text:
\section{Introduction}

A first step in the study of an arbitrary category $\mathcal{C}$ is
to define the appropriated set $\mathcal{S(C)}$ of isomorphisms
classes of {\em simple} objects in $\mathcal{C}$. For example, an
object $y \in \rm{Ob}(\mathcal{C})$ of an abelian category $\mathcal{C}$
(see \cite{Man}) is said to be simple if in any exact sequence
$$0 \to x \to y \to z \to 0,$$
either $x$ is isomorphic to $0$ or $z$ is isomorphic to $0$. It is a
remarkable fact that non-equivalent categories may very well have
equivalent sets of isomorphisms classes of simple objects. Let us
introduce a list of categories that at first seem to be utterly
unrelated and yet the corresponding sets of simple objects are
deeply connected.

\begin{enumerate}
\item $\mathbf{WeylGroup} \subset\mathbf{FinGroup} \subset
\mathbf{LieGroup} \subset \mathbf{Group}.$

We denote by $\mathbf{Group}$ the category whose objects are
groups and whose morphisms are group homomorphisms. We let
$\mathbf{LieGroup}$ denote the subcategory of $\mathbf{Group}$
whose objects are finite dimensional complex Lie groups. Morphism
in $\mathbf{LieGroup}$ are smooth group homomorphisms. We define
$\mathbf{FinGroup}$ to be the full subcategory of $\mathbf{Group}$
whose objects are finite groups and $\mathbf{WeylGroup}$ to be the
full subcategory of $\mathbf{Group}$ whose objects are Weyl
groups.

$S(\mathbf{Group})$ denotes the set of isomorphisms classes of
groups having no proper normal subgroups. $S(\mathbf{LieGroup})$
denotes the set of isomorphisms classes of  Lie groups which are
simple as groups and also are connected and simply connected.
$S(\mathbf{FinGroup})$ denotes the set of isomorphisms classes of
finite simple groups. $S(\mathbf{WeylGroup})$ denotes the set
isomorphisms classes of Weyl groups, which can be taken to be
$A_n=S_{n+1}$, $B_n=C_n=\mathbb{Z}_2^{n}\ltimes S_n$ ,
$D_n=\mathbb{Z}_2^{n-1}\ltimes S_n$, where $S_n$ is the group of
permutations in $n$ letters, and $E_6, E_7, E_8, F_4, G_2$ are so
called exceptional groups.

\item $\mathbf{LieAlg}$ denotes the category whose objects are
finite dimensional complex Lie algebras, morphism are Lie algebra
homomorphism. $S(\mathbf{LieAlg})$ is the set of isomorphisms
classes of simple Lie algebras, i.e., Lie algebras having no proper
ideals.

\item $\mathbf{Root}$ denotes the category of root systems.

\begin{itemize}
\item Objects in $\mathbf{Root}$ are triples $(V,\langle \mbox{ }, \mbox{ }\rangle, \Phi)$ such that

\begin{itemize}

\item{$(V, \langle \mbox{ }, \mbox{ }\rangle)$ is an Euclidean
space.} \item{$\Phi$ is a finite set spanning $V$.} \item{If
$\alpha\in\Phi$ then $-\alpha\in\Phi$, but $k \alpha\not\in \Phi$ if
$k$ is any real number other then $\pm 1$.} \item{For
$\alpha\in\Phi$ the reflection $S_{\alpha}$ in the hyperplane
$\alpha^{\perp}=\{x\in V: \langle x,\alpha\rangle=0\}$ maps $\Phi$
to itself.} \item{For $\alpha, \beta \in\Phi$, $A_{\alpha, \beta}=
{\displaystyle 2\frac{\langle \alpha, \beta\rangle}{ \langle
\alpha, \alpha \rangle}\in \mathbb{Z}}$.}
\end{itemize}

\item Morphisms in $\mathbf{Root}$ from $(V_1,\langle \mbox{ }, \mbox{ }\rangle_1, \Phi_1)$ to
 $(V_2,\langle \mbox{ }, \mbox{ }\rangle _2, \Phi_2)$ consists of
linear transformations $T:V_1 \longrightarrow V_2$ such that
$\langle T(x),T(y)\rangle _2= \langle x,y \rangle _1$ for all
$x,y\in V_1$, and $T(\Phi_1)\subset \Phi_2$.

\item Suppose that $(V_i,\langle \mbox{ }, \mbox{ }\rangle _{V_i}, \Phi_i)$,
$i=1,\dots,n$ are root systems, then the direct sum\\
$V=\bigoplus_{i=1}^{n} V_i$ is an Euclidean vector space with
inner product
$$\langle \mbox{ },\mbox{ }\rangle_V= \sum_{i=1}^{n}\langle \mbox{ },\mbox{ } \rangle _{V_i}$$
Moreover setting $\Phi=\bigsqcup_{i=1}^{n} \Phi_i$, the triple
$(V,\langle \mbox{ }, \mbox{ }\rangle _V, \Phi)$ becomes a root
system called the direct sum of root systems $(V_i,\langle \mbox{
}, \mbox{ }\rangle_{V_i}, \Phi_i)$. $S(\mathbf{Root})$ is the set
of isomorphisms classes of simple root systems, i.e., root systems
which are not isomorphic to the direct sum of two non-vanishing
root systems.

\end{itemize}

\item $\mathbf{Dynkin}$ denotes the category of Dynkin diagrams.
\begin{itemize}
\item Objects in $\mathbf{Dynkin}$ are called Dynkin diagrams and
are non-directed graphs $\Delta$ with the following properties

\begin{itemize}
\item{The set $V_{\Delta}$ of vertices of $\Delta$ is equal to
$\{1,\dots,n\}$ for some $n\geq 1$.} \item{The number of edges
joining two vertices in $\Delta$ is $0,1,2$ or $3$.}
\item{If vertices $i$ and $j$ are joined by $2$ or $3$ edges, then an arrow is chosen
pointing either from $i$ to $j$, or from $j$ to $i$.}
\item{The quadratic form
$$Q(x_{1},x_{2}, \cdots, x_{n})=2\sum_{i=1}^{n}x_{i}^{2}-
\sum_{i\neq j}\sqrt{n_{ij}}x_{i}x_{j}$$ where $(n_{ij})$ is the
adjacency matrix of $\Delta$, i.e., $n_{ij}$ equal the number of
edges from vertex $i$ to vertex $j$ is positive definite.}
\end{itemize}
\item Morphism in $\mathbf{Dynkin}$ from diagram $\Delta_1$ to
diagram $\Delta_2$ consists of maps $\rho: V_{\Delta_1}\rightarrow
V_{\Delta_2}$ such that $Q_2(x_{\rho(1)},x_{\rho(2)}, \cdots,
x_{\rho(n)})=Q_1(x_{1},x_{2}, \cdots, x_{n}).$
\item{
$S(\mathbf{Dynkin})$ denotes the set of of isomorphisms classes of
connected Dynkin diagrams}

\end{itemize}

\end{enumerate}

\noindent Next theorem gives an explicit characterization of
$S(\mathbf{Dynkin})$.
\begin{thm}$S(\mathbf{Dynkin})$ consists of the Dynkin diagrams
included in the following list
\begin{figure}[!h]
\begin{center}
\includegraphics[width=2in]{dynk.eps}
\caption{Simple Dynkin diagrams. \label{dynkin}}
\end{center}
\end{figure}
\end{thm}
\newpage

\noindent We enunciate the following fundamental

\begin{thm}\label{t1}
\begin{enumerate}
\item{$\mathbf{WeylGroup}\subset S(\mathbf{FinGroup})\subset
S(\mathbf{LieGroup})\subset S(\mathbf{Group})$.}
\item{$S(\mathbf{LieGroup})\cong S(\mathbf{LieAlg}) \cong
S(\mathbf{Root})\cong
S(\mathbf{Dynkin})\rightarrow\mathbf{WeylGroup}$. }
\end{enumerate}
\end{thm}

\noindent Part 1 of Theorem \ref{t1} is obvious. Although we shall not give a
complete proof of part 2 the reader will find in the body of this
paper many statements that shed light into its meaning. The map
$S(\mathbf{Dynkin})\longrightarrow\mathbf{WeylGroup} $ is surjective
but if fails to be injective. Diagrams $B_n$ and $C_n$ of the list
above have both $\mathbb{Z}_{2}^{n}\rtimes S_n$ as associated Weyl
groups.
\section{Lie Algebras}

We proceed to consider in details the category of Lie algebras.
First we recall the notion of a Lie group.

\begin{defi}A group $(G,m)$ is said to be a Lie group if
\begin{enumerate}
\item{$G$ is a finite dimensional smooth manifold.} \item{ the map
$m:G \times G \longrightarrow  G$ given by $m(a,b)=ab$, for all $a,b
\in G$, is smooth.}
\item{ The map $I:G \to G$ given by
$I(a)=a^{-1}$, for all $a \in G$, is smooth.}
\end{enumerate}

\end{defi}

\begin{defi}
A Lie algebra $({\mathfrak g}, [\mbox{ },\mbox{ }])$ over a field
$k$ is a vector space ${\mathfrak g}$ together with a binary
operation $[ \mbox{} \ ,\mbox{}\ ]:{\mathfrak g} \times {\mathfrak
g} \longrightarrow {\mathfrak g}$, called the Lie bracket,
satisfying

\begin{enumerate}
\item{$[ \mbox{} \ ,\mbox{}\ ]$ is a bilinear operation.}
\item{Antisymmetry: $[x,y]=-[y,x]\ \mbox{ for each }\ x,y\in
{\mathfrak g}$.} \item{Jacobi identity: $[x,[y,z]]=[[x,y],z]+
[y,[x,z]]\ \mbox{ for each }\ x,y,z\in {\mathfrak g}$.}
\end{enumerate}

\end{defi}

\begin{exa}
A $k$-algebra $A$ may be regarded as a Lie algebra $(A,[\mbox{
},\mbox{ }])$, with bracket $[x,y]=xy-yx$ for all $x,y \in A$. In
particular ${\rm{End}}(V)$ is a Lie algebra for any $k$-vector space
$V$.
\end{exa}

\begin{exa}
Let $M$ be a smooth manifold. The space
$\Gamma(M)=\{X:M\longrightarrow TM,\ X(m)\in T_m M,\  m\in M\}$ of
vector fields on $M$ is a Lie algebra with the Lie bracket
$$[X,Y]=X^{j}\frac{\partial
Y^{i}}{\partial x^{j}}\frac{\partial}{\partial x^{i}}-
Y^{j}\frac{\partial X^{i}}{\partial x^{j}}\frac{\partial}{\partial
x^{i}},  \ \ \mbox{for all}\ \ X,Y\in\Gamma(M).$$
\end{exa}

\begin{exa}
Let $G$ be a Lie group. The space $T_{e}(G)$ tangent to the identity
$e\in G$ is a Lie algebra since $T_{e}(G)\cong \Gamma(G)^G$ is a Lie
subalgebra of $\Gamma(G)$.

\end{exa}

\begin{defi}
A {\em morphism} of Lie algebras $\rho: {\mathfrak g}
\longrightarrow {\mathfrak h}$ is a linear map $\rho$ from
${\mathfrak g}$ to ${\mathfrak h}$ such that
$\rho([x,y])=[\rho(x),\rho(y)]$ for $x,y \in {\mathfrak g}.$ A
{\em representation} $\rho$ of a Lie algebra $\mathfrak g$ on a
$k$-vector space $V$ is a morphism $\rho: {\mathfrak g} \to
{\rm{End}}(V)$ of Lie algebras.
\end{defi}

\noindent The functor
$$ \begin{array}{ccccc}
  T_e: & {\mathbf{LieGroup}} & \longrightarrow & {\mathbf{LieAlg}}& \\
   & G & \longmapsto & T_{e}(G)& \\
   & \varphi:G\rightarrow H& \longmapsto & d_{e}\varphi:& T_{e}(G)\rightarrow T_{e}(H)\\
\end{array}$$
induces an equivalence between $S(\mathbf{LieGroup})$ and
$S(\mathbf{LieAlg})$.

\begin{defi}
For any Lie algebra $\mathfrak g$ the {\em adjoint} representation
${\rm{ad}}: {\mathfrak g} \to {\rm{End}}({\mathfrak g})$ is given by
${\rm{ad}}(x)(y)=[x,y]$ for all $x,y \in {\mathfrak g}$.
\end{defi}

\begin{defi}

\begin{enumerate}
\item{A subspace $I$ of a Lie algebra ${\mathfrak g}$ is called a
{\em Lie subalgebra} if $[x,y] \in I$ for all $x,y \in I$.}
\item{A subalgebra $I$ of ${\mathfrak g}$ is said to be {\em
abelian} if $[x,y]=0$ for all $x,y \in I$.} \item{A subalgebra $I$
of a Lie algebra ${\mathfrak g}$ is called an {\em ideal} if
$[x,y] \in I$ for all $x \in I$ and $y \in \mathfrak g$.}

\end{enumerate}

\end{defi}

\begin{exa}
For any $k$-algebra the space of derivations of A \\
${\rm{Der}}(A)=\{d:A
\longrightarrow A \ \ |\ \  d(xy)= d(x)y+xd(y) \mbox{ for all }
x,y \in A \}$is a Lie subalgebra of ${\rm{End}}(A)$.
\end{exa}

\begin{defi}
\begin{enumerate}
\item{A Lie algebra ${\mathfrak g}$ is called {\em simple} if it
has no ideals other than ${\mathfrak g}$ and $\{ 0\}$.} \item{A
Lie algebra ${\mathfrak g}$ is called {\em semisimple} if it has
no abelian ideals other than $\{0\}$.} \item{A maximal abelian
subalgebra ${\mathfrak h}$ of ${\mathfrak g}$ is called a {\em
Cartan subalgebra}.}
\end{enumerate}

\end{defi}
\noindent Next theorem is due to Cartan. A proof of it may be
found in \cite{Fu}.
\begin{thm} Let $\mathfrak g$ be a finite dimensional simple
Lie algebra over $\mathbb{C}$, then $\mathfrak g$ is isomorphic to
one of those in the list $\mathfrak{sl}_{n}(\mathbb{C})$,
$\mathfrak{sp}_{2n}(\mathbb{C})$, $\mathfrak{so}_{2n}(\mathbb{C})$,
$\mathfrak{so}_{2n+1}(\mathbb{C})$, $E_6, E_7, E_8, F_4$ and $G_2$.
\end{thm}
\noindent Algebras $\mathfrak{sl}_{n}(\mathbb{C})$,
$\mathfrak{sp}_{2n}(\mathbb{C})$, $\mathfrak{so}_{2n}(\mathbb{C})$
and $\mathfrak{so}_{2n+1}(\mathbb{C})$ are called classical and will
be explained using our graphical notation in Sections \ref{sl},
\ref{sp}, \ref{sop} and \ref{soi}. The algebras $E_6, E_7, E_8, F_4$
and $G_2$ are called exceptional and the reader may find their
definitions in \cite{Fu}.
\begin{defi}
The {\em Killing form} on $\mathfrak g$ is the bilinear map
$\langle \mbox{ }, \mbox{ } \rangle : {\mathfrak g}\times
{\mathfrak g} \longrightarrow  \mathbb{C}$ given for all $x,y \in
{\mathfrak g}$ by $\langle x,y \rangle= {\rm{tr}}({\rm{ad}}(x) \circ {\rm{ad}}(y))$,
where $\circ$ denotes the product in ${\rm{End}}({\mathfrak g})$ and
${\rm{tr}}:{\rm{End}}({\mathfrak g}) \to \mathbb{C}$ is the trace map.
\end{defi}
\noindent Denote by ${\mathfrak h}^{\ast}$ the linear dual of vector
space ${\mathfrak h}$. The following proposition describes
representations of abelian Lie algebras.

\begin{prop}\label{DC}
Let $\mathfrak h$ be an abelian Lie algebra and $\rho: {\mathfrak h}
\to {\rm{End}}(V)$ a representation of $\mathfrak h$. Then $V$
admits a decomposition
\begin{equation}\label{card}
V = \bigoplus_{\alpha \in \Phi} V_{\alpha}\end{equation} where for
each $\alpha \in \mathfrak h^{*}$,  $V_{\alpha}=\{x \in V:
\rho(h)(x)=\alpha(h)x,\ \mbox{ for all }\ \ h\in {\mathfrak h}
\}$, and
 $\Phi=\{ \alpha \in \mathfrak h^{*} |\mbox{  }  \mathfrak
h_{\alpha} \neq 0 \}.$
\end{prop}
\noindent Equation (\ref{card}) is called  {\em Cartan
decomposition} of the representation $\rho$ of $\mathfrak h$.
Proposition \ref{DC} yields a map from $S(\mathbf{LieAlg})$ into
$S(\mathbf{Root})$, which turns out to be a bijection, as follows.
Let ${\mathfrak g}$ be a finite dimensional simple Lie algebra over
$\mathbb{C}$ and $\mathfrak h\subset \mathfrak g$ a Cartan
subalgebra. The killing form $\langle \mbox{ }, \mbox{ }
\rangle:\mathfrak h\times \mathfrak h\longrightarrow\mathbb{C}$
restricted to $\mathfrak h$ is non-degenerated and makes the pair
$(\mathfrak h, \langle \mbox{ } , \mbox{ } \rangle)$ an Euclidean
space. The linear dual $\mathfrak h^{\ast}$ has an induced Euclidean
structure, which we still denote by $\langle \mbox{ } , \mbox{ }
\rangle $ induced by the linear isomorphism $f:\mathfrak h
\longrightarrow \mathfrak h^{\ast}$ given by $f(x)(y)=\langle x,y
\rangle$, for all $x,y\in \mathfrak h$.

\noindent The adjoint representation ${\rm{ad}}:\mathfrak h \longrightarrow
{\rm{End}}(\mathfrak g)$ restricted to $\mathfrak h$ give us a Cartan
decomposition ${\displaystyle \mathfrak g=\bigoplus_{\alpha \in
\Phi}
\mathfrak g_{\alpha}}$ where for each $\alpha\in \mathfrak h$\ \
$\mathfrak g_{\alpha}=\{x\in \mathfrak g|\ \ [h,x]=\alpha(h)x,\
\mbox{ for all}\ \ h\in \mathfrak h \}$ and $\Phi=\{\alpha\in
\mathfrak h^{\ast}: \ \mathfrak h_{\alpha}\neq 0\}$. It is not
difficult to show that the triple $(\mathfrak h^{\ast}, \langle
\mbox{ }, \mbox{ } \rangle,\Phi)$ is a root system.
\begin{defi} Let $\mathfrak g$ be a simple Lie algebra and let
$\Phi$ the root systems associated to $\mathfrak g$. The group $W$
generated by all reflections $S_{\alpha}$ with $\alpha\in\Phi$,
where $S_{\alpha}(\beta)=\beta -2\frac{\langle \alpha , \beta
\rangle} {\langle \alpha , \alpha \rangle}\alpha,$ is known as
the {\em Weyl group} associated to $\mathfrak g$.
\end{defi}
 \noindent One can show that there exists a subset
$\Pi=\{\alpha_1,\alpha_2,\dots,\alpha_n\}$ of $\Phi$ such that
 $\Pi$ is a basis of ${\mathfrak h }^{\ast}$ and
each root $\alpha \in \Phi$  can be written as a linear combination
of roots  in  $\Pi$ with coefficients in $\mathbb{Z}$ which are
either all non-negative or all non-positive. The set $\Pi$ is called
a set of {\em fundamental roots}. The integers
\begin{equation}\label{Cartm}
A_{ij}=2\displaystyle{\frac{\langle \alpha_{i},
\alpha_{j}\rangle}{\langle \alpha_{i}, \alpha_{i}\rangle}}
\end{equation}
are called the {\em Cartan integers} and the matrix $A=(A_{ij})$ is
called the {\em Cartan matrix}. Notice that $A_{ii}=2$, and that for
any $\alpha_{i}, \alpha_{j}\in \Pi$ with $i\neq j$,
$S_{\alpha_{i}}(\alpha_{j})$ is a $\mathbb{Z}$-combination of
$\alpha_{i}$ and $\alpha_{j}$. Since the coefficient of $\alpha_{j}$
is $1$, the coefficient associated to $\alpha_{i}$ in
$S_{\alpha_{i}}(\alpha_{j})$  must be a non-positive integer, i.e.,
$A_{ij}\in \mathbb{Z}^{\leq 0}$. The angle $\theta_{ij}$ between
$\alpha_{i}, \ \alpha_{j}$ is given by the cosine formula
$$\langle \alpha_{i},
\alpha_{j}\rangle=\langle \alpha_{i},\alpha_{i} \rangle
^{\frac{1}{2}} \langle \alpha_{j},
\alpha_{j}\rangle^{\frac{1}{2}}\cos(\theta_{ij}).$$  Then we have
$$4 \cos^{2}(\theta_{ij})=2\frac{\langle\alpha_{i},\alpha_{j}\rangle}
{\langle\alpha_{j},\alpha_{j}\rangle}\cdot
2\frac{\langle\alpha_{j},\alpha_{i}\rangle}{\langle\alpha_{i},\alpha_{i}\rangle},$$
and therefore $4 \cos^{2}(\theta_{ij})=A_{ij}A_{ji}.$ Let
$n_{ij}=A_{ij}A_{ji}$ clearly $n_{ij}\in \mathbb{Z}$ and $n_{ij}\geq
0$. Since $-1\leq \cos(\theta_{ij})\leq 1$ the only possible values
for $n_{ij}$ are $n_{ij}=0,1,2\ \mbox{or}\ 3$.

\begin{defi}\label{DD}
The Dynkin diagram $\Delta$ associated to a simple Lie algebra
${\mathfrak g}$ is the graph $\Delta$ with vertices
$\{1,\dots,n\}$ in bijective correspondence  with the set $\Pi$ of
fundamental roots of ${\mathfrak g}$ such that
\begin{enumerate}
\item{Vertices $i,j$ with $i\neq j$ are joined by $n_{ij}=A_{ij}A_{i}$ edges, where
$A_{ij}$ is given by formula $(\ref{Cartm})$.}
\item{Between each double edge or triple we attach the symbol\  $<$,
 or the symbol\   $>$  \ pointing towards the shorter root with
respect to Killing form.}
\end{enumerate}
\end{defi}
\begin{thm}\label{iso}
Consider the root decomposition of the simple Lie algebra $\mathfrak
g$ with respect to $\mathfrak h$. Let $\alpha\in \Phi$ be a root,
for each nonzero $x_{\alpha}\in \mathfrak g_{\alpha}$ there is
$x_{-\alpha}\in \mathfrak g_{\alpha}$ and $ h_{\alpha}\in \mathfrak
h$ such that $\alpha(h_{\alpha})=2$, $[x_{\alpha},x_{-\alpha}]=
h_{\alpha}$, $[ h_{\alpha},x_{\alpha} ]=2x_{\alpha}$ and $[
h_{\alpha},x_{-\alpha} ]=2x_{-\alpha}$.
\end{thm}

\noindent Theorem \ref{iso} implies that for any root
$\alpha\in\Phi$, $x_{\alpha}, x_{-\alpha}$ and $h_{\alpha}$ span a
subalgebra $\mathfrak s_{\alpha}$ such that $\mathfrak
s_{\alpha}\cong \mathfrak {sl}_{2}(\mathbb{C})$, see Section
\ref{sl2} for a definition of $\mathfrak {sl}_{2}(\mathbb{C})$. This
fact explain the distinguished role played by  $\mathfrak
{sl}_{2}(\mathbb{C})$ in the representation theory of simple Lie
algebras.
\begin{defi}
Let $\Phi\subset\mathfrak{h}$ be the root system associated with a
simple Lie algebra $\mathfrak{g}$.

\begin{enumerate}
\item{For any $\alpha \in \Phi$, the Cartan element $h_{\alpha}\in \mathfrak{h}$  given by
Theorem \ref{iso} is called the  {\em coroot} associated to root
$\alpha$. $\Phi_c=\{h_{\alpha}: \alpha\in \Phi\}$ is the coroot
system associated to ${\mathfrak h}$ and $\Pi_c=\{
h_{\alpha}:\alpha\in\Pi\}$ is the set of fundamental coroots. }

\item{The elements
$w_1,\dots,w_n$ in $\mathfrak{h}^{\ast}$ given by the relations
$w_{i}(h_{j})=\delta_{ij}$, for all $1\leq i,j\leq n$, where $h_{j}$
is the coroot associated to fundamental root $\alpha_{j}$, are
called the fundamental weights.}
\end{enumerate}
\end{defi}

\noindent One can recover a simple Lie algebra  $\mathfrak g$ from its associated Dynkin
diagram $\Delta$ as follows: Let $n_{ij}$ be the adjacency matrix of
$\Delta$. The relation $n_{ij}=A_{ij}A_{ji}$ determines univocally
the Cartan matrix $A_{ij}$. Consider the free Lie algebra generated
by the symbols $H_1,\dots,H_n, X_1,\dots,X_n,Y_1,\dots,Y_n$. Form
the quotient of this free Lie algebra  by the relations
$$
\begin{array}{llc}
  [H_i,H_j]=0\ \ (\mbox{all} \ i,j); & [X_i,Y_i]=H_i \ \ (\mbox{all} \ i);
  & [X_i,X_j]=0 \ \ ( i\neq j); \\
  \mbox{} & \mbox{}&\mbox{}\\

  [H_i,X_j]=A_{ij}X_j\ \ (\mbox{all} \ i,j); & [H_i,Y_j]=-A_{ij}Y_{j}\ \ (\mbox{all} \ i,j); &  \mbox{ }\\
\end{array}
$$

and for all $i\neq j$,
$$\begin{array}{lll}
  [X_i,X_j]=0, & [Y_i,Y_j]=0, & \mbox{if}\ \ A_{ij}=0. \\
  \mbox{} & \mbox{} & \mbox{} \\

  [X_i,[X_i,X_j]]=0, & [Y_i,[Y_i,Y_j]]=0 & \mbox{if}\ \ A_{ij}=-1. \\
  \mbox{} & \mbox{} & \mbox{} \\

  [X_i,[X_i,[X_i,X_j]]]=0, & [Y_i,[Y_i,[Y_i,Y_j]]]=0 & \mbox{if}\ \ A_{ij}=-2. \\
  \mbox{} & \mbox{} & \mbox{} \\

  [X_i,[X_i,[X_i,[X_i,X_j]]]]=0, & [Y_i,[Y_i,[Y_i,[Y_i,Y_j]]]]=0 & \mbox{if}\ \ A_{ij}=-3. \\
\end{array}
$$

\noindent Serre shows that the resulting Lie algebra is a
finite-dimensional simple Lie algebra isomorphic to $\mathfrak g$.
See
\cite{SE} for more details.
\subsection {Jacobian Criterion}

Let $k$ be a field of characteristic zero and let $V$ be a finite
dimensional $k$-vector space. Set $V=\langle e_{1},e_{2},\dots,
e_{n}\rangle$ and $V^{\ast}=\langle x_{1},x_{2}, \dots ,
x_{n}\rangle$ such that $x_{i}(e_{j})=\delta_{ij}$. We have
$$x_{k}\Big(  \sum_{i=1}^{n} a_{i} e_{i} \Big)= a_{k}, \ \ 1\leq k \leq n.$$
Any $f\in V^{\ast}$ is written as  $f= b_{1}x_{1}+b_{2}x_{2}+\dots
+b_{n}x_{n}$. Denote by $S=S(V^{\ast})$ the symmetric algebra of the
dual space $V^{\ast}$ which can be identify with the polynomial ring
$k[x_{1},x_{2},\dots, x_{n}]$. Let $G$ be a finite group which acts
on $V$. $G$ also acts on $V^{\ast}$, and thus it acts on
$S=S(V^{\ast})$ as follows
$$\begin{array}{ccc}
  S(V^{\ast})\times G & \longrightarrow & S(V^{\ast}) \\
  (p,g) & \longmapsto & p(g)
\end{array}$$
where $(p g)(v)=p(gv),\ \ \mbox{for all}\ \ g\in G,
\ p\in S(V^{\ast}),\  v\in V.$ The algebra
$$k[x_{1},x_{2},\dots,x_{n}]^{G}=\{ p\in k[x_{1},x_{2},\dots,x_{n}]: \  p(g)=p,\
 \forall g\in G\}$$
is called the $G$-invariant subalgebra of
$k[x_{1},x_{2},\dots,x_{n}]$.
\begin{defi}
Let $K$ be a field and $F$ a extension of $K$. Let $S$ be a subset
of  $F$. The set $S$ is algebraically dependent over $K$ if for
any positive integer $n$ there is a polynomial non-zero $f\in
k[x_1,\dots,x_n]$ such that $f(s_1,\dots,s_n)=0$ for any different
$s_1,\dots,s_n\in S$. In otherwise $S$ is algebraically
independent.
\end{defi}
\begin{thm}\label{Winv}
Let $\mathbb{C}[x_{1},x_{2},\dots,x_{n}]^{W}$ be the subalgebra of
$\mathbb{C}[x_{1},x_{2},\dots,x_{n}]$ consisting of $W$-invariant
polynomials, then $\mathbb{C}[x_{1},x_{2},\dots,x_{n}]^{W}$ is
generated as an  $\mathbb{C}$-algebra by $n$ homogeneous,
algebraically independent elements of positive degree together
with $1$.
\end{thm}
\noindent The idea of proof of Theorem \ref{Winv} goes as follows:
let $I$ be the ideal of $\mathbb{C}[x_{1},x_{2},\dots,x_{n}]$
generated by all homogeneous $W$-invariant polynomials of positive
degree. Using Hilbert's Basis Theorem we may choose a minimal
generating set $f_{1},f_{2},\dots, f_{r}$ for $I$  consisting of
homogeneous $W$-invariant polynomials of positive degree. One can
show that $r=n$ and furthermore
$\mathbb{C}[x_{1},x_{2},\dots,x_{n}]^{W}=
\mathbb{C}[f_{1},f_{2},\dots,f_{n}]$.

\begin{prop}
Let $f_1,\dots, f_n$ and  $g_1,\dots, g_n$ be two sets of
homogeneous, algebraically independent generators  of
$\mathbb{C}[x_1,\dots,x_n]^{W}$  with degrees $d_i$ and $e_i$
respectively, then (after reordering) $d_i=e_i$ for all
$i=1,\dots,n$.
\end{prop}
\noindent The numbers $d_1,\dots,d_n$ written in increasing order
are called the {\em degrees} of  $W$.  Theorem \ref{CJ} below is a
simple criterion for the algebraic independence of polynomials
$f_1,\dots,f_n$ expressed in terms of the Jacobian determinant. We
write $J(f_1,\dots,f_n)$ for the determinant of the $n\times n$
matrix whose $(i,j)$-entry is $\frac{\partial f_i}{\partial x_j}$.

\begin{thm}[Jacobian criterion]\label{CJ}
The set of polynomials $f_1,\dots,f_n\in k[x_1,\dots,x_n]$ are
algebraically independent over a field $k$ of characteristic zero
if and only if $J(f_1,\dots,f_n)\neq 0$.
\end{thm}

\section{Graph and matrices}

We denote by $\textbf{Digraph}^{1}(n,n)$ the vector space
generated by bipartite directed graphs with a unique edge starting
on the set $[n]$ and ending on the set $[n]$.  We describe
$\textbf{Digraph}^{1}(n,n)$ pictorially as follows
\begin{figure}[ht]
\begin{center}
\includegraphics[width=3in]{marie.eps}
\end{center}
\end{figure}

\noindent where the symbol
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.4in]{simbol.eps}
\end{center}
\end{figure}

\noindent denotes the graph whose unique edge stars at vertex at $i$
and ends at vertex $j$. We define a product on
$\textbf{Digraph}^{1}(n,n)$ as follows
\begin{figure}[ht]
\begin{center}
\includegraphics[width=3.5in]{lady.eps}
\end{center}
\end{figure}

\noindent The trace on $\textbf{Digraph}^{1}(n,n)$ is defined as the
linear functional ${\rm{tr}}: \textbf{Digraph}^{1}(n,n) \longrightarrow
\mathbb{C}$ given by

\begin{figure}[!h]
\begin{center}
\includegraphics[width=3in]{traza.eps}
\end{center}
\end{figure}

\noindent Algebra $\textbf{Digraph}^{1}(n,n)$ is isomorphic to
${\rm{End}}(\mathbb{C}^{n})$ through the application
\begin{figure}[!h]
\begin{center}
\includegraphics[width=2.4in]{baila.eps}
\end{center}
\end{figure}

\noindent We will use this isomorphism to give combinatorial
interpretation of results on ${\rm{End}}(\mathbb{C}^{n})$ that are
traditionally express in the language of matrices.

\section{Linear special algebra ${\mathfrak
{sl}}_{2}(\mathbb{C})$}\label{sl2}

We begin studying the special linear algebra ${\mathfrak
{sl}}_{2}(\mathbb{C})$. It plays a distinguished role in the theory
of Lie algebras. By definition we have
$${\mathfrak {sl}}_2(\mathbb{C})=\{A\in {\rm{End}}(\mathbb{C}^{2}): {\rm{tr}}(A)=0\}.$$
As subspace of $\textbf{Digraph}^{1}(2,2)$, ${\mathfrak
{sl}}_{2}(\mathbb{C})$ is the following vector space
\begin{figure}[!h]
\begin{center}
\includegraphics[width=3.3in]{chillin.eps}
\end{center}
\end{figure}

\noindent We fix as Cartan subalgebra of ${\mathfrak
{sl}_2(\mathbb{C})}$ the $1$-dimensional subspace

\begin{figure}[!h]
\begin{center}
\includegraphics[width=2.7in]{sofine.eps}
\end{center}
\end{figure}

\noindent The dual space is ${\mathfrak h}^{\ast}=\langle
a_1,a_2\rangle / \{a_1+a_2=0\}$, where

\begin{figure}[!h]
\begin{center}
\includegraphics[width=1in]{piaz.eps}
\end{center}
\end{figure}

\subsection{Root system of ${\mathfrak {sl}}_2(\mathbb{C})$}

Consider the projection map $\langle a_1,a_{2}\rangle
\longrightarrow \langle a_1, a_2\rangle /\{a_1+a_2=0\}$. We still
denote by $a_i$ the image of $a_i$ under the projection above. Now
each $h\in{\mathfrak{h}}$ is of the form
\begin{figure}[!h]
\begin{center}
\includegraphics[width=2.5in]{hearme.eps}
\end{center}
\end{figure}
\vspace{0.2cm}

\noindent Let us compute the roots

\begin{figure}[!h]
\begin{center}
\includegraphics[width=2in]{know.eps}
\includegraphics[width=3.5in]{know1.eps}
\includegraphics[width=2in]{know2.eps}
\end{center}
\end{figure}

\noindent Therefore the root system of ${\mathfrak
{sl}}_2(\mathbb{C})$ is $\Phi=\{a_1-a_2,a_2-a_1\}.$ Setting
$\alpha=a_1-a_2$ we have that the roots are $\alpha$ and $-\alpha$
and the set of fundamental roots is $\Pi=\{\alpha\}.$ In pictures
\begin{figure}[ht]
\begin{center}
\includegraphics[width=1.5cm]{rootsl2.eps}
\end{center}
\end{figure}

\subsection{Coroot system of ${\mathfrak {sl}_2(\mathbb{C})}$}

Let $x_{\alpha}$ and $x_{-{\alpha}}$ be the covectors associated
with the roots $\alpha$ and $-\alpha$ of ${\mathfrak
{sl}}_2(\mathbb{C})$ respectively.
\begin{figure}[!h]
\begin{center}
\includegraphics[width=10cm]{covsl2.eps}
\end{center}
\end{figure}

\noindent A vector $h_{\alpha}\in {\mathfrak h}$ is said to be the
coroot associated to the root $\alpha\in {\mathfrak h}^{\ast}$, if
$h_{\alpha}=c[x_{\alpha}, x_{-\alpha}]$, $c\in \mathbb{C}$ and
$\alpha(h_{\alpha})=2$.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=8cm]{corsl2.eps}
\end{center}
\end{figure}
\noindent since
\begin{figure}[!h]
\begin{center}
\includegraphics[width=2in]{tango.eps}
\end{center}
\end{figure}

\subsection{Killing form of ${\mathfrak {sl}}_2(\mathbb{C})$}
\begin{eqnarray*}
\langle x,y\rangle & = & \sum_{\alpha\in \Phi}\alpha(x)\alpha(y)\\
\mbox{ } & = &2(x_1-x_2)(y_1-y_2)\\
\mbox{ } & = & 2x_1y_1+2x_2y_2-2x_1y_2-2x_2y_1\\
\mbox{ } & = &
2(x_1y_1+x_2y_2)-2(x_1+x_2)(y_1+y_2)+2(x_1y_1+x_2y_2)\\
\mbox{ } & = &4{\rm{tr}}(xy).
\end{eqnarray*}

\subsection{Dynkin diagram of ${\mathfrak
{sl}}_2(\mathbb{C})$}

We have only one fundamental root, so the Dynkin diagram is just
$\circ$.

\section{Special linear algebra ${\mathfrak {sl}}_{n}(\mathbb{C})$}\label{sl}

Let us recall the special linear algebra ${\mathfrak
{sl}}_{n}(\mathbb{C})$
$${\mathfrak {sl}}_{n}(\mathbb{C})=\{ A\in {\rm{End}}(\mathbb{C}^{n}):\ {\rm{tr}}(A)=0\}$$
${\mathfrak {sl}}_{n}( \mathbb{C} )$ consider as a subspace of
$\textbf{Digraph}^{1}(n,n)$ is following subspace

\begin{figure}[!h]
\begin{center}
\includegraphics[width=4.3in]{manuel.eps}
\end{center}
\end{figure}

\subsection{Root system of ${\mathfrak {sl}}_{n}(\mathbb{C})$}

We take as Cartan subalgebra the subspace of ${\mathfrak
{sl}}_{n}(\mathbb{C})$
\vspace{-1cm}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=4.5in]{nombre.eps}
\end{center}
\end{figure}

\noindent The dual space is ${\mathfrak h}^{\ast}=\langle
a_1,\dots,a_{n}\rangle/(\sum a_i=0)$, where

\begin{figure}[!h]
\begin{center}
\includegraphics[width=1in]{piaz.eps}
\end{center}
\end{figure}
\noindent Consider the projection $\langle a_1,\dots,a_{n}\rangle
\longrightarrow \langle a_1,\dots,a_{n}\rangle /(\sum a_k=0)$. The
image of $a_i$ under the projection above is still denote by $a_i$.
Then vector $h \in {\mathfrak h}$ can be written as
\vspace{-1cm}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=2.8in]{ropa.eps}
\end{center}
\end{figure}

\noindent Let us compute the root system
\vspace{-0.5cm}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=3in]{car1.eps}
\includegraphics[width=2.5in]{vete.eps}
\end{center}
\end{figure}

\noindent Also
\vspace{-0.5cm}
\begin{figure}[!h]
\begin{center}
\includegraphics[width=3in]{car11.eps}
\includegraphics[width=2.5in]{vete1.eps}
\end{center}
\end{figure}

\noindent Thus the root system of ${\mathfrak{sl}}_{n}(\mathbb{C})$ is
$\Phi=\{a_i-a_j,\ a_j-a_i,\ 1\leq i<j\leq n-1\}\subset {\mathfrak
h}^{\ast}$. The set of fundamental roots is $\Pi=\{a_i-a_{i+1},
\ i=1,\dots, n-1\}$. In pictures for $n=2,3\ \mbox{and}\ 4$ the root
systems look like
\begin{figure}[!h]
\begin{center}
\includegraphics[width=2.8in]{raiz.eps}
\end{center}
\end{figure}
\newpage

\noindent Consider  the linear map
$$T: \textbf{Digraph}^{1}(n,n) \longrightarrow
\textbf{Digraph}^{1}(n,n)$$ sending each directed graph into its
opposite graph. Clearly $T$ is an antimorphism, i.e,
$T(ab)=T(b)T(a)$, for all $a,b\in \textbf{Digraph}^{1}(n,n)$. For
example,

\begin{figure}[ht]
\begin{center}
\includegraphics[width=1.5in]{negasl.eps}
\end{center}
\end{figure}

\noindent Notice that negative roots can be obtain from the positive
ones through an application of $T$.

\subsection{Coroots and weights for ${\mathfrak {sl}}_{n}(\mathbb{C})$}

\begin{enumerate}
\item Coroot associated to the root $a_i-a_j$

\begin{figure}[!h]
\begin{center}
\includegraphics[width=2in]{poder.eps}
\end{center}
\end{figure}

\item Coroot associated to the root $a_j-a_i$

\begin{figure}[!h]
\begin{center}
\includegraphics[width=2.5in]{poder1.eps}
\end{center}
\end{figure}

\end{enumerate}

\noindent The set of fundamental coroots has the form
$\Pi_{c}=\{h_{i}-h_{i+1}, 1\leq i\leq n-1\}$ where

\begin{figure}[!h]
\begin{center}
\includegraphics[width=2in]{jes.eps}
\end{center}
\end{figure}

\noindent The set of fundamental weights is $w_{i}=a_1
+a_2+\cdots+a_i$ since

\begin{figure}[!h]
\begin{center}
\includegraphics[width=3in]{pesosl.eps}
\end{center}
\end{figure}
 \newpage

\subsection{The Killing form of ${\mathfrak {sl}}_{n}(\mathbb{C})$}

Let $x$ and $y$ in ${\mathfrak h}$. Set

\vspace{-0.5cm}
\begin{eqnarray*}
\langle x,y\rangle & = & \sum_{\alpha\in \Phi}\alpha(x)\alpha(y)\\
\mbox{ }& = &\sum_{i<j}(x_i-x_j)(y_i-y_j)+\sum_{i<j}(x_j-x_i)(y_j-y_i)\\
\mbox{ }& = &2\sum_{i<j}(x_i-x_j)(y_i-y_j)\\
\mbox{ }& = &2 \big( \sum_{i<j}x_iy_i+\sum_{i<j}x_jy_j-\sum_{i<j}x_iy_j-\sum_{i<j}x_jy_i \big)\\
\mbox{ }& = &2 \big( \sum (n-i)x_iy_i+ \sum (i-1)x_iy_i +\sum x_iy_i \big)\\
\mbox{ }& = &2n{\rm{tr}}(xy).
\end{eqnarray*}

\subsection{Weyl group of ${\mathfrak {sl}}_{n}(\mathbb{C})$}

Consider the fundamental roots $\alpha_i=a_i-a_{i+1}, \ \
i=1,\dots,n-1$ and $S_{\alpha_{i}}$ the reflection associated to the
fundamental root ${\alpha_{i}}$. Let $h\in{\mathfrak h}$ and
$h_{\alpha_i}$  be the coroot associated to the fundamental root
$\alpha_i$. By definition we have $S_{\alpha_{i}}
(h)=h-\alpha_{i}(h)h_{\alpha_i}$

\begin{figure}[!h]
\begin{center}
\includegraphics[width=5.6in]{migra.eps}
\vspace{0.5cm}
\includegraphics[width=3.6in]{sila.eps}
\end{center}
\end{figure}

\noindent so we see that reflections $S_{\alpha_{i}}$ has the form

\begin{figure}[!h]
\begin{center}
\includegraphics[width=2.3in]{letme.eps}
\end{center}
\end{figure}

\noindent Therefore the Weyl group $A_n$ associated with ${\mathfrak
{sl}}_{n+1}(\mathbb{C})$ is the symmetric group on $n$ letters
$$A_{n}=\langle S_{\alpha_{i}}|\ \ i=1,\dots,n-1\rangle=S_{n}.$$

\subsection{Dynkin diagram of ${\mathfrak {sl}}_{n}(\mathbb{C})$ and Cartan matrix.}

Using equation (\ref{Cartm}) one can checks that the Cartan matrix
associated to the Lie algebra ${\mathfrak {sl}}_{n}(\mathbb{C})$ is

$$A_n= \left( \begin{array}{cccccc}
  2 & \!\!\!\!-1 & 0 & 0 & \ldots& 0 \\
 \!\!\!\! -1 & 2 &\! \!\!\!-1 & 0 &   \ldots& 0\\
    0 & \!\!\!\!-1 & 2 & \!\!\!\!-1 &  \ldots& 0   \\
   &  & & \ddots  &  &    \\
   &  &  &   & \ddots &    \\
  0& 0  & \ldots & 0 &\! \!\!\!-1 & 2
\end{array}\right) $$

\noindent  The Dynkin diagram associated to ${\mathfrak {sl}}_{n}(\mathbb{C})$
is
\begin{figure}[ht]
\begin{center}
\includegraphics[width=2.5in]{An.eps}
\end{center}
\end{figure}
\subsection{Invariant polynomials for ${\mathfrak {sl}}_{n}(\mathbb{C})$}

Consider the action of $S_{n+1}$ on $\mathbb{R}^{n+1}$ given by
$$\begin{array}{ccc}
S_{n+1}\times  {\mathbb{R}}^{n+1} & \longrightarrow &{\mathbb{R}^{n+1}}  \\
(\pi,x)& \longmapsto & (\pi x)_i=x_{{\pi}^{-1}(i)}
\end{array}$$
notice that the permutation $(ij)$ acts as a reflection on
$\mathbb{R}^{n+1}$ since
$$(ij)(x_i-x_j)=x_j-x_i=-(x_i-x_j)$$
$$(ij)(x)=x,\ \ \mbox{si}\  \ x\in(x_i-x_j)^{\bot}\ (\mbox{es decir}\  x_i=x_j )$$
Since $S_{n+1}$ is generated by transpositions $(i\ i+1), \ i=1,
\dots,n $, then $S_{n+1}$ is an example of what is called a
reflection group. Recall that a linear action of a group $G$ on a
vector space $V$ is said to be effective if the only fixed point
is $0$. The action of $S_{n+1}$ on $\mathbb{R}^{n+1}$ fixes points
in $\mathbb{R}^{n+1}$ lying on the straight line $\{(x,x,\dots,x)|
\ \ x\in\mathbb{R}\}$. Thus the action of $A_n$ on
$\mathbb{R}^{n+1}$ fails to be effective. If we instead let $A_n$
act on the hyperplane $V=\{(x_1 \dots x_{n+1}) \in
\mathbb{R}^{n+1}|\  x_1+\dots+x_{n+1}=0\}$ then the action becomes
effective. Consider the power symmetric functions
$$f_i=x_1^{i+1}+\cdots+x_{n+1}^{i+1},\ \ 1\leq i\leq n.$$
Each $f_i$ is $S_{n+1}$-invariant, and together the power
symmetric functions form a set of basic invariants. This fact can
be proven as follows: first notice that

$${\rm{gr}}(f_1) {\rm{gr}}(f_2)\cdots {\rm{gr}}(f_n)=2\cdot3 \cdots n(n+1)=(n+1)!=|S_{n+1}|=|A_n|.$$
Next, it is easy to compute the Jacobian $J(f_1,f_2, \cdots ,f_n)$
yielding the nonvanishing polynomial
$$J(f_1,f_2, \cdots ,f_n)={\displaystyle (n+1)!\prod_{1\leq i<j\leq n}(x_j-x_i)\prod_{i=1}^{n}
(x_1+\cdots+2x_i+\cdots +x_n)}$$ Finally, use the Jacobian
criterion.

\section{Symplectic Lie algebra ${\mathfrak
{sp}_{2n}}(\mathbb{C})$}\label{sp}

Recall that the symplectic Lie algebra $\mathfrak {sp}_{2n}$ is
defined as

$${\mathfrak {sp}}_{2n}(\mathbb{C})=\{X:X^{t}S+SX=0\}.$$ Here  $S\in
M_{2n}(\mathbb{C})$ is the matrix
$$S=\left( \begin{array}{cc}
  0 & I_n \\
  -I_n & 0
\end{array}\right) $$
Equivalently,
$${\mathfrak {sp}}_{2n}(\mathbb{C})=\Bigg \{\left(
\begin{array}{cc}
  A & B \\
  C & -A^{t}
\end{array}\right);\ A,B,C\in M_{n}(\mathbb{C})\  \mbox{y}\   B=B^{t},\  C=C^{t} \Bigg \}$$
${\mathfrak {sp}_{2n}}(\mathbb{C})$ as a subspace of
$\textbf{Digraph}^{1}(2n,2n)$ is given by
\begin{figure}[!h]
\begin{center}
\includegraphics[width=6in]{sp2n.eps}
\end{center}
\end{figure}

\noindent We take as a Cartan subalgebra of ${\mathfrak
{sp}}_{2n}(\mathbb{C})$ the following  subspace

\begin{figure}[!h]
\begin{center}
\includegraphics[width=3in]{carso.eps}
\end{center}
\end{figure}

\subsection{Root system of ${\mathfrak {sp}}_{2n}(\mathbb{C})$}

Consider $h\in {\mathfrak h}$

\begin{figure}[!h]
\begin{center}
\includegraphics[width=2in]{carson.eps}
\end{center}
\end{figure}

\noindent where $ \{a_i\}$ denotes de base of ${\mathfrak h}^{\ast}$
dual to the given base of ${\mathfrak h}$.  Let us define $$T:
\textbf{Digraph}^{1}(2n,2n) \longrightarrow
\textbf{Digraph}^{1}(2n,2n)$$ to be the linear map that sends each
directed graph into its opposite. Clearly $T$ es un antimorphism,
i.e, $T(ab)=T(b)T(a)$ for all $a,b\in \textbf{Digraph}^{1}(2n,2n)$.
For example,
\begin{figure}[ht]
\begin{center}
\includegraphics[width=1.5in]{negativo.eps}
\end{center}
\end{figure}
We will compute explicitly the positive roots. To obtain the
negative roots it is enough to apply the transformation $T$ to
each positive root.

\begin{figure}[!h]
\begin{center}
\includegraphics[width=3.4in]{rson.eps}
\includegraphics[width=1.7in]{rson1.eps}
\includegraphics[width=3.2in]{rspn4.eps}
\includegraphics[width=1.9in]{rspn5.eps}
\includegraphics[width=3.2in]{rspn1.eps}
\end{center}
\end{figure}
\noindent Thus the root system of $ {\mathfrak
{sp}}_{2n}(\mathbb{C})$ is $\Phi=\{ a_i-a_j, a_j-a_i, a_i+a_j,
-a_i-a_j, 2a_i, -2a_i \ \ 1\leq i<j\leq n\}.$ The set of fundamental
roots is $\Pi=\{\alpha_1, \dots,\alpha_n\}$ where
$\alpha_i=a_i-a_{i+1},  i=1,\dots,n-1$ and $\alpha_n=2a_{n}$. In
pictures, the root system of ${\mathfrak {sp}}_{6}(\mathbb{C})$\quad
looks like
\begin{figure}[!h]
\begin{center}
\includegraphics[height=2cm]{tetra.eps}
\end{center}
\end{figure}

\subsection{Coroots and weights of ${\mathfrak {sp}}_{2n}(\mathbb{C})$}

\begin{enumerate}
\item Coroot  asociated to the root $a_i-a_j$
\begin{figure}[!h]
\begin{center}
\includegraphics[width=2.5in]{coson.eps}
\includegraphics[width=2.5in]{coson1.eps}
\end{center}
\end{figure}

$h_i-h_j$ is the coroot associated to the root $a_i-a_j$, since
$(a_i-a_j)(h_i-h_j)=2$.

\item Coroot associated to the root $a_i+a_j$
\newpage
\begin{figure}[!h]
\begin{center}
\includegraphics[width=2.5in]{cspn.eps}
\includegraphics[width=3.2in]{cspn1.eps}
\end{center}
\end{figure}
$h_i+h_j$ is the coroot associated to the la root $a_i+a_j$, since
$(a_i+a_j)(h_i+h_j)=2$.

\item{Coroot associated to the root $2a_i$

\begin{figure}[!h]
\begin{center}
\includegraphics[width=3.2in]{cspn2.eps}
\end{center}
\end{figure}
}
\end{enumerate}

\noindent We conclude that $\Phi_{c}=\{h_i-h_j, h_j-h_i, h_i+h_j, -h_i-h_j,h_i,-h_i,\
1\leq i<j\leq n\}$ is the coroot system of ${\mathfrak
{sp}}_{2n}(\mathbb{C})$. The set of fundamental coroots is given by
$\Pi_{c}=\{h_i-h_{i+1}, h_n ;\ \  1\leq i\leq n-1\}$ where

\begin{figure}[!h]
\begin{center}
\includegraphics[width=3.3in]{correr.eps}
\end{center}
\end{figure}

\noindent and

\begin{figure}[!h]
\begin{center}
\includegraphics[width=1.3in]{correr1.eps}
\end{center}
\end{figure}
\noindent The fundamental weights are
$w_{i}=a_1+a_2+\cdots+a_i$ since

\begin{figure}[!h]
\begin{center}
\includegraphics[width=5in]{pesosp.eps}
\end{center}
\end{figure}

\subsection{Killing form of ${\mathfrak {sp}}_{2n}(\mathbb{C})$}

Let $x$ and $y$ be in ${\mathfrak h}$, we have

\begin{eqnarray*}
\langle x,y\rangle & = & \sum_{\alpha\in \Phi}\alpha(x)\alpha(y)\\
\mbox{ }& = &\sum_{i\neq j}(x_i-x_j)(y_i-y_j)+\sum_{i\neq
j}(x_i+x_j)(y_i+y_j)+
2\sum_{i}(2x_i)(2y_i)\\
\mbox{}&=&4(n+1)\sum x_i y_i\\
\mbox{}&=&4(n+1){\rm{tr}}(xy).
\end{eqnarray*}

\subsection{Weyl group of ${\mathfrak {sp}}_{2n}(\mathbb{C})$}

Consider the fundamental roots of the form $\alpha_i= a_i-a_{i+1}$.
Similarly to the ${\mathfrak {sl}}_{n}(\mathbb{C})$, it is easy to
check that they generate a copy of $S_n$. Let us compute the
reflection associated to the root $\alpha_n=2a_n$. Given $h\in
{\mathfrak h}$, we have that
$S_{\alpha_{n}}(h)=h-\alpha_{n}(h)h_{\alpha_n}$, where
$h_{\alpha_n}$ is the coroot associated to the root $\alpha_n$

\begin{figure}[ht]
\begin{center}
\includegraphics[width=3.7in]{fragile3.eps}
\includegraphics[width=3.7in]{weyl2.eps}
\end{center}
\end{figure}
\noindent This reflections are the sign changes and they generate a
copy of the group $\mathbb{Z}_{2}^{n}$. Altogether the Weyl group
associated to  ${\mathfrak {sp}}_{2n}(\mathbb{C})$ is
$$C_{n}=\mathbb{Z}_{2}^{n}\rtimes S_n.$$

\subsection{ Cartan matrix and Dynkin diagram of ${\mathfrak {sp}}_{2n}(\mathbb{C})$}

$$C_n=\left( \begin{array}{cccccc}
  2 &\!\!\!\! -1 & 0 & 0 & \ldots & 0 \\
  \!\!\!\!-1 & 2 & \!\!\!\!-1 & 0 & \ldots  & 0 \\
  \vdots & \vdots & \ddots  & & \vdots & \vdots  \\
  \vdots & \vdots &  & \ddots &  & \vdots  \\
  0 & 0  & \ldots & \!\!\!\!-1 & 2 & \!\!\!\!-1 \\
  0& 0  & \ldots & 0 & \!\!\!\!-2 & 2
\end{array}\right)$$

\noindent There are $n$ vertices in this case, one for each
fundamental root. The Killing form is $\langle \alpha_i,
\alpha_{i+1}\rangle=1$, if $i=1, \dots , n-1$  and $\langle
\alpha_{n-1}, \alpha_{n}\rangle=2$. Moreover $\langle \alpha_{n-1},
\alpha_{n-1}\rangle<\langle \alpha_{n}, \alpha_{n} \rangle$, and
thus the Dynkin diagram of ${\mathfrak {sp}}_{2n}(\mathbb{C})$ has
the form
\begin{figure}[ht]
\begin{center}
\includegraphics[width=2in]{dcn.eps}
\end{center}
\end{figure}

\subsection{Invariant functions under the action of $C_{n}=\mathbb{Z}_{2}^{n}\rtimes
S_{n}$} Let us recall that the group structure on
$\mathbb{Z}_{2}^{n}\rtimes S_n$ is given by
$$(a_,\pi)(b,\sigma)=(a\cdot\pi(b),\pi\circ\sigma)$$
where $(\pi b)_i=b_{\pi^{-1}(i)}$.

\begin{prop}\label{ACC}
$\mathbb{Z}_{2}^{n}\rtimes S_n$ acts on  $\mathbb{R}^{n}$ as
follows
$$\begin{array}{ccc}
  \mathbb{Z}_{2}^{n}\rtimes S_n \times \mathbb{R}^{n} & \longrightarrow & \mathbb{R}^{n} \\
  ((a,\pi)x) & \longmapsto & ((a,\pi)x)_i=a_ix_{\pi^{-1}(i)}
\end{array}$$
\end{prop}

\noindent Consider the polynomials
$$f_i=x_1^{2i}+x_2^{2i}+\cdots+x_n^{2i}, \ \  1\leq i\leq n$$
Each polynomial $f_i$ is invariant under the action of
$(\mathbb{Z}_{2})^{n}\rtimes S_n$ given by
$$(f(a,\pi))(x)=f((a,\pi)x).$$
The set of invariants
$$\begin{array}{ccc}
  f_1 & = & x_1^2 + x_2^2+ \cdots + x_n^2\\
  f_2 & = &  x_1^4 + x_2^4+ \cdots + x_n^4 \\
  \vdots & \mbox{ } & \mbox{ } \\
  f_n & = &  x_1^{2n}+x_2^{2n}+\cdots +x_n^{2n}
\end{array}$$
is a  basic set. This follows from the Jacobian criterion since
$${\rm{gr}}(f_1){\rm{gr}}(f_2)\cdots {\rm{gr}}(f_n)=2\cdot4\cdot6\cdots 2n= 2^n n!=|\mathbb{Z}_{2}^{n}\rtimes S_n|$$
and
$$J=2^n n!\   x_1\cdots x_n \prod_{1\leq i<j\leq n}(x_j^2-x_i^2)\neq 0.$$

\section{Orthogonal Lie Algebra ${\mathfrak
{so}}_{2n}(\mathbb{C})$}\label{sop}

Recall that the $2n$-orthogonal Lie Algebra is defined as follows
$${\mathfrak {so}}_{2n}(\mathbb{C})=\{X:X^{t}S+SX=0\}$$ where $S\in M_{2n}(\mathbb{C})$
is the matrix
$$S=\left(\begin{array}{cc}
  0 & I_n \\
  I_n & 0
\end{array} \right)$$
Eexplicitly
$${\mathfrak {so}}_{2n}(\mathbb{C})=\Bigg \{\left(
\begin{array}{cc}
  A & B \\
  C & -A^{t}
\end{array}\right);\  A,B,C\in M_{n}(\mathbb{C})\ \mbox{y}\  B=-B^{t},\  C=-C^{t} \Bigg \}.$$
${\mathfrak {so}}_{2n}(\mathbb{C})$ as a subspace of
$\textbf{Digraph}^{1}(2n,2n)$ is given by

\begin{figure}[!h]
\begin{center}
\includegraphics[width=6in]{so.eps}
\end{center}
\end{figure}

\noindent where $1\leq i<j\leq n$. We fix ${\mathfrak h}$ the Cartan
subalgebra of ${\mathfrak {so}}_{2n}(\mathbb{C})$ to be

\begin{figure}[!h]
\begin{center}
\includegraphics[width=3in]{carso.eps}
\end{center}
\end{figure}

\newpage

\subsection{Root System of ${\mathfrak {so}}_{2n}(\mathbb{C})$}

Let $h\in {\mathfrak h}$

\begin{figure}[!h]
\begin{center}
\includegraphics[width=1.8in]{cason.eps}
\end{center}
\end{figure}

\noindent where $\{a_i\}$ is the base of ${\mathfrak h}^{\ast}$ dual
to the natural base of ${\mathfrak h}$. As for the case of the
symplectic algebra we define a map $T: \textbf{Digraph}^{1}(2n,2n)
\longrightarrow \textbf{Digraph}^{1}(2n,2n)$. $T$ sends a given
graph to its opposite if it does not cross the vertical line, and to
minus its opposite if it crosses the vertical line. We have again
that $T(ab)=T(b)T(a)$ For example,

\begin{figure}[!h]
\begin{center}
\includegraphics[width=1.5in]{nega1.eps}
\end{center}
\end{figure}

\noindent Let us find out the positive roots

\begin{figure}[!h]
\begin{center}
\includegraphics[width=3in]{rson.eps}
\includegraphics[width=1.7in]{rson1.eps}
\includegraphics[width=3in]{rson4.eps}
\includegraphics[width=1.7in]{rson5.eps}
\end{center}
\end{figure}

\noindent To get the negative roots it is enough to apply $T$ to the
positive roots. Therefore the root system is $\Phi=\{ a_i-a_j,
a_j-a_i, a_i+a_j, -a_i-a_j, \ \ 1\leq i<j\leq n\}$ and the
fundamental roots can be taken to be $\Pi=\{\alpha_1,
\dots,\alpha_n\}$ where $\alpha_i=a_i-a_{i+1}, \ \ i=1,\dots,n-1$\ \
and\ \ $\alpha_n=a_{n-1}+a_{n}$

\subsection{Coroots and weights of ${\mathfrak {so}}_{2n}(\mathbb{C})$}

\begin{enumerate}
\item{Coroot associated to the root $a_i-a_j$
\begin{figure}[!h]
\begin{center}
\includegraphics[width=2.5in]{coson.eps}
\includegraphics[width=3.2in]{coson1.eps}
\end{center}
\end{figure}

in this case $(a_i-a_j)(h_i-h_j)=2$, thus $h_i-h_j$ is the coroot
associated to the root $a_i-a_j$.}

\item{Coroot associated to the root $a_i+a_j$
\newpage
\begin{figure}[ht]
\begin{center}
\includegraphics[width=2.5in]{COSPN.eps}
\includegraphics[width=3.2in]{COSPN1.eps}
\end{center}
\end{figure}
since $(a_i+a_j)(-h_i-h_j)=-2$, $h_i+h_j$ is the coroot associated
to the root $a_i+a_j$.}
\end{enumerate}
We concluded that  $\Phi_{c}=\{h_i-h_j, h_j-h_i, h_i+h_j,
-h_i-h_j, 1\leq i<j\leq n\}$ is the coroot system of ${\mathfrak
{so}}_{2n}(\mathbb{C})$. The set of fundamental coroots is given
by $\Pi_{c}=\{h_i-h_{i+1}, h_{n-1}+h_{n};\ \ 1\leq i\leq n-1\}$
where

\begin{figure}[!h]
\begin{center}
\includegraphics[width=3.2in]{correr.eps}
\end{center}
\end{figure}

\noindent and

\begin{figure}[!h]
\begin{center}
\includegraphics[width=3.4in]{correr2.eps}
\end{center}
\end{figure}

\noindent The fundamental weights are given by $w_i=a_1+\cdots+a_i,
\ \ i=1,\cdots,n-1$ and $
w_{n}=\frac{a_1+a_2+\cdots+a_n}{2}$ In a similar as for $\mathfrak
{sp}_{2n}(\mathbb{C})$ one can prove that
$w_{i}(h_{j}-h_{j+1})=\delta_{ij}$. For $w_{n}$ we get

\begin{figure}[!h]
\begin{center}
\includegraphics[width=4in]{pesoso.eps}
\end{center}
\end{figure}

\subsection{Killing form of ${\mathfrak {so}}_{2n}(\mathbb{C})$}
Let $x$ and $y$ be in ${\mathfrak h}$

\begin{eqnarray*}
\langle x,y\rangle & = & \sum_{\alpha\in \Phi}\alpha(x)\alpha(y)\\
\mbox{ }& = &\sum_{i\leq j}(x_i-x_j)(y_i-y_j)+\sum_{i\leq
j}(x_i+x_j)(y_i+y_j)+
\sum_{j\leq i}(x_i-x_j)(y_i-y_j)\\
\mbox{} &\mbox{}&+\sum_{j\leq i}(x_i+x_j)(y_i+y_j)\\
\mbox{}&=&\sum_{i\neq j}2x_iy_j+2x_iy_j\\
\mbox{}&=&4(n-1)\sum x_i y_i\\
\mbox{}&=&4(n-1){\rm{tr}}(xy).
\end{eqnarray*}

\subsection{Weyl group of ${\mathfrak {so}}_{2n}(\mathbb{C})$}

Consider the fundamental roots $\alpha_i= a_i-a_{i+1}$. Just as for
${\mathfrak {sl}}_{n}(\mathbb{C})$, the associated reflections
associated to these roots generate the group $S_n$. We compute the
reflections associated to the roots $\alpha_n=a_{n+1}+a_n$. Given
$h\in {\mathfrak h}$, we have
$S_{\alpha_{n}}(h)=h-\alpha_{n}(h)h_{\alpha_n}$ where $h_{\alpha_n}$
is the coroot associated to the root $\alpha_n$

\begin{figure}[!h]
\begin{center}
\includegraphics[width=5.5in]{fragile.eps}
\includegraphics[width=5in]{weyl.eps}
\end{center}
\end{figure}

\noindent This reflection correspond to a change of sign. Thus we
have that the Weyl group associated with ${\mathfrak
{so}}_{2n}(\mathbb{C})$ is
$$D_{n}=\mathbb{Z}_{2}^{n-1}\rtimes S_n$$

\subsection{Cartan matrix and Dynkin diagram of ${\mathfrak {so}}_{2n}(\mathbb{C})$.}

$$D_n=\left( \begin{array}{cccccc}
  2 &\!\!\!\! -1 & 0 & 0 & \ldots&  0 \\
  \!\!\!\!-1 & 2 & \!\!\!\!-1 & 0 & \ldots  & 0 \\
  \vdots & \vdots &\ddots   & & & \vdots  \\
  0 & 0  & \ldots & 2 & \!\!\!\!-1 & \!\!\!\!-1 \\
  0 & 0  & \ldots & \!\!\!\!-1 & 2 & 0 \\
  0& 0  & \ldots & \!\!\!\!-1 & 0 & 2
\end{array}\right)$$

\noindent The Dynkin diagram has $n$ vertices  corresponding with
the fundamental roots. The Killing has the form $\langle \alpha_i,
\alpha_{i+1}\rangle=1$, if $i=1, \dots , n-2$, $\langle \alpha_{n-2}, \alpha_{n-1}\rangle=1$,
$\langle\alpha_{n-2}, \alpha_{n}\rangle=1$ and $\langle
\alpha_{n-1}, \alpha_{n}\rangle=0$.
Thus the Dynkin diagram ${\mathfrak {so}}_{2n}(\mathbb{C})$

\begin{figure}[ht]
\begin{center}
\includegraphics[width=2in]{dn.eps}
\end{center}
\end{figure}

\subsection{Invariant functions under the action of $\mathbb{Z}_{2}^{n-1}\rtimes S_n$}

Consider the polynomials
$$\begin{array}{cccc}
  f_i & = & \displaystyle {\sum_{j=1}^{n}x_j^{2i}}, & 1\leq i \leq n-1 \\
  f_n & = & x_1\cdots x_n & \mbox{ }
\end{array}$$
clearly each $f_i$ is invariant under the action of
$\mathbb{Z}_{2}^{n-1}\rtimes S_n$. It is easy to check that
$${\rm{gr}}(f_1){\rm{gr}}(f_2)\dots {\rm{gr}}(f_n)= 2^{n-1} n!=|\mathbb{Z}_{2}^{n-1}\rtimes S_n|$$
and
$$J=(-2)^{n-1}(n-1)! \displaystyle {\prod_{1\leq i<j\leq n}(x_j^2-x_i^2)} \neq 0,$$
so the Jacobian criterion  tell us that ${f_1,\dots, f_n}$ is a
basic set of invariants.

\section{Orthogonal algebra ${\mathfrak
{so}}_{2n+1}(\mathbb{C})$}\label{soi}

We orthogonal odd algebra
$${\mathfrak {so}}_{2n+1}(\mathbb{C})=\{X:X^{t}S+SX=0\}$$ where $S\in
M_{2n+1}(\mathbb{C})$ is of the form
$$S=\left(\begin{array}{ccc}
  0 & I_n & 0\\
  I_n & 0 & 0\\
  0 & 0 & 1
\end{array} \right)$$
In an explicit form
$${\mathfrak {so}}_{2n+1}(\mathbb{C})=\Bigg\{ \left(
\begin{array}{ccc}
  A & B &-H^{t}\\
  C & -A^{t}& -G^{t}\\
  G & H & 0
\end{array}\right);\   A,B,C\in M_{n}(\mathbb{C}), H,G\in M_{1\times n}(\mathbb{C}), 0\in\mathbb{C}
\ \ \mbox{y} \  B=-B^{t},\  C=-C^{t} \Bigg\}$$ ${\mathfrak
{so}}_{2n+1}(\mathbb{C})$ as a subspace of
$\textbf{Digraph}^{1}(2n+1,2n+1)$ is given by

\begin{figure}[!h]
\begin{center}
\includegraphics[width=6in]{so2n.eps}
\end{center}
\end{figure}

\noindent Let us fix ${\mathfrak h}$ a Cartan subalgebra of
${\mathfrak {so}}_{2n+1}(\mathbb{C})$
\begin{figure}[ht]
\begin{center}
\includegraphics[width=2.5in]{cartan.eps}
\end{center}
\end{figure}

\subsection{Root system of ${\mathfrak {so}}_{2n+1}(\mathbb{C})$}

Let $h\in {\mathfrak h}$,

\vspace{-0.5cm}

\begin{figure}[ht]
\begin{center}
\includegraphics[width=1.7in]{cartan1.eps}
\end{center}
\end{figure}

\noindent where $\{a_i\}$ is a base of ${\mathfrak h}^{\ast}$ dual to the
natural base of ${\mathfrak h}$. We compute the positive roots. The
negative roots are obtain applying the following antimorphism to the
positive roots.\\
$T: \textbf{Digraph}^{1}(2n+1,2n+1)
\longrightarrow \textbf{Digraph}^{1}(2n+1,2n+1)$. For example

\begin{figure}[!h]
\begin{center}
\includegraphics[width=1.5in]{nega1.eps}
\end{center}
\end{figure}

\begin{figure}[!h]
\begin{center}
\includegraphics[width=3.2in]{cartan2.eps}
\includegraphics[width=1.8in]{cartan3.eps}
\includegraphics[width=3.2in]{cartan6.eps}
\includegraphics[width=1.8in]{cartan7.eps}
\includegraphics[width=3.2in]{cartan10.eps}
\includegraphics[width=1.5in]{cartan11.eps}
\end{center}
\end{figure}

\noindent Thus the set of roots is $\Phi=\{a_i-a_j,
a_j-a_i,a_i+a_j,a_i,-a_i\}.$ The fundamental roots are
$\Pi=\{\alpha_1,\dots,\alpha_n\}$ where $\alpha_i=a_i-a_{i+1}, \ \
i=1,\dots,n-1$\ \ and\ \ $\alpha_n=a_{n}$. In pictures the root
system of ${\mathfrak {so}}_{7}(\mathbb{C})$ looks like
\begin{figure}[!h]
\begin{center}
\includegraphics[height=2cm]{cubo1.eps}
\end{center}
\end{figure}

\subsection{Coroots and weights of ${\mathfrak {so}}_{2n+1}(\mathbb{C})$}

\begin{enumerate}
\item{Coroots associated to the root $a_i-a_j$

\begin{figure}[!h]
\begin{center}
\includegraphics[width=2.5in]{luz1.eps}
\includegraphics[width=3in]{coson1.eps}
\end{center}
\end{figure}

Since $(a_i-a_j)(h_i-h_j)=2$, we see that  $h_i-h_j$ is the coroot
associated to the root $a_i-a_j$.}

\item{Coroot associated to the root $a_i+a_j$

\begin{figure}[!h]
\begin{center}
\includegraphics[width=2.5in]{luz2.eps}
\includegraphics[width=3.2in]{luz3.eps}
\end{center}
\end{figure}

Here $(a_i+a_j)(-h_i-h_j)=-2$, and thus $h_i+h_j$ is the coroot
associated to the root $a_i+a_j$.}

\item{Coroot associated to the root $a_i$

\begin{figure}[!h]
\begin{center}
\includegraphics[width=2.5in]{luz4.eps}
\includegraphics[width=2 in]{luz5.eps}
\end{center}
\end{figure}

Thus $2h_i$ is the coroot associated to the root $a_i$.}
\end{enumerate}
We have that  $\Phi_{c}=\{h_i-h_j, h_j-h_i, h_i+h_j, -h_i-h_j,
2h_i,-2h_i, \ 1\leq i<j\leq n \}$ is the coroot system of
${\mathfrak {so}}_{2n+1}(\mathbb{C})$. The set of fundamental coroot
has the form $\Pi_{c}=\{h_{i}-h_{i-1}, 2h_{n};\ \ 1\leq i\leq n-1\}$
where

\begin{figure}[!h]
\begin{center}
\includegraphics[width=3.4in]{correrlo.eps}
\end{center}
\end{figure}
\noindent and

\begin{figure}[!h]
\begin{center}
\includegraphics[width=3.4in]{correr2.eps}
\end{center}
\end{figure}

\noindent The fundamental weights are $w_{i}=a_1+\cdots+a_{i},\ \
i=1,\cdots,n-1$ and ${\displaystyle
w_{n}=\frac{a_1+\cdots+a_n}{2}.}$ In a similar fashion to the
${\mathfrak {so}}_{2n}(\mathbb{C})$ case we have that
$w_{i}(h_j-h_{j+1})=\delta_{ij}$ and $w_{i}(2h_n)=\delta_{in}$.

\subsection{Killing form of ${\mathfrak {so}}_{2n+1}(\mathbb{C})$}
Given $x$ and $y$ in ${\mathfrak h}$

\begin{eqnarray*}
\langle x,y\rangle & = & \sum_{\alpha\in \Phi}\alpha(x)\alpha(y)\\
\mbox{ }& = &\sum_{i\leq j}(x_i-x_j)(y_i-y_j)+\sum_{i\leq
j}(x_i+x_j)(y_i+y_j)+
\sum_{j\leq i}(x_i-x_j)(y_i-y_j)\\
\mbox{} &\mbox{}&+\sum_{j\leq i}(x_i+x_j)(y_i+y_j)+\sum_{i}x_i y_i+\sum_{ i}(2x_i)(2y_i)\\
\mbox{}&=&(4n-2)\sum x_i y_i\\
\mbox{}&=&(4n-2){\rm{tr}}(xy)
\end{eqnarray*}

\subsection{Weyl group of ${\mathfrak {so}}_{2n+1}(\mathbb{C})$}

Consider the fundamental roots $\alpha_i= a_i-a_{i+1}$. Just like
for ${\mathfrak {sl}}_{n+1}(\mathbb{C})$, the reflections associated
to these roots generate the symmetric group $S_n$. Let us analyze
the reflection associated to the root $\alpha_n=a_n$. Let $h\in
{\mathfrak h}$, we have
$S_{\alpha_{n}}(h)=h-\alpha_{n}(h)h_{\alpha_n}$ where $h_{\alpha_n}$
is the coroot associated to the root $\alpha_n$

\begin{figure}[!h]
\begin{center}
\includegraphics[width=4in]{fragile1.eps}
\includegraphics[width=3.3in]{weyl1.eps}
\end{center}
\end{figure}

\noindent This reflections represent sign changes and generate the
group $\mathbb{Z}_{2}^{n}$, therefore the Weyl group associated with
${\mathfrak {so}}_{2n+1}(\mathbb{C})$ is
$$B_n=\mathbb{Z}_{2}^{n}\rtimes S_n$$

\subsection{Cartan matrix and Dynkin diagram of ${\mathfrak {so}}_{2n+1}(\mathbb{C})$.}

$$B_n=\left( \begin{array}{cccccc}
  2 &\!\!\!\! -1 & 0 & 0 & \ldots &  0 \\
  \!\!\!\!-1 & 2 &\!\!\!\! -1 & 0 & \ldots &  0 \\
  \vdots & \vdots & \ddots &  &  & \vdots  \\
  \vdots & \vdots &  &\ddots  &  & \vdots  \\
  0 & 0 & \ldots &\!\!\!\! -1 & 2 & \!\!\!\!-2 \\
   0 & 0 & \ldots & 0 & \!\!\!\!-1 & 2
\end{array} \right)$$
The diagram has $n$ vertices, one for each fundamental root. The
killing form is given by $\langle \alpha_i, \alpha_{i+1}\rangle=1$,
if $i=1, \dots , n-1$ and $\langle \alpha_{n-1},
\alpha_{n}\rangle=2$. Furthermore $\langle \alpha_{n-1},
\alpha_{n-1}\rangle>\langle \alpha_{n}, \alpha_{n} \rangle$, and
thus, the Dynkin diagram of ${\mathfrak {so}}_{2n+1}(\mathbb{C})$
has form
\begin{figure}[ht]
\begin{center}
\includegraphics[width=2in]{dbn.eps}
\end{center}
\end{figure}

\subsection*{Acknowledgment}
Thanks to Manuel Maia for helping us with Latex.

redpajama_set_name: RedPajamaArXiv


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Q: Please add an option to search for posts < X votes in search I'm sometimes looking for very low scored answers/question in order to  find some VERY LOW QUALITY ones.
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If I fill "votes:-15" into internal search box, and I sort it by votes, the latest available post has 60 upvotes.
Can "search for posts with less than X score" be implemented, as opposite for already implemented special phrase votes: or at least deleted limit of pages ?

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