$R$-matrix for a geodesic flow associated with a new integrable peakon equation
Darryl D. Holm$^1$ and Zhijun Qiao$^{1,2}$
$^1$Theoretical Division and CNLS
MS B-284
Los Alamos National Laboratory
Los Alamos
NM 87545
USA
Shanghai 200433
PR China
We use the $r$-matrix formulation to show the integrability of geodesic flow on an $N$-dimensional space with coordinates $q_k$, with $k=1,...,N$, equipped with the co-metric $g^{ij}=e^{-|q_i-q_j|}\big(2-e^{-|q_i-q_j|}\big)$. This flow is generated by a symmetry of the integrable partial differential equation (pde) $m_t+um_x+3mu_x=0, \ m=u-\alpha^2u_{xx}$ ($\al $ is a constant). This equation -- called the Degasperis-Procesi (DP) equation -- was recently proven to be completely integrable and possess peakon solutions by Degasperis, Holm and Hone \cite{DHH[2002]}. The isospectral eigenvalue problem associated with the integrable DP equation is used to find a new $L$-matrix, called the Lax matrix, for the geodesic dynamical flow. By employing this Lax matrix we obtain the $r$-matrix for the integrable geodesic flow.
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\title{\bf  $R$-matrix for a geodesic flow associated with
a new integrable peakon equation}

\author{ Darryl D. Holm$^1$ and Zhijun Qiao$^{1,2}$ \\
$^1$Theoretical Division and CNLS, MS B-284 \\
Los Alamos National Laboratory\\
Los Alamos, NM 87545, USA\\
$^2$Institute of Mathematics, Fudan University\\
   Shanghai 200433, PR
China\\
{\footnotesize  E-mail:  qiao@cnls.lanl.gov}
}
\date{November 27, 2001; revised Jan. 22, 2002}
\begin{document}
\maketitle
\begin{abstract}
We use the $r$-matrix formulation to show the integrability of geodesic
flow on an $N$-dimensional space with coordinates $q_k$, with
$k=1,...,N$, equipped with the co-metric
$g^{ij}=e^{-|q_i-q_j|}\big(2-e^{-|q_i-q_j|}\big)$. This flow is generated
by a symmetry of the integrable partial differential equation (pde)
$m_t+um_x+3mu_x=0,
\ m=u-\alpha^2u_{xx}$ ($\al
$ is a constant). This equation -- called the Degasperis-Procesi (DP)
equation --  was recently proven to be completely integrable and possess
peakon solutions by Degasperis, Holm and Hone \cite{DHH[2002]}. The
isospectral eigenvalue problem associated with the integrable DP equation
is used to find a new $L$-matrix, called the Lax matrix, for the geodesic
dynamical flow. By employing this Lax matrix we obtain the $r$-matrix for
the integrable  geodesic flow.
   \end{abstract}
{\bf Keywords} \ \
Peakon equation, Hamiltonian, Lax matrix,
$r$-matrix structure.

     $ $\\
{\it AMS Subject: 35Q53; 58F07; 35Q35\\
       PACS: 03.40.Gc; 03.40Kf; 47.10.+g}


\section{Introduction}
   \subsection*{The $b=3$ peakon equation and its isospectral problem}

\setcounter{equation}{0}
We begin with the case $b=3$ of the $b-$weighted peakon equation. This is
the evolutionary equation defined on the real line as,
\begin{equation}\label{b-peakon-eqn}
m_t+um_x+b\,mu_x=0
\,, \quad
m=u-\alpha^2u_{xx}
\,, \quad
\lim_{|x|\to\infty}m=0
\,,
\end{equation}
in which the subscripts denote partial derivatives with respect to the
independent variables $x$ and $t$. For
any values of the dimensionless constant
$b>0$ and constant lengthscale
$\alpha$, this equation admits exact
$N$-peakon solutions
\begin{equation}
u(x,t)=\sum_{i=1}^Np_i(t)\, e^{-|x-q_i(t)|/\alpha}
\,,
\end{equation}
in which the $2N$ time-dependent functions $p_i(t)$ and $q_i(t)$,
$i=1,2,\dots,N$, satisfy a system of ordinary differential equations whose
character depends on the value of the bifurcation parameter $b$. The case
$b=2$ is the dispersionless limit of the integrable Camassa-Holm (CH)
equation that was discovered for shallow water waves in \cite{CH1}. As
shown in  \cite{DGH[2001]}, the CH equation with dispersion is one full
order more accurate in asymptotic approximation beyond Korteweg-de Vries
(KdV) for shallow water waves, yet it still preserves KdV's soliton
properties such as complete integrability via the inverse scattering
transform (IST) method.

An equation equivalent to the case $b=3$ of the peakon equation
(\ref{b-peakon-eqn}) was singled out for special attention among a family
of related equations by Degasperis and Procesi in \cite{dega}. The peakon
equation (\ref{b-peakon-eqn}) was shown to be completely integrable for the
case $b=3$ by Degasperis, Holm and Hone \cite{DHH[2002]}, who found
the following Lax pair consisting of a third order eigenvalue problem and a
second-order evolutionary equation for the eigenfunction,
\beq\label{isospectral-prob}
\psi_{xxx}&=&\frac{1}{\al^2}\psi_x-\la m\psi,\\
\psi_{t}&=&-\frac{1}{\al^2\la}\psi_{xx}-u\psi_x+
(u_x+\frac{2}{3\la})\psi.
\eeq
Compatibility $\psi_{xxxt}=\psi_{txxx}$ implies Eq. (\ref{b-peakon-eqn})
with $b=3$ provided $d\lambda/dt=0$. Thus, equation (\ref{b-peakon-eqn})
with $b=3$, namely,
\begin{equation}\label{b3peakon-eqn}
m_t+um_x+3\,mu_x=0
\,, \quad
m=u-\alpha^2u_{xx}
\,,
\end{equation}
was shown to be integrable by the inverse spectral transform for the
isospectral eigenvalue problem (\ref{isospectral-prob}) and to
possess an infinite sequence of conservation laws in Degasperis, Holm and
Hone \cite{DHH[2002]}. Henceforth, we shall consider the case for
$\al=1$ for the sake of simplicity in writing. (The lengthscale $\alpha$ is
easily restored from dimensional considerations.)

The first few conservation laws are given in the notation of
\cite{DHH[2002]} with $\alpha=1$ by
\beq
\begin{array}{cclccl}
H_{-1} & = & \frac{1}{6}\int u^3\, dx\,, &
H_{0} & = & \int m\, dx\,, \\
H_1& =& \frac{1}{2}\int ( v_{xx}^2 +5 v_x^2 +4v^2)\, dx \,,&
H_5 & = & \int m^{1/3}\, dx\,.
\end{array}
\label{eq:hamils}
\eeq
We shall pay special attention to the quadratic conservation law $H_1$,
in which the quantity $v$ is defined as
\beq
v:=(4-\partial_x^2)^{-1}u\equiv(4-\partial_x^2)^{-1}
(1-\partial_x^2)^{-1}m\,.
\eeq


\subsection*{Lax matrix for $N$-peakon dynamics}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Substituting the $N$-peakon solution,
\beq\label{Npkn-soln}
u(x,t)=\sum_{j=1}^Np_j(t)e^{-|x-q_j(t)|}, \ \
m(x,t)=2\sum_{j=1}^Np_j(t)\delta(x-q_j(t)),
\eeq
into the isospectral eigenvalue problem (\ref{isospectral-prob}) yields
\cite{DHH[2002]}
\beq
\frac{1}{\la}\psi(x,t)=\frac{1}{2}\sum_{j=1}^N
\left[1+sgn(x-q_j(t))\left(
1-e^{-|x-q_j(t)|}\right)\right]p_j\psi(q_j(t))
\,.
\eeq
Setting $\psi(q_i(t),t)=\psi_i(t)$ then gives the following matrix
eigenvalue problem,
\beq
\frac{2}{\al^2}\psi_i=\sum_{j=1}^N\tilde{L}_{ij}\psi_j,
\eeq
where
\beq
\tilde{L}_{ij}=\left[1+sgn(q_i-q_j)\left(
1-e^{-|q_i-q_j|}\right)\right]p_j
\,.
\eeq
Let $\tilde{L}$ denote the $N\times N$ matrix $\tilde{L}_{ij}$. In Ref.
\cite{DHH[2002]}, the authors used the two conserved quantities
tr$\tilde{L}$ and tr$\tilde{L}^2$
to solve the 2-peakon subdynamics of the the $N$-peakon dynamics
$q_k,p_k$, with $k=1,...,N, \ \al=1$, satisfying
\beq
\begin{array}{ccl}
\dot{p}_j &=&
2\,\sum_{k=1}^Np_j p_k\, sgn(q_j-q_k)e^{-|q_j-q_k|} ,
\\
\dot{q}_j &=&
\sum_{k=1}^Np_ke^{-|q_j-q_k|}
\,.
\end{array}
\label{eq:tdODEeqn}
\eeq
Amongst other results, the authors in \cite{DHH[2002]} discovered the 
two-peakon
collision rules for $N=2$ and gave explicit formulas for its phase shifts
as functions of the asymptotic speeds of the two peakons.

\section{A geodesic flow for $b=3$ peakons}

\setcounter{equation}{0}

The quantity used for determining the two-peakon $N=2$
collision laws  in \cite{DHH[2002]},
\begin{equation}\label{H1def}
H_1
=
\frac{1}{2}{\rm tr} \tilde{L}^2
=
\frac{1}{2} \sum_{i,j=1}^N p_i p_j
e^{-|q_i-q_j|}\left(2-e^{-|q_i-q_j|}\right)
\,,
\end{equation}
is also the quadratic conservation law $H_1$ in
(\ref{eq:hamils}) for the $b=3$ peakon equation (\ref{b3peakon-eqn}), when
$H_1$ is evaluated on the $N-$peakon solution (\ref{Npkn-soln}).

\rem{\comment{Cancel: with
$\al=1$.}}

The canonical Hamiltonian dynamics generated by $H_1$ is geodesic motion on
an $N$-dimensional space with co-metric
$g^{ij}=e^{-|q_i-q_j|}\big(2-e^{-|q_i-q_j|}\big)$.
As we shall show by finding its $r-$matrix structure in the remainder of the
present paper, this geodesic motion canonically generated by the
conservation law $H_1=Tr \tilde{L}^2$ in (\ref{H1def}) provides a new
$2N$-dimensional integrable system,
\beq
\dot{q}_k&=&\frac{\pa H_1}{\pa p_k}=\sum_{j=1}^Np_j
e^{-|q_k-q_j|}\left(2-e^{-|q_k-q_j|}\right), \label{H11}\\
\dot{p}_k&=&-\frac{\pa H_1}{\pa q_k}=-2p_k\sum_{j=1}^Np_j
sgn(q_j-q_k)e^{-|q_k-q_j|}\left(1-e^{-|q_k-q_j|}\right)
\,. \label{H12}
\eeq
We emphasize that these geodesic $H_1$-dynamics for $p_k, q_k$, are not the
same as the $N-$peakon dynamics in (\ref{eq:tdODEeqn}). Rather, we are
studying the canonical Hamiltonian dynamics on the restriction to the
peakon sector of the $H_1$-flow in the hierarchy of integrable equations
associated with the isospectral problem for equation (\ref{b3peakon-eqn}).

\subsection*{$R$-matrix results for the geodesic $H_1$-dynamics}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
To show integrability of the canonical dynamics in (\ref{H11}) and
(\ref{H12}) produced by Hamiltonian (\ref{H1def}), we shall use the
$r$-matrix method, which reveals that $trL^n$ for a certain Lax matrix $L$
are constants in involution
\cite{BV}.


To find the $r$-matrix structure for these $H_1$-dynamics for $p_k, q_k$, we
shall introduce an alternative Lax matrix for the peakon dynamics of Eqn.
(\ref{b3peakon-eqn}),
\beq
L&=&\sum_{i,j=1}^NL_{ij}E_{ij}, \label{L1}
\eeq
where
\beq
L_{ij}&=&\sqrt{p_ip_j}A_{ij},\label{Laxij}\\
A_{ij}&=&A(q_i-q_j)=\sqrt{(2-e^{-|q_i-q_j|})e^{-|q_i-q_j|}}.\label{Aij}
\eeq
The Lax matrix (\ref{L1}) also satisfies,
\beq
H_1=\frac{1}{2}{\rm tr} L^2=\frac{1}{2}\sum_{i,j=1}^Np_ip_je^{-|q_i-q_j|}
\left(2-e^{-|q_i-q_j|}\right),
\eeq
which is the Hamiltonian for the canonical dynamics in
Eqs.  (\ref{H11}) and (\ref{H12}).
In Eq. (\ref{Aij}), we have
\beq
A(x)=\sqrt{(2-e^{-|x|})e^{-|x|}}, \label{Ax}
\eeq
and the function $A(x)$ satisfies the following relations,
\beq
A'(x)&=&-sgn(x)\frac{1-e^{-|x|}}{2-e^{-|x|}}A(x),\\
A_{ij}&=&A_{ji}, \ \ A_{ii}=1,\\
A'_{ij}=A'(q_i-q_j)&=&-A'(q_j-q_i)=-A'_{ji}, \ \ A'_{ii}=0,
\\
(\frac{\pa}{\pa x}+\frac{\pa}{\pa y})A(x)A(y)&=&A'(x)A(y)+A(x)A'(y)\nonumber\\
&&\hspace{-30mm}
=-A(x)A(y)
\left[sgn(x)\frac{1-e^{-|x|}}{2-e^{-|x|}}
+sgn(y)\frac{1-e^{-|y|}}{2-e^{-|y|}}
\right],
\label{Aij1}\\
(\frac{\pa}{\pa x}+\frac{\pa}{\pa y})A(x)A(-x)&=&0.
\eeq

We shall work in the canonical matrix basis $E_{ij}$,
\beqq
(E_{ij})_{kl}=\delta_{ik}\delta_{jl}, \quad i,j,k,l=1,...,N.
\eeqq
To find the $r$-matrix structure for the $H_1$-dynamics in Eqs. (\ref{H11})
and (\ref{H12}), we consider the so-called fundamental Poisson bracket
\cite{FT1}:
\beq
\{L_1,L_2\}&=&[r_{12},L_1]-[r_{21},L_2], \label{FPB}
\eeq
where
\beqq
L_1&=&L\otimes {\bf 1}=\sum_{i,j=1}^N L_{ij}E_{ij}\otimes {\bf 1},\\
L_2&=&{\bf 1}\otimes L=\sum_{k,l=1}^N L_{kl}{\bf 1}\otimes E_{kl},\\
r_{12}&=&\sum_{i,j,s,t}^N r_{ij;st} E_{ij}\otimes E_{st},\\
r_{21}&=&\sum_{i,j,s,t}^N r_{ij;st} E_{st}\otimes E_{ij},\\
\{L_1,L_2\}&=&\sum_{i,j,k,l=1}^N \{L_{ij},L_{kl}\}E_{ij}\otimes E_{kl}.
\eeqq
Here $\{L_{ij},L_{kl}\}$ is the standard Poisson
bracket of two functions, {\bf 1} is the $N\times N$ unit matrix,
and the quantities $r_{ij;st}$ are to  be determined. In Eq.
(\ref{FPB}), $[\cdot ,\cdot]$ denotes the usual commutator of matrices.


After a lengthy calculation for both sides of Eq. (\ref{FPB}),
we obtain the following key equalities (whose detailed verification is
given in the Appendix):
\beqq
\left[r_{ll}, L\right]_{ll}&=&0,\\
\left[r_{jj}, L\right]_{ll}&=&\left[r_{ll}, L\right]_{jj}, \ j\not=l,\\
\left[r_{jl}, L\right]_{lj}&=&\left[r_{lj}, L\right]_{lj}=0, \\
\left[r_{ll}, L\right]_{lj}&=&\left[r_{ll},
L\right]_{jl}=-\sqrt{p_jp_l}A'_{jl}, \\
\left[r_{lj}, L\right]_{ll}&=&\left[r_{jl},
L\right]_{ll}=-\sqrt{p_jp_l}A'_{jl}, \\
\left[r_{ll}, L\right]_{kj}&=&\left[r_{jl}, L\right]_{kk}=0, \
j\not=l,k; k\not=l,\\
\left[r_{jl}, L\right]_{lk}&=&\left[r_{lj},
L\right]_{kl}=\frac{1}{2}\sqrt{p_kp_j}(A_{jl}A_{lk})',\ j\not=l,k; k\not=l, \\
\left[r_{jl}, L\right]_{kl}&=&\left[r_{lj},
L\right]_{lk}=\frac{1}{2}\sqrt{p_kp_j}(A_{jl}A_{lk})',\ j\not=l,k; k\not=l, \\
\left[r_{st}, L\right]_{jl}&=&0, \ {\rm for \ different}\  s,t,j,l.
\eeqq
where $r_{jl}=\sum_{k,m}r_{km,jl}E_{km}, \
r_{ll}=\sum_{k,m}r_{km,ll}E_{km},$
are two $N\times N$ matrices whose entries are to be determined,
$L$ is the Lax matrix, and $\left[\cdot, L\right]_{kl}$ stands for
the $k$-th row and the $l$-th colum element of $\left[\cdot, L\right]$.

In matrix notation, all the above equalities
can be rewritten as
\beq
\left[r_{jl}, L\right]&=&B^{jl}, \quad  j\not = l, \label{m11} \\
\left[r_{ll}, L\right]&=&B^{ll}, \label{m21}
\eeq
where $B^{jl}, B^{ll}$ are the following two $N\times N$ matrices:
{\footnotesize \beqq
B^{jl}&=&\left(\begin{array}{ccccccc}
0&...&\frac{1}{2}\sqrt{p_1p_l}(A_{lj}A_{j1})'& ... &
\frac{1}{2}\sqrt{p_1p_j}(A_{jl}A_{l1})'& ... & 0\\
\vdots &\ddots&\vdots&\vdots&\vdots&\vdots&\vdots\\
   \frac{1}{2}\sqrt{p_1p_l}(A_{lj}A_{j1})' & ... &
   -\sqrt{p_jp_l}A'_{lj}&...&
0&...& \frac{1}{2}\sqrt{p_Np_l}(A_{lj}A_{jN})'\\
\vdots &\vdots&\vdots&\ddots&\vdots&\vdots&\vdots\\
   \frac{1}{2}\sqrt{p_1p_j}(A_{jl}A_{l1})' & ... &
0&...&  -\sqrt{p_jp_l}A'_{jl}&...&
\frac{1}{2}\sqrt{p_Np_j}(A_{jl}A_{lN})'\\
\vdots &\vdots&\vdots&\vdots&\vdots&\ddots&\vdots\\
0&...&\frac{1}{2}\sqrt{p_Np_l}(A_{lj}A_{jN})'& ... &
\frac{1}{2}\sqrt{p_Np_j}(A_{jl}A_{lN})'& ... & 0
\end{array}
\right),
\eeqq}
and
\beqq
B^{ll}&=&\left(\begin{array}{ccccc}
0&...&-\sqrt{p_1p_l}A'_{1l}& ... & 0\\
\vdots &\ddots&\vdots&\vdots&\vdots\\
   -\sqrt{p_1p_l}A'_{1l} & ... & 0& ... & -\sqrt{p_Np_l}A'_{Nl}\\
\vdots &\vdots&\vdots&\ddots&\vdots\\
0&...&-\sqrt{p_Np_l}A'_{Nl}& ... & 0
\end{array}
\right).
\eeqq

By solving Eqs. (\ref{m11}) and (\ref{m21}),
we have the following $r$-matrix structure:
\beq
r_{12}&=&\sum_{j,l=1}^N\left(\frac{A'_{lj}}{A_{lj}}E_{jl}\otimes
(E_{jl}+E_{lj})+\frac{A'_{lj}}{A_{lj}}E_{ll}\otimes
E_{jj}\right)\nonumber\\
&&+\frac{1}{2}\sum_{j,k,l=1}^N
\sqrt{\frac{p_k}{p_j}}\left(\frac{A'_{kj}A_{kl}}{A_{kj}A_{lj}}
+\frac{(A_{jk}A_{lj})'}{A_{lj}}\right)E_{ll}\otimes
E_{jk}.
\eeq

Perhaps not unexpectedly, this non-contant $r$-matrix for the geodesic
$H_1$-dynamics differs from the constant  $r$-matrix associated with the
CH equation ($b=2$) discovered by Ragnisco and Bruschi in \cite{RB1}.
According to the $r$-matrix method \cite{BV},
the auxiliary matrix $M$ corresponding to $L$
may be also constructed. But, in our case
the $r$-matrix is dynamical, and therefore
the construction of $M$ is a rather complicated process,
which we defer to another time.

\subsection*{Concluding remarks}
In this paper, we found the $r$-matrix formulation for the
integrable geodesic motion generated canonically by the quadratic
quantity $H_1$ in (\ref{H1def}).  This quantity arises by restriction to
the peakon sector of a quadratic conservation law in the hierarchy
of integrable equations associated with the isospectral problem for equation
(\ref{b3peakon-eqn}). The quadratic quantity $H_1$ is also conserved  for
the $2$-peakon dynamics of the 1+1 integrable partial differential equation
(\ref{b3peakon-eqn}) that was singled out in
\cite{dega} and was proven to be completely integrable by the isospectral
transform in \cite{DHH[2002]}. We also introduced a new Lax matrix $L$
for the $N$-peakon flows of the integrable equation (\ref{b3peakon-eqn})
that facilitated the $r$-matrix calculations and for which $H_1=
\frac{1}{2}{\rm tr}L^2$.  In later work, we shall discuss additional flows
in the hierarchy of integrable equations associated with the isospectral
problem for equation (\ref{b3peakon-eqn}) and study their relationships to
classical finite-dimensional integrable systems
\cite{QH1}.

\subsection*{Acknowledgments}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
D. D. Holm is grateful to A. Degasperis and A. N. W. Hone for
their help and collaboration in many discussions that parallel the work
reported here. Z. Qiao would like to express his sincere thanks to
J. M. Hyman and L. G. Margolin  for their warm
invitations and friendly academic discussions.
This work was supported by the U.S. Department of Energy under
contracts W-7405-ENG-36 and the Applied Mathematical Sciences Program
KC-07-01-01; and also the Special Grant of National
Excellent Doctorial Dissertation of China.


\begin{thebibliography}{99}
\parskip=0.01cm

\bibitem{BV}O. Babelon and
C.M. Viallet, Phys. Lett. B 237 (1990) 411-416.
\bibitem{dega}
A. Degasperis and M. Procesi, Asymptotic integrability, in
{\it Symmetry and Perturbation Theory}, edited by A. Degasperis and G.
Gaeta, World Scientific (1999) pp.23-37.

\bibitem{CH1}R. Camassa and D. D. Holm,
          An integrable shallow water
          equation with peaked solitons,
                     Phys. Rev. Lett.  71 (1993) 1661-1664.

\bibitem{DGH[2001]}
H. Dullin, G. Gottwald and D. D. Holm,
An integrable shallow water equation
with linear and nonlinear dispersion.
Phys. Rev. Lett. 87 (2001) 1945-1940.

\bibitem{DHH[2002]}
A. Degasperis, D. D. Holm and A. N. W. Hone,
A new integrable equation with peakon
solutions. Submitted to NEEDS Proceedings.

\bibitem{RB1}
O. Ragnisco and M. Bruschi,
Peakons, $r$-matrix and Toda lattice,
Physica A 228 (1996) 150-159.

\bibitem{FT1}
L. D. Faddeev and L. A. Takhtajan, {\it Hamiltonian Methods
in the Theory of Solitons}   (Springer-Verlag, Berlin, 1987).

\bibitem{QH1}
D. D. Holm and Z. Qiao,
On the relationship between the b=3 peakon equation
and finite-dimensional integrable systems. In preparation
(2001).

\end{thebibliography}

\newpage
{\Large \bf Appendix}

$ $

The following computations are needed in verifying Eq. (\ref{FPB}).

First, we calculate the left hand side of Eq. (\ref{FPB}).
\beqq
\frac{\pa L_{ij}}{\pa
    q_m}&=&\sqrt{p_ip_j}A'_{ij}(\delta_{im}-\delta_{jm}),\\
\frac{\pa L_{kl}}{\pa
    p_m}&=&\frac{A_{kl}}{2\sqrt{p_kp_l}}(p_l\delta_{km}+p_k\delta_{lm}),
\eeqq
\beqq
\{L_{ij},L_{kl}\}&=&\sum_{m=1}^N\left(\frac{\pa L_{ij}}{\pa
      q_m}\frac{\pa L_{kl}}{\pa p_m}-\frac{\pa L_{kl}}{\pa q_m}\frac{\pa
      L_{ij}}{\pa p_m}\right)\\
&=&\frac{1}{2}\sum_{m=1}^N
\left[\sqrt{p_ip_j}A'_{ij}\frac{A_{kl}}{\sqrt{p_kp_l}}
(\delta_{im}-\delta_{jm})(p_l\delta_{km}+p_k\delta_{lm})\right.\\
& & -\left.\sqrt{p_kp_l}A'_{kl}\frac{A_{ij}}{\sqrt{p_ip_j}}
(\delta_{km}-\delta_{lm})(p_j\delta_{im}+p_i\delta_{jm})\right]\\
&=&\frac{1}{2}
\left[\sqrt{p_ip_j}
(-A'_{ij}A_{kj}+A'_{kj}A_{ij})
\delta_{jl}+\sqrt{p_lp_j}
(A'_{ij}A_{il}-A'_{il}A_{ij})
\delta_{ik}\right.\\
& & -\left.\sqrt{p_ip_l}
(A_{ij}A_{jl})'
\delta_{kj}+\sqrt{p_kp_j}
(A_{ij}A_{ki})'
\delta_{il}
\right]\\
&=&\frac{1}{2}
\left[\sqrt{p_kp_i}
(A_{kj}A_{ji})'
\delta_{jl}+\sqrt{p_lp_j}
(A_{li}A_{ij})'
\delta_{ik}\right.\\
& & +\left.\sqrt{p_ip_l}
(A_{lj}A_{ji})'
\delta_{kj}+\sqrt{p_kp_j}
(A_{ki}A_{ij})'
\delta_{il}
\right],
\eeqq
where the superscript $\prime$ means Eq. (\ref{Aij1}) with the argument.

   Thus, we obtain the following formula,
\beq
\{L_1,L_2\}&=&\sum_{i,j,k,l=1}^N \{L_{ij},L_{kl}\}E_{ij}\otimes
E_{kl}\nonumber\\
&=&\frac{1}{2}\sum_{j,k,l=1}^N \left[\sqrt{p_kp_j}
(A_{kl}A_{lj})'E_{jl}\otimes
E_{kl}+\sqrt{p_kp_j}
(A_{kl}A_{lj})'E_{lj}\otimes
E_{lk}\right.\nonumber\\
& &+\left.\sqrt{p_kp_j}
(A_{kl}A_{lj})'E_{jl}\otimes
E_{lk}+\sqrt{p_kp_j}
(A_{kl}A_{lj})'E_{lj}\otimes
E_{kl}\right]\nonumber\\
&=&\frac{1}{2}\sum_{j,k,l=1}^N \sqrt{p_kp_j}
(A_{kl}A_{lj})'\left[E_{jl}\otimes
E_{kl}+E_{lj}\otimes
E_{lk}+E_{jl}\otimes
E_{lk}+E_{lj}\otimes
E_{kl}\right]\nonumber\\
&=&\frac{1}{2}\sum_{j,k,l=1,j\not=k,l;k\not=l}^N \sqrt{p_kp_j}
(A_{kl}A_{lj})'\left(E_{jl}+E_{lj}\right)\otimes
\left(E_{kl}+E_{lk}\right)\nonumber\\
& & +\sum_{k,l=1}^N\sqrt{p_kp_l}
A'_{kl}\left(E_{ll}\otimes
(E_{kl}+E_{lk})-(E_{kl}+E_{lk})\otimes E_{ll}\right). \label{L12}
\eeq

Next, we compute the right hand side of Eq. (\ref{FPB}),
\beq
& & [r_{12},L_1]-[r_{21},L_2]\nonumber\\
&=&
\sum_{i,j,s,t,k,l=1}^N r_{ij;st}L_{kl}\left[(E_{ij}\otimes
    E_{st})(E_{kl}\otimes {\bf 1})-(E_{kl}\otimes {\bf 1})(E_{ij}\otimes
    E_{st})\right.\nonumber\\
& & \left.
-(E_{st}\otimes E_{ij})( {\bf 1}\otimes E_{kl})+({\bf 1}\otimes
E_{kl})(E_{st}\otimes E_{ij})
\right]\nonumber\\
&=&\sum_{i,j,s,k,l=1}^N \left[r_{ij;sk}\left(L_{jl}(E_{il}\otimes
    E_{sk})- L_{li}(E_{lj}\otimes
E_{sk})\right)-r_{ij;ss}L_{kl}\left((E_{ss}\otimes
    E_{ij}E_{kl})-(E_{ss}\otimes
    E_{kl}E_{ij})\right)
\right]\nonumber\\
&=&\sum_{i,j,s,k,l=1}^N \left[r_{ij;sk}\left(L_{jl}(E_{il}\otimes
    E_{sk})- L_{li}(E_{lj}\otimes E_{sk})\right)\right]\nonumber\\
& &+\sum_{i,j,s,l=1}^N\left[-r_{ji;ss}L_{il}(E_{ss}\otimes
    E_{jl})+r_{ij;ss}L_{li}(E_{ss}\otimes
    E_{lj})\right]\nonumber\\
&=&\sum_{i,j,s,t,l, s\not=t,j\not=l} \left[r_{ji;st}
L_{il}(E_{jl}\otimes
    E_{st})-r_{ij;st} L_{li}(E_{lj}\otimes E_{st})\right]\nonumber\\
& &+
     \sum_{i,j,s,l,s\not=j,l,j\not=l} \left[r_{ji;ss}
     L_{il}(E_{jl}\otimes
    E_{ss}-E_{ss}\otimes E_{jl})-
    r_{ij;ss} L_{li}(E_{lj}\otimes E_{ss}-E_{ss}\otimes
    E_{lj})\right.\nonumber\\
& &+\left. (r_{si;lj}
     L_{is}-r_{is;lj}
     L_{si})E_{ss}\otimes E_{lj}) \right]\nonumber\\
& &+\sum_{i,j,l,j\not=l} \left[(r_{li;lj}
     L_{il}-r_{il;lj}L_{li}+r_{ij;ll}
     L_{li}-r_{li;ll}L_{ij})E_{ll}\otimes  E_{lj}\right.\nonumber\\
& &+(r_{li;jl}
     L_{il}-r_{il;jl}L_{li}-r_{ji;ll}
     L_{il}+r_{il;ll}L_{ji})E_{ll}\otimes  E_{jl}\nonumber\\
& &+\left. (r_{ji;ll}L_{il}-r_{il;ll}L_{ji})(E_{jl}\otimes  E_{ll}
-E_{ll}\otimes  E_{jl})-(r_{ij;ll}L_{li}-r_{li;ll}L_{ij})(E_{lj}\otimes  E_{ll}
-E_{ll}\otimes  E_{lj}) \right]\nonumber\\
& &+\sum_{i,j,l,j\not=l} \left[(r_{li;jj}
     L_{il}-r_{il;jj}L_{li}+
r_{ij;ll}
     L_{ji}-r_{ji;ll}L_{ij})E_{ll}\otimes  E_{jj}\right.\nonumber\\
& &+\left.(r_{li;jj}
     L_{il}-r_{il;jj}
     L_{li})(E_{ll}\otimes  E_{jj}-E_{jj}\otimes
     E_{ll})\right]\nonumber\\
& &+\sum_{i,l} 4(r_{li;ll}
     L_{il}-r_{il;ll}L_{li})E_{ll}\otimes  E_{ll} \label{P12}
\eeq
The first term of Eq. (\ref{P12}) is:
\beqq
& & \sum_{i,j,s,t,l, s\not=t,j\not=l} \left(r_{ji;st}
     L_{il}-r_{il;st} L_{ji}\right)E_{jl}\otimes E_{st}\\
&=& \sum_{i,j,s,t,l; s\not=t,j,l;t\not=j,l; j\not=l} \left(r_{ji;st}
      L_{il}-r_{il;st} L_{ji}\right) E_{jl}\otimes E_{st}\\
& &  +\sum_{i,j,k,l, j\not=k,l;k\not=l} \left[\left(r_{li;kl}
      L_{ij}-r_{ij;kl} L_{li}\right)E_{lj}\otimes E_{kl}+\left(r_{ki;lj}
      L_{il}-r_{il;lj} L_{ki}\right)E_{kl}\otimes E_{lj}\right.\\
& & + \left.\left(r_{ji;kl}
      L_{il}-r_{il;kl} L_{ji}\right)E_{jl}\otimes E_{kl}+\left(r_{li;lk}
      L_{ij}-r_{ij;lk} L_{li}\right)E_{lj}\otimes E_{lk} \right]\\
& &  +2\sum_{i,j,l, j\not=l} \left[
   \left(r_{li;jl}
      L_{ij}-r_{ij;jl} L_{li}\right)E_{lj}\otimes E_{jl}+\left(r_{li;lj}
      L_{ij}-r_{ij;lj} L_{li}\right)E_{lj}\otimes E_{lj} \right].
\eeqq

Therefore, we have
\beq
& & [r_{12},L_1]-[r_{21},L_2]\nonumber\\
&=&\sum_{i,j,s,t,l; s\not=t,j,l;t\not=j,l; j\not=l} \left(r_{ji;st}
      L_{il}-r_{il;st} L_{ji}\right) E_{jl}\otimes E_{st}\nonumber\\
& &  +\sum_{i,j,k,l, j\not=k,l;k\not=l} \left[\left(r_{li;kl}
      L_{ij}-r_{ij;kl} L_{li}\right)E_{lj}\otimes E_{kl}+\left(r_{ji;lk}
      L_{il}-r_{il;lk} L_{ji}\right)E_{jl}\otimes E_{lk}\right.\nonumber\\
& & + \left.\left(r_{ji;kl}
      L_{il}-r_{il;kl} L_{ji}\right)E_{jl}\otimes E_{kl}+\left(r_{li;lk}
      L_{ij}-r_{ij;lk} L_{li}\right)E_{lj}\otimes E_{lk} \right]\nonumber\\
& &+
     \sum_{i,j,k,l;k\not=j,l,j\not=l} \left[(-r_{ji;kk}L_{il}+r_{il;kk} L_{ji}+
r_{ki;jl}
     L_{ik}-r_{ik;jl}
     L_{ki})E_{kk}\otimes E_{jl}
\right. \nonumber\\
& &  +\left.(r_{ji;kk} L_{il}-r_{il;kk} L_{ji})E_{jl}\otimes E_{kk}
     \right]\nonumber\\
& &+\sum_{i,j,l;j\not=l} \left[(r_{li;lj}
  L_{il}-r_{il;lj}L_{li}+2\left(r_{ij;ll}L_{li}-r_{li;ll}L_{ij}\right))E_{ll}\otimes
     E_{lj}\right.
\nonumber\\
& & +\left(r_{li;jl}
     L_{il}-r_{il;jl}L_{li}+2(-r_{ji;ll}L_{il}+r_{il;ll}L_{ji})\right)
E_{ll}\otimes  E_{jl}\nonumber\\
& &  +\left(r_{li;ll}
      L_{ij}-r_{ij;ll} L_{li}\right)E_{lj}\otimes E_{ll}
      +2 \left(r_{li;jl}
      L_{ij}-r_{ij;jl} L_{li}\right)E_{lj}\otimes E_{jl}\nonumber\\
& & +\left. \left(r_{ji;ll}
      L_{il}-r_{il;ll} L_{ji}\right)E_{jl}\otimes E_{ll}+2\left(r_{li;lj}
      L_{ij}-r_{ij;lj} L_{li}\right)E_{lj}\otimes E_{lj} \right]\nonumber\\
& &+2\sum_{i,j,l,j\not=l} \left(
     r_{li;jj}L_{il}-r_{il;jj}L_{li}
    -r_{ji;ll}L_{ij}+r_{ij;ll}L_{ji}\right)E_{ll}\otimes  E_{jj} \nonumber\\
& &+4\sum_{i,l} (r_{li;ll}
     L_{il}-r_{il;ll}L_{li})E_{ll}\otimes  E_{ll} \nonumber\\
&=&\{L_1,L_2\}
\quad\hbox{by Eqs. (\ref{L12}) and (\ref{P12}). This finishes the proof of Eq.
(\ref{FPB}).}
\nonumber
\eeq



\end{document}


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