪮���
��� ��ࠬ��� � ���쭥�襬 �㤥� ����� ஫� ����ﭭ�� {\Large �}�����, ��
�㤥� ��������� ��� ��� ${\hbar}$. ����� ��ࠧ��, ����� $j$
��������� ���������
\begin{eqnarray}
{\hbar} j \equiv s, \label{coord1}
\end{eqnarray}
� �� ����� ���� ��稭��� ������� �� ��ࠬ��� ${\hbar}$.
�����, �����뢭� (�����������᪨�) �।�� ����� ���� ��।����
���
\begin{eqnarray}
{\hbar}\rightarrow 0, \quad s = {\lim}_{{\hbar}\rightarrow 0, ~j
>> 1}({\hbar} j), \label{displimit1}
\end{eqnarray}
� $s$ ����㯠�� � ஫� �����뢭�� "���筮�" ���न����.

��� ⮣�, �⮡� ��⮪� (\ref{equationss1}--\ref{equationss4b})
�뫨 ���ਢ����묨 � ॣ���묨 � �।��� \p{displimit1},
����室��� �������⥫쭮 �����⢨�� ������� ����⠡��
�८�ࠧ������ ����ᨬ�� � ������ᨬ�� ��६�����, �ਭ��������
��⥬�. ���, �� ����㫨�㥬 ᫥���騥 �ࠢ��� ���室� � ����
�����樮��� �६����
\begin{eqnarray}
t^{\pm}_{2n+1} \rightarrow \frac{1}{\sqrt{\hbar}} t^{\pm}_{2n+1},
~~ t^{\pm}_{2n} \rightarrow \frac{1}{{\hbar}} t^{\pm}_{2n}
~\Leftrightarrow ~ D^{\pm}_{2n+1}  \rightarrow \sqrt{\hbar}
D^{\pm}_{2n+1}, ~~ D^{\pm}_{2n}  \rightarrow {\hbar}
D^{\pm}_{2n}~~~~~ \label{displimit2}
\end{eqnarray}
� ���� ����娨
\begin{eqnarray}
&&u_{2k,j} \quad ~\rightarrow ~~~~u_{2k}({\hbar} j), \quad
\quad ~~~v_{2k,j} ~\quad \rightarrow ~~~~v_{2k}({\hbar} j),\nonumber\\
&&u_{2k+1,j}~\rightarrow ~ \frac{1}{\sqrt{\hbar}}~u_{2k+1}({\hbar}
j), \quad ~~v_{2k+1,j}~~\rightarrow ~
\frac{1}{\sqrt{\hbar}}~v_{2k+1}({\hbar} j), \label{displimit3}
\end{eqnarray}
������� �� ��ᨭ����묨 �� ${\hbar}=0$.

������ ����� ��⠭����� ��� ���ਢ����� ���祢� ᢮��⢠
�����������᪮�� �।��� (\ref{displimit1}--\ref{displimit3}),
����� ����� ������ ���祭�� ��� ���쭥�襣� ��ᬮ�७�� �
�஢������� ���묨 ���᫥��ﬨ.

{\Large �}�ࢮ� ᢮��⢮ ��⮨� � ⮬, �� ���� ��������� ����,
��।������ �� ᫥���騬 �ࠢ����
\begin{eqnarray}
&&u^{(m)}_{2k,j} \quad ~\rightarrow ~~~~u^{(m)}_{2k}({\hbar} j),
\quad \quad ~v^{(m)}_{2k,j} \quad \rightarrow
~~~~~v^{(m)}_{2k}({\hbar} j), \nonumber\\
&&u^{(2m+1)}_{2k+1,j}\rightarrow ~\frac{1}{\sqrt{\hbar}}~
u^{(2m+1)}_{2k+1}({\hbar}j), \quad v^{(2m+1)}_{2k+1,j}\rightarrow
~ \frac{1}{\sqrt{\hbar}}~v^{(2m+1)}_{2k+1}({\hbar} j),\nonumber\\
&&u^{(2m)}_{2k+1,j}~\rightarrow ~\sqrt{\hbar}~
u^{(2m)}_{2k+1}({\hbar} j), \quad \quad
v^{(2m)}_{2k+1,j}~\rightarrow ~\sqrt{\hbar}~
v^{(2m)}_{2k+1}({\hbar}j), \label{displimit4}
\end{eqnarray}
ॣ���� � �����������᪮� �।���.

{\Large �}� (\ref{displimit1}--\ref{displimit4}) � �祢����� ⮦����
\begin{eqnarray}
(L^{{\alpha}})^{2(m+1)}_{*}:= (L^{{\alpha}})^{2}_{*}
(L^{{\alpha}})^{2m}_{*}, \quad (L^{{\alpha}})^{(2m+1)}_{*}:=
L^{{\alpha}} (L^{{\alpha}})^{2m}_{*}, \label{ident1}
\end{eqnarray}
᫥����� �� \p{rule2}, ����� �������, ���ਬ��, ����� ४��७��
ᮮ⭮襭�� ��� ��������� 童��� �㭪樮�����
$u^{(m)}_{k}\equiv u^{(m)}_{k}(s)$
\begin{eqnarray}
&& u^{(2(l+1))}_{2k}
=\sum^{k}_{n=0}u^{(2)}_{2n}u^{(2l)}_{2(k-n)},\quad
u^{(2(l+1))}_{2k+1}=\sum^{2k+1}_{n=0}u^{(2)}_{n}u^{(2l)}_{2k-n+1},
\nonumber\\
&& u^{(2l+1)}_{2k} ~=\sum^{2k}_{n=0}u_{n}u^{(2l)}_{2k-n},\quad
\quad ~~u^{(2l+1)}_{2k+1}=\sum^{k}_{n=0}u_{2n+1}u^{(2l)}_{2(k-n)},
\label{rec1}
\end{eqnarray}
\begin{eqnarray}
&& u^{(2)}_{2k} =\sum^{k}_{n=0}u_{2n}u_{2(k-n)} +2\sum^{k-1}_{n=0}
(k-n-1)u_{2(k-n)-1}{\partial}_su_{2n+1}, \nonumber\\
&& u^{(2)}_{2k+1} =\sum^{k}_{n=0}\Big[(1-2n)u_{2n}{\partial}_s
u_{2(k-n)+1} +2(k-n)u_{2(k-n)+1} {\partial}_su_{2n}\Big],
\label{rec2}
\end{eqnarray}
��� ${\partial}_s:= \frac{{\partial}}{{\partial s}}$.

{\Large �}�஥ ᢮��⢮ ᢮����� � ⮬�, �� � �����������᪮� �।���
(\ref{displimit1}--\ref{displimit4}) ��⮪�
(\ref{equationss1}--\ref{equationss4b}) $N=(1|1)$ 2DTL ����娨
���ਢ����� � ॣ����. ���, � ��ࠦ���� ��� �� ���������
童��� ᫥���騥:
\begin{eqnarray}
�� \quad \{m=2l, ~&k&=2r\}, \quad ��� \quad \{m=2l+1, ~k=2r+1\} \nonumber\\
D^{-}_{2n+1} u^{(m)}_{k}&=&\sum^{n-1}_{p=0}
\Bigl[2(p-n)v^{(2n+1)}_{2p+1}{\partial}_s u^{(m)}_{k+2(p-n)}\nonumber\\
&+&(k-m+2(p-n))({\partial}_sv^{(2n+1)}_{2p+1})u^{(m)}_{k+2(p-n)}\Bigr]
\nonumber\\
&+&2(-1)^k\sum^{n}_{p=0}v^{(2n+1)}_{2p}u^{(m)}_{k+2(p-n)-1},
\nonumber\\
D^{-}_{2n} u^{(m)}_{k}&=&\sum^{n-1}_{p=0}
\Bigl[2(p-n)v^{(2n)}_{2p}{\partial}_s u^{(m)}_{k+2(p-n)}\nonumber\\
&+&(k-m+2(p-n))({\partial}_sv^{(2n)}_{2p})u^{(m)}_{k+2(p-n)}
\nonumber\\
&+&2(-1)^kv^{(2n)}_{2p+1}u^{(m)}_{k+2(p-n)+1}\Bigr], \nonumber\\
D^{+}_{2n} u^{(m)}_{k}&=&\sum^{n}_{p=0}
\Bigl[2(n-p)u^{(2n)}_{2p}{\partial}_s u^{(m)}_{k+2(n-p)}\nonumber\\
&+&(k-m+2(n-p))({\partial}_su^{(2n)}_{2p})u^{(m)}_{k+2(n-p)}\Bigr]
\nonumber\\&+& 2(-1)^k \sum^{n-1}_{p=0}
u^{(2n)}_{2p+1}u^{(m)}_{k+2(n-p)-1}, \nonumber\\
D^{+}_{2n+1} v^{(m)}_{k}&=&\sum^{n}_{p=0}
\Bigl[2(n-p)u^{(2n+1)}_{2p+1}{\partial}_s v^{(m)}_{k+2(p-n)}\nonumber\\
&-&(k-m+2(p-n))({\partial}_su^{(2n+1)}_{2p+1})v^{(m)}_{k+2(p-n)} \nonumber\\
&+& 2(-1)^k u^{(2n+1)}_{2p}v^{(m)}_{k-1+2(p-n)}\Bigr], \nonumber\\
D^{+}_{2n} v^{(m)}_{k}&=&\sum^{n}_{p=0}
\Bigl[2(n-p)u^{(2n)}_{2p}{\partial}_s v^{(m)}_{k+2(p-n)}\nonumber\\
&-&(k-m+2(p-n))({\partial}_su^{(2n)}_{2p})v^{(m)}_{k+2(p-n)}\Bigr]
\nonumber\\&+& 2(-1)^k \sum^{n-1}_{p=0}
u^{(2n)}_{2p+1}v^{(m)}_{k+1+2(p-n)}, \nonumber\\
D^{-}_{2n} v^{(m)}_{k}&=&\sum^{n-1}_{p=0}
\Bigl[2(p-n)v^{(2n)}_{2p}{\partial}_s v^{(m)}_{k+2(n-p)}\nonumber\\
&-&(k-m+2(n-p))({\partial}_sv^{(2n)}_{2p})v^{(m)}_{k+2(n-p)}\nonumber\\
&+&2(-1)^kv^{(2n)}_{2p+1}v^{(m)}_{k+2(n-p)-1}\Bigr],
\label{equ1+}
\end{eqnarray}
�
\begin{eqnarray}
�� \quad \{m=2l, ~&k&=2r+1\}, \quad ��� \quad \{m=2l+1, ~k=2r\} \nonumber\\
D^{-}_n u^{(m)}_{k}&=&\sum^{n-1}_{p=0}
(-1)^{m(p+n)}\Bigl[(p-n)v^{(n)}_{p}{\partial}_s u^{(m)}_{k+p-n}\nonumber\\
&+&(k+p-n-m)({\partial}_sv^{(n)}_{p})u^{(m)}_{k+p-n}\Bigr],
\nonumber\\ D^{+}_{2n} u^{(m)}_{k}&=&\sum^{2n}_{p=0}
(-1)^{mp}\Bigl[(2n-p)u^{(2n)}_{p}{\partial}_s u^{(m)}_{k-p+2n}\nonumber\\
&+&(k-p+2n-m)({\partial}_su^{(2n)}_{p})u^{(m)}_{k-p+2n}\Bigr],
\nonumber\\ D^{+}_{n} v^{(m)}_{k}&=&\sum^{n}_{p=0}
(-1)^{m(p+n)}\Bigl[(n-p)u^{(n)}_{p}{\partial}_s v^{(m)}_{k+p-n}\nonumber\\
&-&(k+p-n-m)({\partial}_su^{(n)}_{p})v^{(m)}_{k+p-n}\Bigr],
\nonumber\\
D^{-}_{2n} v^{(m)}_{k}&=&\sum^{2n-1}_{p=0}
(-1)^{mp}\Bigl[(p-2n)v^{(2n)}_{p}{\partial}_s v^{(m)}_{k-p+2n}\nonumber\\
&-&(k-p+2n-m)({\partial}_sv^{(2n)}_{p})v^{(m)}_{k-p+2n}\Bigr],
\label{equ2+}
\end{eqnarray}
� ⠪��
\begin{eqnarray}
D^{+}_{2n+1} u^{(2m)}_{2k}&=&2\sum^{n}_{p=0}
\Bigl[(n-p)u^{(2n+1)}_{2p+1}{\partial}_s u^{(2m)}_{2(k-p+n)}\nonumber\\
&+&(k-p+n-m)({\partial}_su^{(2n+1)}_{2p+1})u^{(2m)}_{2(k-p+n)}
\nonumber\\ &+& u^{(2n+1)}_{2p}u^{(2m)}_{2(k-p+n)+1}\Bigr],
\nonumber\\ D^{+}_{2n+1} u^{(2m+1)}_{2k+1}&=&2\sum^{k}_{p=1}
\Bigl[pu^{(2n+1)}_{2(p+n)+1}{\partial}_s u^{(2m+1)}_{2(k-p)+1}\nonumber\\
&+&(m+p-k)({\partial}_su^{(2n+1)}_{2(p+n)+1})u^{(2m+1)}_{2(k-p)+1}\Bigr]
\nonumber\\ &+& 2\sum^{k}_{p=0}
u^{(2n+1)}_{2(p+n+1)}u^{(2m+1)}_{2(k-p)},\nonumber\\
D^{+}_{2n+1} u^{(2m+1)}_{2k}&=&\sum^{2k}_{p=1}
(-1)^p\Bigl[pu^{(2n+1)}_{p+2n+1}{\partial}_s u^{(2m+1)}_{2k-p}\nonumber\\
&+&(2(m-k)+p+1)({\partial}_su^{(2n+1)}_{p+2n+1})u^{(2m+1)}_{2k-p}\Bigr],
\nonumber\\ D^{+}_{2n+1} u^{(2m)}_{2k+1}&=&\sum^{2n+1}_{p=0}
\Bigl[(2n-p+1)u^{(2n+1)}_{p}{\partial}_s u^{(2m)}_{2(k+n+1)-p}\nonumber\\
&+&(2(k+n-m+1)-p)({\partial}_su^{(2n+1)}_{p})u^{(2m)}_{2(k+n+1)-p}\Bigr],
\nonumber\\D^{-}_{2n+1} v^{(2m)}_{2k}&=&2\sum^{n-1}_{p=0}
\Bigl[(p-n)v^{(2n+1)}_{2p+1}{\partial}_s v^{(2m)}_{2(k-p+n)}\nonumber\\
&-&(k-p+n-m)({\partial}_sv^{(2n+1)}_{2p+1})v^{(2m)}_{2(k-p+n)}\Bigr]
\nonumber\\ &+&2\sum^{n}_{p=0}
v^{(2n+1)}_{2p}v^{(2m)}_{2(k-p+n)+1}, \nonumber\\D^{-}_{2n+1}
v^{(2m+1)}_{2k+1}&=&2\sum^{k}_{p=0}
\Bigl[-pv^{(2n+1)}_{2(p+n)+1}{\partial}_s v^{(2m+1)}_{2(k-p)+1}\nonumber\\
&+&(k-p-m){\partial}_sv^{(2n+1)}_{2(p+n)+1})v^{(2m+1)}_{2(k-p)+1}
\nonumber\\ &+& v^{(2n+1)}_{2(p+n+1)}v^{(2m+1)}_{2(k-p)}\Bigr],
\nonumber\\D^{-}_{2n+1} v^{(2m+1)}_{2k}&=&\sum^{k}_{p=0}
(-1)^p\Bigl[-pv^{(2n+1)}_{p+2n+1}{\partial}_s v^{(2m+1)}_{2k-p}\nonumber\\
&-&(2(m-k)+p+1)({\partial}_sv^{(2n+1)}_{p+2n+1})v^{(2m+1)}_{2k-p}\Bigr],
\nonumber\\D^{-}_{2n+1} v^{(2m)}_{2k+1}&=&\sum^{2n}_{p=0}
\Bigl[-(2n-p+1)v^{(2n+1)}_{p}{\partial}_s v^{(2m)}_{2(k+n+1)-p}\nonumber\\
&-&(2(k+n-m+1)-p)({\partial}_sv^{(2n+1)}_{p})v^{(2m)}_{2(k+n+1)-p}\Bigr],
\label{eqvv6-}
\end{eqnarray}
��� � �ࠢ�� ����� �� ���� $\{u^{(m)}_{k},~v^{(m)}_{k}\}$ � $k < 0$
������ ���� �������� ࠢ�묨 ���. {\Large �}���祭�� ⠪�� ᯮᮡ�� ��⮪�
(\ref{equ1+}--\ref{eqvv6-}) �� ���뢠�� �����ᯥ�ᨮ���� $N=(1|1)$
2DTL ����娥�. {\Large �}।�⠢����� {\Large �}��� ��� ��� �㤥� ����஥�� �
��ࠣ�� 3.3.

\subsection{{\Large �}����ᯥ�ᨮ���� N=(1$|$1) 2DTL �ࠢ����� � ��� �������
ᨬ���ਨ}

$\quad$ {\Large �}����ᯥ�ᨮ��� �।��� $N=(1|1)$ 2DTL �ࠢ�����
\p{todabs} � ��� �������� ᨬ���਩ (\ref{flowss+-bos}) ����� ����
����祭� ��� ��㤠, �᫨ �ᯮ�짮���� ᮮ⭮襭��
(\ref{relation}--\ref{displimit4}). {\Large �}���砥�, �⢥��⢥���:
\begin{eqnarray}
D^{+}_1D^{-}_1 \ln v_{0}= 2 {\partial}_s v_{0}
\label{disptoda}
\end{eqnarray}
�
\begin{eqnarray}
D^{\pm}_{2n} v_{0}&=&v_{0}{\partial}_s
u^{(2n)\pm}_{2n}, \quad u^{(2n)\pm}_{0,j}=1, \nonumber\\
\mp D^{\mp}_1 u^{(2n)\pm}_{2k+1}&=&v_{0}{\partial}_s
u^{(2n)\pm}_{2k} + 2(n-k)({\partial}_sv_{0})u^{(2n)\pm}_{2k},
\nonumber\\ \pm D^{\mp}_1
u^{(2n)\pm}_{2k}&=&2v_{0}u^{(2n)\pm}_{2k-1}.
\label{disp-flowss+-bos}
\end{eqnarray}

{\Large �}᪫��� $u^{(2n)\pm}_{2k+1}$ �� ��⥬� �ࠢ�����
\p{disp-flowss+-bos}, �����⥫쭮 ����砥� ᫥������ ४��७���
��⥬� �ࠢ����� ��� �����樨 �������� ᨬ���਩
�����ᯥ�ᨮ����� $N=(1|1)$ 2DTL �ࠢ����� \p{disptoda}
\begin{eqnarray}
D^{\pm}_{2n} v_{0}&=&v_{0}{\partial}_s
u^{(2n)\pm}_{2n}, \quad u^{(2n)\pm}_{0,j}=1, \nonumber\\
-D^{\mp}_1 u^{(2n)\pm}_{2k}&=&2v_{0} {(D^{\mp}_1)}^{-1} \Bigl[
v_{0}{\partial}_s u^{(2n)\pm}_{2(k-1)} +
2(n-k+1)({\partial}_sv_{0})u^{(2n)\pm}_{2(k-1)}\Bigr].
\label{disp-flowss+-bos1}
\end{eqnarray}
�⬥⨬, �� ᨬ���ਨ ��������� �����ᯥ�ᨮ����� 2DTL �ࠢ�����
�뫨 ������� � ࠡ�� \cite{fl} (�. ⠪�� ࠡ��� \cite{hp}) ��⥬
�襭�� ᮮ⢥�����饣�, �����筮 ᫮����� �ࠢ����� ᨬ���ਨ.

\subsection{{\Large �}।�⠢����� {\Large �}���}
%\subsection{�㯥�᪮��� ���ᮭ�}

$\quad$ {\Large �} ��ࠣ�� 3.1 �뫨 ����஥�� ��⮪�
(\ref{equ1+}--\ref{eqvv6-}) �����ᯥ�ᨮ���� $N=(1|1)$ 2DTL
����娨. {\Large �}�襩 ����祩 ᥩ�� �㤥� ��宦����� ᮮ⢥�����饣�
�� �।�⠢����� {\Large �}���. {\Large �}�᪮��� �।�⠢����� {\Large �}���
\p{laxreprs1} $N=(1|1)$ 2DTL ����娨 ��ࠦ����� � �ନ���
����饭��� �ࠤ�஢����� ᪮��� \p{brack}, �� �।��饣� ����
����⢥��� �������, �� ��� ����祭�� ��� �����������᪮��
�।���, ᮮ⢥�����饣� �����ᯥ�ᨮ���� $N=(1|1)$ 2DTL
����娨, �����筮 �������� � ��� ��� ᪮��� �� ��।�������
�㯥�᪮��� {\Large �}��ᮭ�, � ������� - �� �� ᨬ����, ������� ��
ᮮ⢥�����饬 䠧���� ����࠭�⢥.

{\Large �}�襩 ������襩 ����祩 �㤥� ������஢���� 䠧����� ����࠭�⢠
� �㯥�᪮��� {\Large �}��ᮭ�, ��室� �� �������� ᢮���, ����묨 ���
������ ���� ��������. ���, �ᯮ�����, �� ������� {\Large �}���
$N=(1|1)$ 2DTL ����娨, ���� �����묨 �� ����࠭�⢥
�����஢, �ࠤ�஢����� ���� ࠧ�묨 $Z_2$--�ࠤ�஢����
(\ref{grad1}) � (\ref{grad2}), �������� ⮫쪮 ����� ��������쭮�
$Z_2$--�ࠤ�஢��� \p{gradop}, ����� ��������, �� 䠧����
����࠭�⢮ 㭠᫥��� �� ᢮��⢠. {\Large �}ਭ���� �� ��������
$Z_2$--�ࠤ�஢�� $d_{e^{{\partial}}}=1, ~d_{e^{2{\partial}}}=0$
(\ref{grad2}) ������ ���筮�� ᤢ��� $e^{l{\partial}}$, ��
�।��������, �� �� 䠧���� ����࠭�⢥ ��� ᮮ⢥������ ���
���न����, � ������, ���ᬠ���� ���न��� $\pi$ ($\pi^2=0$) �
�������� ���न��� $p$, � ᫥���騬� �ࠢ����� ᮮ⢥��⢨�:
\begin{eqnarray}
e^{{\partial}}~\rightarrow ~ \frac{1}{{\sqrt\hbar}}~\pi, \quad
 e^{2{\partial}}~\rightarrow  ~p . \label{displax1}
\end{eqnarray}
{\Large �}�ᬮ��� �� ���ᬠ������� ���न���� $\pi$, ��� ������ ��������
{\it "��⨯���"} ��� ���ᬠ����� ���न��� ᢮��⢮� ������஢��� ��
⮫쪮 � ������묨, �� � �ନ���묨 ���ﬨ ����娨, �� ᫥���
�� �����뢭��� �।��� ᮮ⭮襭�� \p{rule1}.

�祢����, ��, �஬� ���न��� $\pi$ � $p$, 䠧���� ����࠭�⢮
������ ⠪�� ������� �����뢭�� "������" ���न���� $s$
\p{coord1}. {\Large �}�������饥 � १���� 䠧���� �㯥����࠭�⢮
$\{\pi, ~p,~ s\}$ �����ᯥ�ᨮ���� $N=(1|1)$ 2DTL ����娨
ᮤ�ন� 䠧���� ����࠭�⢮ $\{p,~ s\}$ �����ᯥ�ᨮ���� 2DTL
����娨 ��� �������࠭�⢮.

{\Large �}�᫥ ⮣�, ��� ���न���� 䠧����� ����࠭�⢠ ��⠭������, ��
������ ����ந�� ��� ��� �㯥�᪮��� {\Large �}��ᮭ�. {\Large �}ᯮ�����, ��
�㯥�᪮��� {\Large �}��ᮭ� ������ ��室����� � ᮮ⢥��⢨� �
c㯥ࠫ���ன $Z_2$--�ࠤ�஢����� �����஢
$\{\sqrt{\hbar}e^{{\partial}}, ~e^{2{\partial}},~ {\hbar} j\}$,
����� �⢥��� ���न���� 䠧����� �㯥����࠭�⢠ $\{\pi,
~p,~ s\}$, �� �㯥�᪮��� ����� ���� �ࠢ��⥫쭮 �����
᪮�����஢���. {\Large �}ਢ���� ����� �㯥�᪮��� {\Large �}��ᮭ� ����� ����
�ந�����묨 �㭪�ﬨ ${\bF}_{1,2}\equiv {\bF}_{1,2}(\pi,p,s)$
�� 䠧���� ����࠭�⢥, ����祭�� ᯮᮡ��, ��������� ���:
\begin{eqnarray}
\Bigl\{{\bF}_1,{\bF}_2\Bigr\}&=& 2p~\Bigl(\frac{\partial
{\bF}_1}{\partial p} ~\frac{\partial {\bF}_2}{\partial s}-
\frac{\partial {\bF}_1}{\partial s} ~\frac{\partial
{\bF}_2}{\partial p}+ \frac{\partial {\bF}_1}{\partial \pi}~
\frac{\partial {\bF}_2}{\partial \pi}\Bigr)\nonumber\\
&+&\pi ~\Bigl(\frac{\partial {\bF}_1}{\partial \pi}~
\frac{\partial {\bF}_2}{\partial s}- \frac{\partial
{\bF}_1}{\partial s}~ \frac{\partial {\bF}_2}{\partial \pi}\bigr).
\label{poissbrack}
\end{eqnarray}

{\Large �}���� 㬥�⭮ �⬥���, �� ��᫥ ���室� � ������ ������
$\{{\widetilde s}, \widetilde p,\widetilde \pi\}$ 䠧�����
����࠭�⢠, ����������� ��㫠��
\begin{eqnarray}
{\widetilde s}&:=&\frac{s}{2}, \quad \widetilde p := \ln p, \quad
{\widetilde \pi} := \frac{\pi}{\sqrt{2p}}, \nonumber\\
s&:=&2{\widetilde s}, \quad p := e^{\widetilde p}, \quad \pi :=
\sqrt{2}{\widetilde \pi} e^{\frac{\widetilde p}{2}},
 \label{poissbrack2}
\end{eqnarray}
�㯥�᪮��� {\Large �}��ᮭ� \p{poissbrack} �ਭ����� ���
\begin{eqnarray}
\Bigl\{{\bF}_1,{\bF}_2\Bigr\}&=& \frac{\partial {\bF}_1}{\partial
\widetilde p} ~\frac{\partial {\bF}_2}{\partial \widetilde s}-
\frac{\partial {\bF}_1}{\partial \widetilde s} ~\frac{\partial
{\bF}_2}{\partial \widetilde p}+ \frac{\partial {\bF}_1}{\partial
\widetilde \pi} ~\frac{\partial {\bF}_2}{\partial \widetilde \pi},
\label{poissbrack1}
\end{eqnarray}
�� ᮮ⢥����� �������᪮� ���ᨬ������᪮� �������
䠧����� �㯥����࠭�⢠.

������ �� ��३��� � ᫥���饬� �⠯� ����祭�� �।�⠢�����
{\Large �}��� �����ᯥ�ᨮ���� $N=(1|1)$ 2DTL ����娨 � �ਢ����
������᪨� ���� ��।������ � ����஥��� ᨬ����� ${\cal
L}^{{\pm}}$ � $({\cal L}^{{\pm}})^{2m}_{*}$ �����஢ {\Large �}���
$L^{\pm}$ \p{laxs1} � ���������� �����஢ $(L^{\pm})^{n}_{*}$
(\ref{rule2}--\ref{laxs2})
\begin{eqnarray}
L^{{\pm}}~ &\rightarrow & ~ \frac{1}{{\sqrt\hbar}}~{\cal
L}^{{\pm}},  \nonumber\\
{\cal L}^{+}=\sum^{\infty}_{k=0} (u_{2k+1}+u_{2k}\pi)p^{-k},
&\quad &{\cal L}^{-}=\sum^{\infty}_{k=0}
(v_{2k-1}+v_{2k}\pi)p^{k-1}
\label{displaxs1}
\end{eqnarray}
�
\begin{eqnarray}
&&(L^{{\pm}})^{2m}_{*}~ \rightarrow ~({\cal L}^{{\pm}})^{2m}_{*},
\quad (L^{{\pm}})^{2m+1}_{*}~ \rightarrow ~
\frac{1}{{\sqrt\hbar}}~({\cal L}^{{\pm}})^{2m+1}_{*}, \nonumber\\
&&\quad \quad \quad ({\cal L}^{+})^{2m}_{*}~~~:=~
\sum^{\infty}_{k=0} (u^{(2m)}_{2k}+u^{(2m)}_{2k-1}\pi)p^{m-k}, \nonumber\\
&&\quad \quad \quad ({\cal L}^{-})^{2m}_{*}~~~:=~
\sum^{\infty}_{k=0} (v^{(2m)}_{2k}+v^{(2m)}_{2k+1}\pi)p^{k-m}, \nonumber\\
&&\quad \quad \quad ({\cal L}^{+})^{2m+1}_{*}:=~
\sum^{\infty}_{k=0}
(u^{(2m+1)}_{2k+1}+u^{(2m+1)}_{2k}\pi)p^{m-k},\nonumber\\
&&\quad \quad \quad ({\cal L}^{-})^{2m+1}_{*}:=~
\sum^{\infty}_{k=0}
(v^{(2m+1)}_{2k-1}+v^{(2m+1)}_{2k}\pi)p^{k-m-1}, \label{diplaxs2}
\end{eqnarray}
ᮮ⢥��⢥���. {\Large �}� ��।������, �� ����
$\{u^{(m)}_{k},~v^{(m)}_{k}\}$ � $k < 0$ ������ ���� ��������
ࠢ�묨 ���. {\Large �}� ����祭�� ��� ��ࠦ���� �� �ᯮ�짮����
����⠭���� \p{displimit4} � \p{displax1}.
{\Large �}���⨬, �� �� ᨬ���� � ��饬 ��砥 ��������⨢��
\begin{eqnarray}
({\cal L}^{{\alpha}})^{k}_{*} ({\cal L}^{{\beta}})^{m}_{*}=
(-1)^{km}(({\cal L}^{{\beta}})^{m}_{*})^{*(k)} (({\cal
L}^{{\alpha}})^{k}_{*})^{*(m)}, \quad {\alpha},{\beta} =+,-,
\label{dispprop3}
\end{eqnarray}
�� �易�� � {\it "��⨯���"} ᢮��⢮� ���ᬠ����� ���न���� $\pi$,
�⬥祭�� ��� (�. ��ࠣ�� ��᫥ \p{displax1}).

{\Large �}, �������, ������� ������� {\Large �}��� �� �� ᨬ����
(\ref{displaxs1}--\ref{diplaxs2}), � ����饭��� �ࠤ�஢�����
᪮��� (\ref{brack}) �� �㯥�᪮��� {\Large �}��ᮭ� \p{poissbrack} �
�।�⠢������ {\Large �}��� (\ref{laxreprs1}) � \p{laxreprs2} ���
$N=(1|1)$ 2DTL ����娨 � � ��।������ ��� ᮮ⭮襭���
(\ref{rule2}) �� �ࠢ���
\begin{eqnarray}
\Bigl[~ \ldots ~, ~\ldots ~\Bigr\} ~\rightarrow  ~
{\hbar}~\Bigl\{~\ldots~,~\ldots~\Bigr\}, \label{dispbrac}
\end{eqnarray}
�����⢫�� ��⥬ ����⠭���� \p{displimit2} � ���室� � �।���
\p{displimit1}, �� ����砥� ��ࠦ����
%\newpage
\begin{eqnarray}
D^{\pm}_n {\cal L}^{{\alpha}}&=& \mp {\alpha} (-1)^n \Bigr\{
{((({\cal L}^{\pm})^{n}_{*})}_{-{\alpha} })^{*}, {\cal
L}^{{\alpha}}\Bigl\}, \quad n \in {\bN}, \quad {\alpha} =+,-,
\label{displaxreprs1}
\end{eqnarray}
\begin{eqnarray}
D^{\pm}_n ({\cal L}^{{\alpha}})^{m}_{*}&=& \mp {\alpha} (-1)^{nm}
\Bigr\{ {((({\cal L}^{\pm})^{n}_{*})}_{-{\alpha} })^{*(m)}, ({\cal
L}^{{\alpha}})^{m}_{*}\Bigl\},\quad n,m \in {\bN},
\label{displaxreprs2}
\end{eqnarray}
\begin{eqnarray}
({\cal L}^{{\alpha}})^{2m}_{*}:= \Bigl(~\frac{1}{2}~\Bigl\{({\cal
L}^{{\alpha}})^{*},{\cal L}^{{\alpha}}\Bigr\}~\Bigr)^{m},\quad
({\cal L}^{{\alpha}})^{2m+1}_{*}:={\cal L}^{{\alpha}}~({\cal
L}^{{\alpha}})^{2m}_{*} \label{disprule2}
\end{eqnarray}
��� �����ᯥ�ᨮ���� $N=(1|1)$ 2DTL ����娨. {\Large �}��� ���᫥���
���⢥ত���, �� ����祭�� ᮮ⭮襭��
(\ref{displaxreprs2}--\ref{disprule2}) �ࠢ��쭮 ���ந������
�����ᯥ�ᨮ��� ��⮪� (\ref{rec1}--\ref{eqvv6-}), �.�. �����
{\it �᪮�� �।�⠢������ {\Large �}��� ��� �����ᯥ�ᨮ���� $N=(1|1)$
2DTL ����娨.}

\section{{\Large �}����祭��}

$\quad$ {\Large �} ������ ࠡ�� �� �।������ ᪮����� ������
(\ref{brack}) � ᢮��⢠�� (\ref{prop1}--\ref{jacobi}) ��
����࠭�⢥ �ࠤ�஢����� �����஢ � ������樥�, ���������
�ࠤ�஢���� �������� �㯥ࠫ����. {\Large �}�⥬ �� ����稫� �����
��� �।�⠢����� {\Large �}��� (\ref{laxreprs1}--\ref{rule2}) �
\p{laxreprs2} ��� ��㬥୮� $N=(1|1)$ �㯥�ᨬ����筮� ���筮�
����娨 ���� � �ନ��� ����饭��� �ࠤ�஢����� ᪮���. �����
�� �ᯮ�짮���� �� �।�⠢����� �� ����஥���
�����������᪮�� �।��� (\ref{displimit1}--\ref{displimit4})
�⮩ ����娨 --- �����ᯥ�ᨮ���� $N=(1|1)$ ����娨 ����
(\ref{equ1+}--\ref{eqvv6-}), � ��� �।�⠢����� {\Large �}���
(\ref{displaxs1}--\ref{diplaxs2}),
(\ref{displaxreprs1}--\ref{disprule2}) �� �ࠤ�஢����� 䠧����
�㯥����࠭�⢥ � ᪮���� {\Large �}��ᮭ� \p{poissbrack}. {\Large �}, �������,
����⭮ �� ��⠭����� ������� ᨬ���ਨ \p{disp-flowss+-bos1}
�����ᯥ�ᨮ����� $N=(1|1)$ �㯥�ᨬ����筮�� �ࠢ����� ����
\p{disptoda}.

{}~

{\it ���� �� ��� ({\Large �.�.}) ����� ������������ � {\Large �.�. �}��㭮��, ��
���� ��㤥�⮬ II ���� {\Large �}����᪮�� 䠪���� {\Large ���} (1956), � ��
���殮��� ��襤�� ����⨫�⨩ ���頫 � ��� ��஭� ��஬���
�����প� ��� � �����, �易���� � ��㪮�, ⠪ � � �����. {\Large �}���
���஢�, ��ண�� {\Large �}��⮫�� {\Large �}���ᥥ���!}


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\end{thebibliography}

\end{document}

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%\section{�������}

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M.V. Saveliev and P. Sorba, {\it Solution of the Cauchy problem
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superextension}, {\it Lett. Math. Phys.} {\bf 22} (1991) 119.
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V.G. Kadyshevsky, R.M. Mir-Kasimov and N.B. Skachkov, {\it
Quasi-potential approach and the expansion in relativistic
spherical functions}, {\it Nuovo Cimento} {\bf A55} (1968) 233.
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V.G. Kadyshevsky, R.M. Mir-Kasimov and N.B. Skachkov, {\it
Relativistic two-body problem and finite-difference calculus},\\
{\it Sov. J. Nucl. Phys.} {\bf 9} (1969) 212.
\bibitem{11}
V.G. Kadyshevsky, R.M. Mir-Kasimov and N.B. Skachkov, {\it The
Schroedinger difference equation for two relativistic particles in
the simplest cases}, {\it Sov. J. Nucl. Phys.} {\bf 9} (1969) 462.
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V.G. Kadyshevsky, R.M. Mir-Kasimov and M. Freeman, {\it
Hypergeometric difference equations and Coulomb relativistic
problem}, \\{\it Sov. J. Nucl. Phys.} {\bf 9} (1969) 646.
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C. Itzykson, V.G. Kadyshevsky and I.T. Todorov, {\it
Three-dimensional formulation of the relativistic two-body problem
and infinite-component wave equations}, {\it Phys. Rev.} {\bf D1}
(1970) 2823.
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V.G. Kadyshevsky,  {\it Quantum field theory and the momentum
space of a constant curvature}, In the book {\it "Problems of
Theoretical Physics"} dedicated to the memory of I.E. Tamm, M.,
"Nauka", (1972) 52.
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V.G. Kadyshevsky, {\it Fundamental length as a new scale in
quantum field theory}, {\it Proceedings of the International
Conference on High Energy Physics and Elementary Particles},
Warsaw, (1975) 305.
\bibitem{16}
D.V. Fursaev and V.G. Kadyshevsky, {\it Difference equations and
gauge symmetry}, {\it CRM Proceedings and Lecture Notes} {\bf 9}
(1996) 125.
\bibitem{17}
K. Takasaki and T. Takebe, {\it SDIFF(2) Toda equation --
hierarchy, tau functions and symmetries}, {\it Lett. Math. Phys.}
{\bf 23} (1991) 205, \\ hep-th/9112042.
\bibitem{18}
K. Takasaki and T. Takebe, {\it Quasi--classical limit of Toda
hierarchy and W--infinity symmetries}, {\it Lett. Math. Phys.}
{\bf 28} (1993) 165, \\ hep-th/9301070.
\bibitem{19}
K. Takasaki and T. Takebe, {\it Integrable hierarchies and
dispersionless limit}, {\it Rev. Math. Phys.} {\bf 7} (1995) 743,
hep-th/9405096.
\bibitem{20}
C.P. Boyer and J.D. Finley, {\it Killing vectors in self--dual
Euclidean Einstein spaces}, {\it J. Math. Phys.} {\bf 23} (1982)
1126.
\bibitem{21}
I. Bakas, {\it The structure of the $W_{\infty}$ algebra}, {\it
Commun. Math. Lett.} {\bf 134} (1990) 487.
\bibitem{22}
Q-Han Park, {\it Extended conformal symmetries in real heavens},\\
{\it Phys. Lett.} {\bf B236} (1990) 429.
\bibitem{23}
R. S. Ward, {\it Einstein--Weyl spaces and SU$({\infty})$ Toda
fields}, {\it Class. Quantum Grav.} {\bf  7} (1990) L95.
\bibitem{24}
I. Krichever, {\it Dispersionless Lax equations and topological
minimal models}, {\it Commun. Math. Phys.} {\bf 143} (1992) 415.
\bibitem{25}
F. Delduc, M.V. Saveliev and P. Sorba, {\it Continuum Lie algebras
and the scalar--flat K\"ahler surfaces}, {\it Phys. Lett.} {\bf
B277} (1992) 411.
\bibitem{26}
J. Hoppe and Q-Han Park, {\it Infinite charge algebra of
gravitational instantons}, {\it Phys. Lett.} {\bf B231} (1994)
333, hep-th/9312020.
\bibitem{27}
K. Takasaki, {\it Dispersionless Toda hierarchy and two-dimensional
string theory}, {\it Commun. Math. Phys.} {\bf 170} (1995) 101,
hep-th/9403190.
\bibitem{28}
L. Bonora and C.S. Xiong, {\it Two--matrix model and $c=1$ string
theory}, {\it Phys. Lett.} {\bf B347} (1995) 41, hep-th/9405004.
\bibitem{29}
D. M. Calderbank and P. Tod, {\it Einstein metrics, hypercomplex
structures and the Toda field equation}, {\it Differ. Geom. Appl.}
{\bf 14} (2001) 199, math.DG/9911121.
\bibitem{30}
V. Kazakov, I. Kostov and D. Kutasov, {\it A matrix model for the
two dimensional black hole}, {\it Nucl. Phys.} {\bf B622} (2002)
141, hep-th/0101011.
\bibitem{31}
S. Alexandrov and V. Kazakov, {\it Correlators in 2D string theory
with vortex condensation}, {\it Nucl. Phys.} {\bf B610} (2001) 77,
hep-th/0104094.
\bibitem{32}
D.B. Fairlie and A.N. Leznov, {\it Infinite series solutions of
the symmetry equation for the 1+2 dimensional continuous Toda
chain}, \\ hep-th/9512217.

\end{thebibliography}