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\begin{document}
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\begin{center}
{\footnotesize Available at:
{\tt http://publications.ictp.it}}\hfill IC/2010/006\\
\vspace{1cm}
United Nations Educational, Scientific and Cultural Organization\\
and\\
International Atomic Energy Agency\\
\medskip
THE ABDUS SALAM INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS\\
\vspace{2.5cm}
{\bf FROBENIUS MANIFOLDS \\
FROM REGULAR CLASSICAL $W$-ALGEBRAS}\\
\vspace{2cm}
Yassir Ibrahim Dinar\footnote{dinar@ictp.it}\\
{\it Faculty of Mathematical Sciences, University of Khartoum, Sudan\\
and\\
The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.}\\
\end{center}
\vspace{1cm}
\centerline{\bf Abstract}
\baselineskip=18pt
\bigskip
We obtain polynomial Frobenius manifolds from classical $W$-algebras associated to regular nilpotent elements in simple Lie algebras using the related opposite Cartan
subalgebras.
\vfill
\begin{center}
MIRAMARE -- TRIESTE\\
February 2010\\
\end{center}
\vfill
\newpage
\section{Introduction}
This work is a continuation of \cite{mypaper} where we began to develop a construction of algebraic Frobenius manifolds from Drinfeld-Sokolov reduction to support a conjecture of Dubrovin.
A Frobenius manifold is a manifold $M$ with the structure of Frobenius algebra on the tangent space $T_t$ at any point
$t \in M $ with certain compatibility conditions \cite{DuRev}. We say $M$ is semisimple or massive if $T_t$ is semisimple for generic $t$. This structure locally
corresponds to a potential satisfying a system of partial differential equations known in topological field theory as the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations. We say $M$ is algebraic if, in the flat coordinates, the potential is an algebraic function. Dubrovin conjecture is stated as follows: Semisimple irreducible algebraic Frobenius manifolds
with positive degrees correspond to quasi-Coxeter (primitive) conjugacy classes in finite Coxeter groups. We discussed in \cite{mypaper} how the examples of algebraic Frobenius manifolds constructed from Drinfeld-Sokolov reduction support this conjecture.
Let $e$ be a \textbf{regular nilpotent element} in a simple Lie algebra $\g$ over $\mathbb{C}$ of rank $r$. By definition a nilpotent element is called regular if its centralizer in $\g$ is of dimension $r$. We fix, by using the Jacobson-Morozov theorem, a semisimple element $h$ and a nilpotent element $f$ such that $\A=\{e,h,f\}$ is an $sl_2$-triple. Then $\A$ is called \textbf{regular $sl_2$-triple}. Let $\kappa+1$ be the Coxeter number of $\g$. We prove the following
\begin{thm}\label{main thm}
The \textbf{Slodowy slice}
\beq
Q':=e+\ker \ad ~f
\eeq
has a natural structure of polynomial Frobenius manifold of degree $\kappa-1\over \kappa+1$.
\end{thm}
By natural structure we mean that it can be formulated entirely in terms of the representation theory of the regular $sl_2$-triple $\A$ along with the closely related opposite Cartan subalgebra. Let us recall some structures related to $\A$. The element $h\in \A$ defines a $\mathbb{Z}$-grading on $\g$ called the Dynkin grading given as follows
\begin{equation}
\g=\oplus_{i\in \Z} \g_i, ~~\g_i=\{q\in \g: ad~h(q)=i q\}.
\end{equation}
Fix nonzero element $a\in \g_{-2\kappa}$. Then it follows from the work of Kostant \cite{kostBetti} that $y_1=e+a$ is regular semisimple. The Cartan subalgebra $\h'=\ker \ad~ y_1$ is called \textbf{the opposite Cartan subalgebra} and it is one of the main ingredients in our work. Let \beq
1=\eta_1\leq \eta_2\leq \ldots \leq \eta_{r-1}<\eta_r=\kappa
\eeq
be the exponents of $\g$. Then the element $y_1$ can be completed to a basis $y_1,\ldots,y_r$ for $\h'$ having the form \beq \label{elemtry} y_i=v_i+u_i, ~~u_i\in \g_{2\eta_i},~ v_i\in \g_{2\eta_i-2(\kappa+1)}.\eeq
We define the coordinates $(z^1,...,z^r)$ on $Q'$ by setting for $q\in Q'$
\beq
z^i(q):=\bil {q}{u_i}.
\eeq
Here $\bil . .$ is the invariant bilinear form on $\g$ normalized such that $\bil e f =1 $.
Our main idea is to use the theory of local bihamiltonian structures on a loop space to construct the polynomial
Frobenius manifold on $Q'$. Recall that a bihamiltonian structure on a manifold $M$ is two compatible Poisson brackets on $M$. It is well known that the dispersionless limit of a local bihamiltonian structure
on the loop space $\lop M$ of a finite dimensional manifold $M$ (if it exists) always
gives a bihamiltonian structure of hydrodynamic type on $\lop M$:
\begin{eqnarray}\label{ini eq} \{t^i(x),t^j(y)\}_{1} &=& g^{ij}_{1}(t(x))
\delta'(x-y) + \Gamma^{ij}_{1;k}(t(x)) t^k_x \delta(x-y),\\\nonumber
\{t^i(x),t^j(y)\}_{2} &=& g^{ij}_{2}(t(x))
\delta'(x-y) + \Gamma^{ij}_{2;k}(t(x)) t^k_x \delta(x-y),\end{eqnarray}
This gives a flat pencil of
metrics $g^{ij}_{1,2}$ on $M$ provided that the two matrices $g_1^{ij}$ and $g_2^{ij}$ are nondegenerate. A flat pencil of metrics, under the quasihomogeneity and the regularity conditions,
corresponds to a Frobenius structure on $M$ \cite{DFP} (see section 2.1 for details).
We obtain a bihamiltonian structure on the affine loop space \beq Q=e+ \lop {\ker \ad\, f}.\eeq
by using the Drinfeld-Sokolov reduction \cite{DS} (see also \cite{mypaper} or \cite{BalFeh1}). This reduction depends only on the representation theory of $\A$. It begins by defining a bihamiltonian structure $P_1$ and $P_2$ in $\lop \g$. The Poisson structure $P_2$ is the standard Lie-Poisson structure and $P_1$ depends on the adjoint action of $a$. Main while, the space $Q$ will be transversal to an action of the adjoint group of $\lop {\nneg}$ on a suitable affine subspace \beq S:=e+\lop\bneg\eeq of $\lop\g$. Here
\beq\nneg:=\bigoplus_{i\leq -2}\g_i,~~~~\bneg:=\bigoplus_{i\leq 0}\g_i.\eeq
It turns out that the space of local functionals with densities in the ring $R$ of invariant differential polynomials of this action is closed under $P_1$ and $P_2$. This defines the Drinfeld-Sokolov bihamiltonian structure $P_1^Q$ and $P_2^Q$ on $Q$ since the coordinates $z^i(x)$ of $Q$ can be interpreted as generators of the ring $R$. The second reduced Poisson structure on $Q$ is known in the literature as \textbf{classical $W$-algebras} associated to principal nilpotent elements in $\g$. Therefore, we call it \textbf{regular classical $W$-algebras}. For a general definition for classical $W$-algebras see \cite{fehercomp}.
In \cite{BalFeh} they proved the Drinfeld-Sokolov reduction of $P_2$ on $Q$ is the same as Dirac reduction of $P_2$ to $Q$. In particular, they obtained the following
\begin{prop} \cite{BalFeh}\label{walg1} The second Poisson bracket on $Q$ takes the form
\begin{eqnarray}\label{leading terms1}
\{z^1(x),z^1(y)\}_2^Q&=& \eps \delta^{'''}(x-y) +2 z^1(x) \delta'(x-y)+ z^1_x\delta(x-y) \\\nonumber
\{z^1(x),z^i(y)\}_2^Q &=& (\eta_i+1) z^i(x) \delta'(x-y)+ \eta_i z^i_x \delta(x-y).
\end{eqnarray}
\end{prop}
We use this result and some facts about the structure of Lie-Poisson brackets on $\g$ to prove the following
\begin{thm}
The Drinfeld-Sokolov bihamiltonian structure on $Q$ admits a dispersionless limit. The corresponding bihamiltonian structure of hydrodynamic type gives a flat pencil of metrics on the Slodowy slice $Q'$.
\end{thm}
A large portion of this work is devoted to prove the nondegeneracy condition for the matrices $g_1^{ij}$ and $g_2^{ij}$ obtained from the dispersionless limit of $P_1^Q$ and $P_2^Q$, respectively. For this end we use mainly two facts. First, the basis $y_i$ for $\h'$ can be normalized in such away that the elements $u_i$ in \eqref{elemtry} are the highest weight vectors for irreducible $\A$-submodules $V^i$ satisfying
\beq
\g=\bigoplus_{i=1}^r V^i,~~\bil {V^i}{V^j}=0 \textrm{ if } i\neq j.
\eeq
Using this decomposition we introduce a basis \beq X^i_I;~~i=1,\ldots, r;~~~I=-\eta_i,-\eta_i+1,\ldots,\eta_i\eeq for $\g$ compatible with the adjoint action of $\A$. Second, in the coordinates corresponding to this basis $X^i_I$, it is very easy to obtain the linear terms of the generators $z^i(x)$ written as differential polynomials in the coordinates of $S$. In the end we are able to prove
\begin{prop}
The matrix $g^{ij}_1$ is nondegenerate and its determinant is equal to the determinant of the matrix $A_{ij}=\bil {y_i} {y_j}$.
\end{prop}
The nondegeneracy condition for $g_2^{ij}$ will follow from a certain differential relation between the entries of two matrices. Namely we have
\beq
\partial_{z^r} g^{ij}_2=g^{ij}_1.
\eeq
The quasihomogeneity and the regularity conditions for the flat pencil of metrics fellows from proposition \ref{walg1} and the quasihomogeneity of the entries of $g^{ij}_2$ when we assign degree $2 \eta_i+2$ to $z^i$. Finally we get the promised polynomial Frobenius manifold by using the work of \cite{DFP}.\\
We mention that from the work of Dubrovin \cite{DCG} and Hertling \cite{HER} semisimple polynomial Frobenius manifolds with positive degrees are already classified. They correspond to Coxeter conjugacy classes in Coxeter groups. Dubrovin constructed all these polynomial Frobenius manifolds on the orbit spaces of Coxeter groups using the results of \cite{saito}. There is another method to obtain the classical $W$-algebra associated to regular nilpotent elements known in the literature as Muira type transformation \cite{DS}. It was used in \cite{DubCentral} (see also \cite{dinarphd}) to prove that the dispersionless limit of the Drinfeld-Sokolov bihamiltonian structure gives the polynomial Frobenius manifold defined on the orbit space of the corresponding Weyl group \cite{DCG}. The proof depends also on the invariant theory of Coxeter groups. In the present work we give a new method to obtain polynomial Frobenius manifolds from the Drinfeld-Sokolov reduction which depending only on the representation theory of principal $sl_2$-triples.
\section{Preliminaries}
\subsection{Frobenius manifolds and local bihamiltonian structures}
Starting we want to recall some definitions and review the construction of Frobenius manifolds from local bihamiltonian structure of hydrodynamics type.
A \textbf{Frobenius manifold} is a manifold $M$ with
the structure of Frobenius algebra on the tangent space $T_t$ at
any point $t \in M $ with certain compatibility conditions \cite{DuRev}. This structure locally corresponds to a potential
$\mathbb{F}(t^1,...,t^r)$ satisfying the WDVV equations
\begin{equation} \label{frob}
\partial_{t^i}
\partial_{t^j}
\partial_{t^k} \mathbb{F}(t)~ \eta^{kp} ~\partial_{t^p}
\partial_{t^q}
\partial_{t^n} \mathbb{F}(t) = \partial_{t^n}
\partial_{t^j}
\partial_{t^k} \mathbb{F}(t) ~\eta^{kp}~\partial_{t^p}
\partial_{t^q}
\partial_{t^i} \mathbb{F}(t)
\end{equation}
where $(\eta^{-1})_{ij} = \partial_{t^r} \partial_{t^i} \partial_{t^j}
\mathbb{F}(t)$ is a constant matrix. Here we assume that the quasihomogeneity condition takes the form
\begin{equation}
\sum_{i=1}^r d_i t_i \partial_{t^i} \mathbb{F}(t) = \left(3-d \right) \mathbb{F}(t)
\end{equation}
where $d_r=1$. This condition defines {\bf the degrees} $d_i$ and {\bf the charge} $d$ of the Frobenius structure on $M$. If $\mathbb{F}(t)$ is an algebraic function we call $M$ an
\textbf{algebraic Frobenius manifold}.
Let $\lop M$ denote the loop space of $M$, i.e the space of smooth maps from the circle to $M$. A local Poisson bracket $\{.,.\}_1$ on $\lop M$ can be written in the form \cite{DZ}
\begin{equation} \label{genLocPoissBra}\{u^i(x),u^j(y)\}_1=
\sum_{k=-1}^\infty \epsilon^k \{u^i(x),u^j(y)\}^{[k]}_1.
\end{equation}
Here $\epsilon$ is just a parameter and
\begin{equation}\label{genLocBraGen}
\{u^i(x),u^j(y)\}^{[k]}_1=\sum_{s=0}^{k+1} A_{k,s}^{i,j}
\delta^{(k-s+1)}(x-y),
\end{equation}
where $A_{k,s}^{i,j}$ are homogenous polynomials in $\partial_x^j
u^i(x)$ of degree $s$ (we assign degree $j$ to
$\partial_x^j u^i(x)$) and $\delta(x-y)$ is the Dirac delta function defined by
\[\int_{S^1} f(y) \delta(x-y) dy=f(x).\]
The first terms can be written
as follows
\begin{eqnarray}
% \nonumber to remove numbering (before each equation)
\{u^i(x),u^j(y)\}^{[-1]}_1 &=& F^{ij}_1(u(x))\delta(x-y) \\
\{u^i(x),u^j(y)\}^{[0]}_1 &=& g^{ij}_{1}(u(x)) \delta' (x-y)+ \Gamma_{1k}^{ij}(u(x)) u_x^k \delta (x-y)
\end{eqnarray}
Here the entries $g^{ij}_1(u)$, $F^{ij}_1(u)$ and $\Gamma_{1k}^{ij}(u)$ are smooth functions on the finite dimension space $M$. We note that, under the change of coordinates on $M$ the matrices $g^{ij}_1(u)$, $F^{ij}_1(u)$ change as a $(2,0)$-tensors.
The matrix $F^{ij}_1(u)$ defines a Poisson structure on
$M$. If $F^{ij}_1(u(x))=0$ and $\{u^i(x),u^j(y)\}^{[0]}_1\neq 0$ we say the Poisson bracket admits a \textbf{dispersionless limit}. If the Poisson bracket admits a dispersionless limit then $\{u^i(x),u^j(y)\}^{[0]}_1$ defines a
Poisson bracket on $\lop M$ known as \textbf{Poisson bracket of
hydrodynamic type}. By nondegenerate Poisson bracket of
hydrodynamic type we mean those with the matrix $g^{ij}_1$ is nondegenerate. In
this case the matrix $g^{ij}_1(u)$ defines a contravariant flat metric on the cotangent space $T^*M$ and
$\Gamma_{1k}^{ij}(u)$ is its contravariant Levi-Civita connection \cite{DN}.
Assume there are two Poisson structures $\{.,.\}_2$ and $\{.,.\}_1$ on $\lop M$ which form a bihamiltonian structure, i.e $\{.,.\}_\lambda:= \{.,.\}_2+\lambda \{.,.\}_1$ is a Poisson structure on $\lop M$ for every $\lambda$. Consider the notations for the leading terms of $\{.,.\}_1$ given above and write the leading terms of $\{.,.\}_2$ in the form
\begin{eqnarray}
% \nonumber to remove numbering (before each equation)
\{u^i(x),u^j(y)\}^{[-1]}_2 &=& F^{ij}_2(u(x))\delta(x-y) \\
\{u^i(x),u^j(y)\}^{[0]}_2 &=& g^{ij}_{2}(u(x)) \delta' (x-y)+ \Gamma_{2k}^{ij}(u(x)) u_x^k \delta (x-y)
\end{eqnarray}
Suppose that $\{.,.\}_1$ and $\{.,.\}_2$ admit a dispersionless limit. In addition, assume the corresponding Poisson brackets of hydrodynamics type are nondegenerate as well as the dispersionless limit of $\{.,.\}_\lambda$ for generic $\lambda$. Then by definition $g_1^{ij}(u)$ and $g_2^{ij}(u)$ form what is called \textbf{flat pencil of metrics} \cite{DFP}, i.e $g_\lambda^{ij}(u):=g_2^{ij}(u)+\lambda g_1^{ij}(u)$ defines a flat metric on $T^*M$ for generic $\lambda$ and its Levi-Civita connection is given by $\Gamma_{\lambda k}^{ij}(u)=\Gamma_{2k}^{ij}(u)+\lambda \Gamma_{1k}^{ij}(u)$.
\begin{defn} \label{def reg} A contravariant flat pencil of metrics on a manifold $M$ defined by the matrices $g_1^{ij}$ and $g_2^{ij}$ is
called \textbf{ quasihomogenous of degree} $d$ if there exists a
function $\tau$ on $M$ such that the vector fields
\begin{eqnarray} E&:=& \nabla_2 \tau, ~~E^i
=g_2^{is}\partial_s\tau
\\\nonumber e&:=&\nabla_1 \tau, ~~e^i
= g_1^{is}\partial_s\tau \end{eqnarray} satisfy the following
properties
\begin{enumerate}
\item $ [e,E]=e$.
\item $ \Lie_E (~,~)_2 =(d-1) (~,~)_2 $.
\item $ \Lie_e (~,~)_2 =
(~,~)_1 $.
\item
$ \Lie_e(~,~)_1
=0$.
\end{enumerate}
Here for example $\Lie_E$ denote the Lie derivative along the vector field $E$ and $(~,~)_1$ denote the metric defined by the matrix $g^{ij}_1$. In addition, the quasihomogenous flat pencil of metrics is called \textbf{regular} if the
(1,1)-tensor
\begin{equation}\label{regcond}
R_i^j = {d-1\over 2}\delta_i^j + {\nabla_1}_i
E^j
\end{equation}
is nondegenerate on $M$.
\end{defn}
The connection between the theory of Frobenius manifolds and flat pencil of metrics is encoded in the following theorem
\begin{thm}\cite{DFP}\label{dub flat pencil}
A contravariant quasihomogenous regular flat pencil of metrics of degree $d$ on a manifold $M$ defines a Frobenius structure on $M$ of the same degree.
\end{thm}
It is well known that from a Frobenius manifold we always have a flat pencil of metrics but it does not necessary satisfy the regularity condition \eqref{regcond}. In the notations of \eqref{frob} from a Frobenius structure on $M$, the flat pencil of metrics is
found from the relations \begin{eqnarray}\label{frob eqs} \eta^{ij}&=&g_1^{ij} \\
\nonumber g_2^{ij}&=&(d-1+d_i+d_j)\eta^{i\alpha}\eta^{j\beta}
\partial_{t^\alpha}
\partial_{t^\beta} \mathbb{F}
\end{eqnarray}
This flat pencil of metric is quasihomogenous of degree $d$ with $\tau =t^1$. Furthermore we have
\begin{equation}
E=\sum_i d_i t^i {\partial_{t^i}},~~~e={\partial_{t^n}}
\end{equation}
\subsection{Regular nilpotent element and opposite Cartan subalgebra}
We review some facts about regular nilpotent elements in simple Lie algebra we need to perform the Drinfeld-Sokolov reduction. In particular, we recall the concept of the opposite Cartan subalgebra and we introduce a particular basis for $\g$ compatible with the action of a given regular $sl_2$-triple.
Let $\g$ be a simple Lie algebra over $\mathbb C$ of rank $r$. We fix a regular nilpotent element $e\in \g$. By definition a nilpotent element is called regular if $\g^e:=\ker \ad ~e$ has dimension equals to $r$. Using the Jacobson-Morozov theorem we fix a semisimple element $h$ and a nilpotent element $f$ in $\g$ such that $\{e,h,f\}$ generate $sl_2$ subalgebra $\A\subset \g$, i.e
\begin{equation}
[h,e]=2 e, ~~~ [h,f]=-2f,~~~[e,f]=h.
\end{equation}
Then $\A$ is called regular $sl_2$-triple.
We normalize the invariant bilinear from $\bil . . $ on $\g$ such that $\bil e f=1$. The affine space
\beq Q'=e+\g^f\eeq is called the \textbf{Slodowy slice}. Let
\begin{equation}
1=\eta_1<\eta_2\leq \eta_3\ldots\leq\eta_{r-1}<\eta_r.
\end{equation}
the exponents of the Lie algebra $\g$. We will refer to the number $\eta_r$ by $\kappa$. Recall that $\kappa+1$ is the Coxeter number of $\g$ and the exponents satisfy the relation
\beq \eta_i+\eta_{r-i+1}=\kappa+1.\eeq
We also recall that for all simple Lie algebras the exponents are different except for the Lie algebra of type $D_{2n}$ the exponent $n-1$ appears twice.
Consider the restriction of the adjoint representation of $\g$ to $\A$. Under this restriction $\g$ decomposes to irreducible $\A$-submodules
\begin{equation}
\g=\oplus V^i
\end{equation}
with $ \dim V^i=2 \eta_i+1$ \cite{HumLie}. We normalize this decomposition by using the following proposition
\begin{prop}\label{reg:sl2:normalbasis}
There exists a decomposition of $\g$ into a sum of irreducible $\A$-submodules $\g=\oplus_{i=1}^{r} V^i$ in such a way that there is a basis $X_I^i, I=-\eta_i,-\eta_i+1,...,\eta_i$ in each $V^i, ~i=1,\ldots,r$ satisfying the following relations
\begin{equation}\label{sl2expand}
X_{I}^i={1\over (\eta_i+I)!} \ad \,e^{\eta_i+I}~X_{-\eta_i}^i~ ,~~~~I=-\eta_i,-\eta_i+1,\ldots, \eta_i.
\end{equation}
and
\beq\label{sl2bilinear}
=\delta_{i,j}\delta_{I,-J} (-1)^{\eta_i-I+1}{2\eta_i\choose \eta_i-I}.
\eeq
Furthermore
\begin{eqnarray}\label{sl2relation}
\ad \, h\,X_I^i&=& 2I X_I^i.\\\nonumber
\ad \, e\,X_I^i&=& (\eta_i+I+1) X_{I+1}^i.\\\nonumber
\ad \, f\, X_I^i &=& (\eta_i -I+1) X_{I-1}^i.
\end{eqnarray}
\end{prop}
\begin{proof}
The proof that one could compose the Lie algebra as irreducible $\A$-submodules satisfying \eqref{sl2expand} and \eqref{sl2relation} is standard and can be found in \cite{HumLie} or \cite{kostBetti}. Let $\g=\oplus_{i=1}^{r} V^i$ be such decomposition. It is easy to prove $\bil {V^i}{V^j}=0$ in the case $\eta_i\neq\eta_j$ by applying the step operators $\ad~ e$ and using the invariance of the bilinear form. Hence the proof is reduced to the case of irreducible $\A$-submodules of the same dimension. But there is at most two irreducible submodules of the same dimension. Assume $V^{i_1}$ and $V^{i_2}$ have the same dimension and denote the corresponding basis $X^{i_1}_I$ and $X^{i_1}_J$, respectively. Then one can prove by using the step operator $\ad~ e$ that the subspaces $V^{i_1}$ and $V^{i_2}$ are orthogonal if and only if $\bil {X^{i_1}_0}{X^{i_2}_0}= 0$. But it obvious that the restriction of the invariant bilinear form to ${X^{i_1}_0}$ and ${X^{i_2}_0}$ is nondegenerate. Hence by applying the Gram-Schmidt procedure we can assume that $\bil {X^{i_1}_0}{X^{i_2}_0}= 0$.
Therefore, we can assume that the given decomposition satisfying $\bil {V^i}{V^j}=0$ if $i\neq j$. It remains to obtain the normalization \eqref{sl2bilinear}. From the invariance of the bilinear form we have
\begin{equation}
\bil{h.X_{I}^i}{X_J^i}=(2 I) \bil{X_I^i}{X_J^i}
\end{equation}
while
\begin{equation}
-\bil{X_I^i}{h.X_J^i}=-(2 J)\bil{X_I^i}{X_J^i}
\end{equation}
Therefore $\bil{X_I^i}{X_J^j}=0$ if $I+J\neq 0$. We calculate using the step operator $\ad ~e$ where $I\geq 0$ the value
\begin{eqnarray}
\bil{X_I^i}{X_{-I}^i}&=&{1\over (\eta_i-I) } \bil{X_{I}^i}{e.X_{-I-1}^i} \\\nonumber
&=&{-1\over \eta_i-I } \bil{e.X_{I}^i}{X_{-I-1}^i} \\\nonumber
&=&{(-1)(\eta_i-I+1)\over \eta_i-I } \bil{X_{I+1}^i}{X_{-I-1}^i} \\\nonumber
&=&{(-1)^{\eta_i-I}(\eta_i-I+1)(\eta_i-I+2)\ldots 2\eta_i\over (\eta_i-I)(\eta_i-I-1)\ldots(1) } \bil{X_{\eta_i}^i}{X_{-\eta_i}^i}\\\nonumber
&=& (-1)^{\eta_i-I} {2\eta_i\choose \eta_i-I} \bil{X_{\eta_i}^i}{X_{-\eta_i}^i}.
\end{eqnarray}
The result follows by multiplying $X_I^i$ by the value of $-\bil{X_{\eta_i}^i}{X_{-\eta_i}^i}^{-1}$. We note that the formula \eqref{sl2bilinear} will give the same result when replacing $I$ with $-I$. This ends the proof.
\end{proof}
Note that the normalized basis for $V^1$ are $X_1^1=-e,~X_0^1=h,~X_{-1}^1=f$ since it is isomorphic to $\A$ as a vector subspace.
The element $h$ define a $\mathbb{Z}$-grading on $\g$ called the Dynkin grading given as follows
\begin{equation}
\g=\oplus_{i\in \Z} \g_i, ~~\g_i=\{q\in \g: ad~h(q)=i q\}.
\end{equation}
It is well known that $\g_i=0$ if $i$ is odd and \beq \bneg=\oplus_{i\leq 0} \g_i\eeq is a Borel subalgebra with \beq \nneg=\oplus_{i\leq
{-2}}\g_i=[\bneg,\bneg]\eeq is a nilpotent subalgebra. Note that the subalgebra $\g^f$ has a basis $X_{-\eta_i}^i, ~i=1,\ldots,r$ and
\beq \bneg=\g^f\oplus \ad ~e (\nneg).
\eeq Hence $Q'$ is transversal to the orbit of $e$ under the adjoint group action.
In order to introduce the concept of opposite Cartan subalgebra we need to summarize Kostant results about the relation between the regular nilpotent element $e$ and Coxeter conjugacy class in Weyl group of $\g$.
\begin{thm}\cite{kostBetti}
The element $y_1=e+ X_{-2\kappa}^r$ is regular semisimple. Denote $\h'$ the Cartan subalgebra containing $y_1$, i.e $\h':=\ker \ad~y_1$ and consider the adjoint group element $w$ defined by $w:=\exp {\pi \imn\over \kappa+1}\ad~h$. Then $w$ acts on $\h'$ as a representative of the Coxeter conjugacy class in the Weyl group acting on $\h'$. Furthermore, the element $y_1$ can be completed to a basis $y_i,~i=1,\ldots,r$ for $\h'$
having the form
\[ y_i=v_i+u_i, ~~u_i\in \g_{2\eta_i},~ v_i\in \g_{2\eta_i-2(\kappa+1)}\]
and such that $y_i$ is an eigenvector of $w$ with eigenvalue $\exp {\pi \imn\eta_i\over \kappa+1}$.
\end{thm}
\begin{rem}
Kostant proved this theorem by writing the regular nilpotent element $e$ as the sum of the root vectors corresponding to simple roots. It will follows then $X_{-2\kappa}^r$ is a constant multiplication of the root vector corresponding to the the negative of the maximum root. These assumptions will follow easily if we choose the root vectors with respect to the Cartan subalgebra $\h$ contains $h$ and ordering the roots with respect to $h$ \cite{COLMC}.
\end{rem}
Let $a$ denote the element $X_{-2\kappa}^r$. The element $y_1=e+a$ is called a \textbf{cyclic element} and the Cartan subalgebra $\h'=\ker \ad ~y_1$ is called the \textbf{opposite Cartan subalgebra}. We fix a basis $y_i$ for $\h'$ satisfying the theorem above. It is easy to see that $u_i, i=1,...,r$ form a homogenous basis for $\g^e$. We assume the basis $y_i$ are normalized such that
\begin{equation}
u_i=-X_{\eta_i}^i.
\end{equation}
Form construction this normalization does not effect $y_1$.
Let us define the matrix of the invariant bilinear form on $\h'$
\beq
A_{ij}:=\bil {y_i}{y_j}=-\bil {X_{\eta_i}^i}{v_j}-\bil {v_i}{X_{\eta_j}^i},~i,j=1,\ldots,r.
\eeq
The following proposition summarize some useful properties we need in the following sections.
\begin{prop}\label{Gold}
The matrix $A_{ij}$ is nondegenerate and antidiagonal with respect to the exponents $\eta_i$, i.e $A_{ij}=0, ~{\rm if}~\eta_i+\eta_j\neq\kappa+1$. Moreover, the commutators of $a$ and $X_{\eta_i}^i$ satisfy the relations
\begin{equation}
{\bil {[a,X_{\eta_i}^i]}{X_{\eta_j-1}^j}\over 2\eta_j }+{\bil {[a,X_{\eta_j}^j]}{X_{\eta_i-1}^i}\over 2 \eta_i}= A_{ij}
\end{equation}
for all $i,j=1,\ldots,r$.
\end{prop}
\begin{proof}
The matrix $A_{ij}$ is nondegenerate since the restriction of the invariant bilinear form to a Cartan subalgebra is nondegenerate. The fact that it is anidiagonal with respect to the exponents follows from the identity
\beq
\bil {y_i}{y_j}=\bil {w y_i}{w y_j}=\exp {(\eta_i+\eta_j)\pi \imn\over \kappa+1}\bil {y_i}{y_j}
\eeq
where $w:=\exp {\pi \imn\over \kappa+1}\ad~h$. For the second part of the proposition we note that the commutator of $y_1= e+a$ and $y_i=v_i-X_{\eta_i}^i$ gives the relation
\begin{equation}
[e,v_i]=[a,X_{\eta_i}^i], ~ i=1,...,r.
\end{equation}
Which in turn give the following equality for every $i,~j=1,...,r$
\begin{eqnarray}
\bil {[a,X_{\eta_i}^i]}{X_{\eta_j-1}^j}&=&\bil {[e,v_i]}{X_{\eta_j-1}^j}=-\bil {v_i}{[e,X_{\eta_j-1}^j]}\\\nonumber &=& -2 \eta_j \bil {v_i}{X_{\eta_j}^j}
\end{eqnarray}
but then
\begin{equation}
{\bil {[a,X_{\eta_i}^i]}{X_{\eta_j-1}^j}\over 2\eta_j }+{\bil {[a,X_{\eta_j}^j]}{X_{\eta_i-1}^i}\over 2 \eta_i}= -\bil {v_i}{X_{\eta_j}^j}-\bil {v_j}{X_{\eta_i}^i}= A_{ij}.
\end{equation}
\end{proof}
\section{Drinfeld-Sokolov reduction}
We will review the standard Drinfeld-Sokolov reduction associated with the regular nilpotent element \cite{DS} (see also \cite{mypaper}).
We introduce the following bilinear form on the loop algebra $\lop\g$:
\begin{equation} (u|v)=\int_{S^1}\bil {u(x)}{v(x)} dx,~ u,v \in \lop M,
\end{equation}
and we identify $\lop\g$ with $\lop \g^*$ by means of this bilinear form. For a functional $\f$ on $\lop\g$ we
define the gradient $\delta \f (q)$ to be the unique element in
$\lop\g$ such that
\begin{equation}
\frac{d}{d\theta}\f(q+\theta
\dot{s})\mid_{\theta=0}=\int_{S^1}\langle\delta \f|\dot{s}\rangle dx
~~~\textrm{for all } \dot{s}\in \lop\g.
\end{equation}
Recall that we fixed an element $a\in \g$ such that $y_1=e+a$ is a cyclic element. Let us introduce a bihamiltonian structure on $\lop \g$ by means of Poisson tensors
\begin{eqnarray}\label{bih:stru on g}
P_2(v)(q(x))&=&{1\over \eps}[\eps \partial_x+q(x),v(x)].\\\nonumber
P_1(v)(q(x))&=&{1\over \eps} [a,v(x)].
\end{eqnarray}
It is well known fact that these define a bihamiltonian structure on $\lop \g$ \cite{MRbook}.
We consider the gauge transformation of the adjoint group $G$ of $\lop\g$ given by
\begin{eqnarray}
q(x)&\rightarrow& \exp \ad~ s(x)( \partial_x+q(x))-\partial_x
\end{eqnarray}
where $s(x),~q(x)\in \lop \g$. Following Drinfeld and Sokolov \cite{DS}, we consider the restriction of this action to the adjoint group $\gauge$ of $\lop {\nneg}$.
\begin{prop}(\cite{mypaper}, \cite{Pedroni2})\label{DS as momentum }
The action of $\gauge$ on $\lop\g$ with Poisson tensor \beq P_\lambda:=P_2+\lambda P_1\eeq is
Hamiltonian for all $\lambda$. It admits a momentum map $J$ to be
the projection
\[J:\lop\g\to\lop \npos\]
where $\npos$ is the image of $\nneg$ under the Killing map. Moreover, $J$ is $\Ad^*$-equivariant.
\end{prop}
We take $e$ as regular value of $J$. Then
\beq
S:=J^{-1}(e)=\lop\bneg+e,
\eeq
since $\bneg$ is the orthogonal complement to $\nneg$. It follows from the Dynking grading that the isotropy group of $e$ is $\gauge$. Let $R$ be the ring of invariant differential polynomials of $S$ under the action of $\gauge$. Then the set $\mathcal{R} $ of functionals
on $\s$
which have densities in the ring $R$ is closed under $P_2$ and $P_1$. Another proof of this result can be found in \cite{DS}.
Recall that the space $Q$ is defined as
\beq
Q:=e+\lop {\g^f}.
\eeq
The following proposition identified $S/\gauge$ with the space $Q$.
\begin{prop}\cite{DS}
The space $Q$
is a cross section for the action of $\gauge$ on $\s$, i.e for any element $q(x)+e\in \s$ there is a unique element $s(x) \in
\lop \nneg$ such that \begin{equation}\label{gauge fix} z(x)+e=(\exp \ad~s(x))( \partial_x+q(x))-\partial_x \in Q.\end{equation} The entries of $z(x)$ are generators of the ring
$R$ of differential polynomials on $S$ invariant under the action of $\gauge$.
\end{prop}
Hence we have an isomorphism between the set of functionals on $Q$ and the set $\mathcal{R} $. Therefore, $Q$ has a bihamiltonian structure $P_1^Q$ and $P_2^Q$ from $P_1$ and $P_2$, respectively. The reduced Poisson structure $P_2^Q$ is known as \textbf{classical $W$-algebra} associated to the regular nilpotent element $e$. For a formal definition of classical $W$-algebras see \cite{fehercomp}. We obtain the reduced bihamiltonian structure as follows. We write the coordinates of $Q$ as differential polynomials in the coordinates of $S$ by means of equation \eqref{gauge fix} and then apply the Leibnitz rule. For $u,v \in R$ the Leibnitz rule have the following form
\begin{equation}
\{u(x),v(y)\}_\lambda={\partial u(x)\over \partial (q_i^I)^{(m)}}\partial_x^m\Big({\partial v(y)\over \partial (q_j^J)^{(n)}} \partial_y^n\big(\{q_i^I(x),q_j^J(y)\}_\lambda\big)\Big)
\end{equation}
The generators of the invariant ring $R$ will have nice properties when we use the normalized basis we developed in last section. Let us begin by writing the equation of gauge fixing \eqref{gauge fix} after introducing a parameter $\tau$ as follows
\begin{eqnarray*}
q(x)+ e&=& \tau \sum_{i=1}^{r} \sum_{I=0}^{\eta_i} q_{i}^I X_{-I}^{i}+e\in \s\\
z(x)+e &=&\tau\sum_{i=1}^{r} z^i(x) X_{-\eta_i}^i+e\in Q\\
s(x)&=&\tau\sum_{i=1}^{r}\sum_{I=1}^{\eta_i} s_{i}^{I}(x) X_{-I}^{i} \in \lop\nneg.
\end{eqnarray*}
Then equation \eqref{gauge fix} expands to
\begin{equation}\label{gauge fixing data }
\begin{split}
\sum_{i=1}^{r} z^i(x) X_{-\eta_i}^i+&\sum_{i=1}^{r}\sum_{I=1}^{\eta_i}(\eta_i-I+1) s_{i}^{I} X_{-I+1}^{i}=\\&\sum_{i=1}^{r} \sum_{I=0}^{\eta_i} q_{i}^I(x) X_{-I}^{i}-\sum_{i=1}^{r}\sum_{I=1}^{\eta_i} \partial_x s_{i}^{I}(x) X_{-I}^{i}+\mathcal{O}(\tau).
\end{split}
\end{equation}
It obvious that any invariant $z^i(x)$ has the form
\begin{eqnarray}\label{leading terms1}
z^i(x)&=&q_i^{\eta_i}-\partial_x s_i^{\eta_i}+\mathcal{O}(\tau)\\\nonumber
&=& q_i^{\eta_i}(x)-\partial_x q_{i}^{\eta_i-1}+\mathcal{O}(\tau).
\end{eqnarray}
That is, we obtained the linear terms of each invariant $z^i(x)$. Furthermore, since $\bil {e}{f}=1$ then $z^1(x)$ has the expression
\beq
\begin{split}
z^1(x)= q_1^1(x) -\partial_x s_1^1+& \tau \bil {e}{[s_i^1(x) X_{-1}^i, q_{i}^0 X_0^i]}\\&+{1\over2}\tau \bil{e}{[s_i^1(x) X_{-1}^i,[s_i^I(x) X_{-1}^i,e]]}.
\end{split}
\eeq
Which is simplified by using the identity
\beq
[s_i^1(x) X_{-1}^i,[s_i^I(x) X_{-1}^i,e]]=-[s_i^1(x) X_{-1}^i,q_i^0(x) X_0^i]
\eeq
and
\beq
\bil {e}{[s_i^1(x) X_{-1}^i, q_{i}^0 X_0^i]}=-\bil {[s_i^1(x) X_{-1}^i,e]}{q_i^0(x) X_0^i}=(q_i^0(x))^2\bil{X_0^i}{ X_0^i}
\eeq
with $s^1_1(x)=q_1^0(x)$ to the expression
\begin{equation}
z^1(x)= q_1^1(x) -\partial_x q_1^0(x)+{1\over 2} \tau\sum_i (q_i^0(x))^2\bil{X_0^i}{ X_0^i}
\end{equation}
The invariant $z^1(x)$ is called a \textbf{Virasoro density} and the expression above agree with \cite{BalFeh}.
Our analysis will relay on the quasihomogeneity of the invariants $z^i(x)$ in the coordinates of $q(x)\in \lop \bneg$ and their derivatives. This property is summarized in the following corollary
\begin{cor}\label{lin inv poly}
If we assign degree $2 J+2l+2$ to $\partial_x^l(q_i^J(x))$ then $z^i(x)$ will be quasihomogenous of degree $2\eta_i+2$. Furthermore, each invariant $z^i(x)$ depends linearly only on $q_i^{\eta_i}(x)$ and $\partial_x q_{i}^{\eta_i-1}(x)$. In particular, $z^i(x)$ with $i1\] such that the matrix $g_1^{ij}(t)$ is constant antidiagonal.
\end{prop}
For the remainder of this section, we fix a coordinates $(t^1,...,t^n)$ satisfying the proposition above. The following proposition emphasis that under this change of coordinates some entries of the matrix $g_2^{ij}$ remain invariant.
\begin{prop}
The second metric $g^{ij}_2(t)$ have the following entries
\begin{equation}
g^{1,n}_2(t)= {\eta_i+1\over \kappa+1} t^i.
\end{equation}
\end{prop}
\begin{proof}
We know from proposition \ref{walg} that in the coordinates $z^i$ the matrix $g^{ij}_2(z)$ have the following entries
\begin{equation}
g^{1,n}_2(z)= (\eta_i+1) z^i
\end{equation}
Let $E'$ denote the Euler vector field give by
\beq
E'=\sum_i {\eta_i+1\over \kappa+1} z^i { \partial_{z^i}}.
\eeq
Then from the quasihomogeneity of $t^i$ we have $E'(t^i)={\eta_i+1\over \kappa+1} t^i$. The formula for change of coordinates and the fact that $t^1={1\over \kappa+1} z^1$ give the following
\beq
g^{1j}(t)={\partial_{z^a} t^1 } {\partial_{z^b} t^j}~ g_2^{a b}(z)= E'(t^j)={\eta_j+1\over \kappa+1} t^j.
\eeq
\end{proof}
We arrive to our basic result
\begin{thm}\label{my thm}
The flat pencil of metrics on the Slodowy slice $Q'$ obtained from the dispersionless limit of Drinfeld-Sokolov bihamiltonian structure on $Q$ (see theorem \ref{the pencil}) is regular quasihomogenous of degree $\kappa-1\over \kappa+1$.
\end{thm}
\begin{proof}
In the notations of definition \ref{def reg} we take $\tau =t^1$ then
\begin{eqnarray}
E&=& g^{ij}_2 {\partial_{t^j} \tau }~{\partial_{t^i} }={1\over \kappa+1} \sum_{i} (\eta_i+1) t^i{\partial_{t^i} },\\\nonumber
e &=& g^{ij}_1 {\partial_{t^j} \tau }~{\partial_{t^i} }={\partial_{t^r} }.
\end{eqnarray}
We see immediately that \[ [e,E]=e\]
The identity \begin{equation} \Lie_e (~,~)_2 =
(~,~)_1 \end{equation} follows from and the fact that ${\partial_{t^r} }={\partial_{ z^r}}$ and proposition \ref{diff relaton}. The fact that
\begin{equation}
\Lie_e(~,~)_1
=0.
\end{equation}
is a consequence from the quasihomogeneity of the matrix $g_1^{ij}$ (see lemma \ref{homg of g1}). We also obtain from proposition \ref{homg of g2}
\begin{equation} \Lie_E (~,~)_2 =(d-1) (~,~)_2
\end{equation}
since
\begin{equation}
\Lie_E(~,~)_2(dt^i,dt^j)= E(g^{ij}_2)-{\eta_i+1\over \kappa+1}g^{ij}_2-{\eta_j+1\over \kappa+1} g^{ij}_2={-2\over \kappa+1} g^{ij}_2.
\end{equation}
The (1,1)-tensor
\begin{equation}
R_i^j = {d-1\over 2}\delta_i^j + {\nabla_1}_iE^j = {\eta_i \over \kappa+1} \delta_i^j.
\end{equation}
is obviously nondegenerate. This complete the proof.
\end{proof}
Now we are ready to prove theorem \ref{main thm}.
\begin{proof}{[Theorem \ref{main thm}]}
It follows from theorem \ref{my thm} and \ref{dub flat pencil} that $Q'$ has a Frobenius structure of degree $\kappa-1\over \kappa+1$ from the dispersionless limit of Drinfeld-Sokolov bihamiltonian structure. This Frobenius structure is polynomial since in the coordinates $t^i$ the potential $\mathbb{F}$ is constructed from equations \eqref{frob eqs} and we know from proposition \ref{mat poly} that the matrix $g^{ij}_2$ is polynomial.
\end{proof}
\subsection{Conclusions and remarks}
The results of the present work can be generalized to a certain class of distinguished nilpotent elements in simple Lie algebras. In particular, we notice that the existence of opposite Cartan subalgebras is the main reason behind the examples of algebraic Frobenius manifolds constructed in \cite{mypaper} which are associated to distinguished nilpotent elements in the Lie algebra of type $F_4$. In \cite{mypaper} we discussed how these examples support Dubrovin conjecture. Our goal is to develop a method to uniformize the construction of all algebraic Frobenius manifolds that can be obtained from distinguished nilpotent elements in simple Lie algebras by performing the generalized Drinfeld-Sokolov reduction. Similar treatment of the present work for algebraic Frobenius manifolds that can be obtained from subregular nilpotent elements in simple Lie algebras is now under preparation.\\
\vskip 0.5truecm \noindent{\bf Acknowledgments}
The author thanks B. Dubrovin for useful discussions.
\newpage
\begin{thebibliography}{99}
\bibitem{BalFeh} Balog, J.; Feher, L.; O'Raifeartaigh, L.; Forgacs, P.; Wipf, A., Toda theory and $W$-algebra from a gauged WZNW point of view. Ann. Physics 203 , no. 1, 76--136 (1990).
\bibitem{CMP} Casati, Paolo; Magri, Franco; Pedroni, Marco, Bi-Hamiltonian manifolds and $\tau$-function. Mathematical aspects of classical field theory, 213--234 (1992).
\bibitem{CP}
Casati, Paolo; Pedroni, Marco; Drinfeld-Sokolov reduction on a simple Lie algebra from the bi-Hamiltonian point of view. Lett. Math. Phys. 25, no. 2, 89--101 (1992).
\bibitem{COLMC} Collingwood, David H.; McGovern, William M., Nilpotent orbits in semisimple Lie algebras. Van Nostrand Reinhold Mathematics Series. ISBN: 0-534-18834-6 (1993).
\bibitem{DamSab} Damianou, P. A., Sabourin, H., Vanhaecke, P., Transverse Poisson structures to adjoint orbits in semisimple Lie algebras. Pacific J. Math., no. 1, 111--138 232 (2007).
\bibitem{mypaper} Dinar, Yassir, On classification and construction of algebraic Frobenius manifolds. Journal of Geometry and Physics, Volume 58, Issue 9, September (2008).
\bibitem{dinarphd} Dinar, Yassir, PhD Thesis, Title: Algebraic Frobenius manifolds and primitive conjugacy classes in Weyl groups, SISSA (July 2007).
\bibitem{DS} Drinfeld, V. G.; Sokolov, V. V., Lie algebras and equations of Korteweg-de Vries type. (Russian) Current problems in mathematics, Vol. 24, 81--180, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, (1984).
\bibitem{DCG} Dubrovin, Boris, Differential geometry of the space of orbits of a Coxeter group. Surveys in differential geometry IV: integrable systems, 181--211 (1998).
\bibitem{DuRev} Dubrovin, Boris, Geometry of $2$D topological field theories. Integrable systems and quantum groups (Montecatini Terme, 1993), 120--348, Lecture Notes in Math., 1620, Springer, Berlin, (1996).
\bibitem{DFP} Dubrovin, Boris, Flat pencils of metrics and Frobenius manifolds. Integrable systems and algebraic geometry (Kobe/Kyoto, 1997), 47--72, World Sci. Publ. (1998).
\bibitem{DN} Dubrovin, B. A.; Novikov, S. P., Poisson brackets of hydrodynamic type. (Russian) Dokl. Akad. Nauk SSSR 279, no. 2, 294--297 (1984).
\bibitem{DubCentral} Dubrovin, B., Liu Si-Qi; Zhang, Y., Frobenius manifolds and central invariants for the Drinfeld-Sokolov biHamiltonian structures. Adv. Math. 219, no. 3,780--837(2008).
\bibitem{DZ} Dubrovin, B. , Zhang, Y., Normal forms of hierarchies of integrable PDEs, Frobenius
manifolds and Gromov-Witten invariants, www.arxiv.org,math/0108160.
\bibitem{HER} Hertling, Claus, Frobenius manifolds and moduli spaces for singularities. Cambridge Tracts in Mathematics, 151. Cambridge University Press, ISBN: 0-521-81296-8 (2002).
\bibitem{HumLie} Humphreys, James E. Introduction to Lie algebras and representation theory. Graduate Texts in Mathematics, 9. Springer-Verlag, ISBN: 0-387-90053-5 (1978).
\bibitem{BalFeh1} Feher, L.; O'Raifeartaigh, L.; Ruelle, P.; Tsutsui, I.; Wipf, A. On Hamiltonian reductions of the Wess-Zumino-Novikov-Witten theories. Phys. Rep. 222, no. 1 (1992).
\bibitem{fehercomp} Feher, L.; O'Raifeartaigh, L.; Ruelle, P.; Tsutsui, I., On the completeness of the set of classical $ W$-algebras obtained from DS reductions. Comm. Math. Phys. 162 , no. 2, 399--431 (1994).
\bibitem{kostBetti} Kostant, B., The principal three-dimensional subgroup and the Betti numbers of a
complex simple Lie group, Amer. J. Math. 81, 973(1959).
\bibitem{MRbook} Marsden, Jerrold E.; Ratiu, Tudor S., Introduction to mechanics and symmetry. Springer-Verlag, ISBN: 0-387-97275-7; 0-387-94347-1 (1994).
\bibitem{Pedroni2} Pedroni, Marco, Equivalence of the Drinfeld-Sokolov reduction to a bi-Hamiltonian reduction. Lett. Math. Phys. 35, no. 4, 291--302 (1995).
\bibitem{saito} Saito, K.; Yano, T.; Sekiguchi, J. , On a certain generator system of the ring of invariants of a finite reflection group. Comm. Algebra 8 , no. 4, 373--408 (1980).
\end{thebibliography}
\end{document}