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| Title of object: |
root system |
| Canonical Name: |
RootSystem |
| Type: |
Definition |
| Created on: |
2002-12-04 01:40:28 |
| Modified on: |
2004-09-05 05:24:28 |
| Classification: |
msc:17B20 |
| Defines: |
reduced root system, root, root space, root decomposition, Cartan subalgebra |
Revision comment (for changes between this and next version):
| Changes for correction #5958 ('Formula for orthogonal reflection'). |
Preamble:
\usepackage{amsmath}
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\newcommand{\cnums}{\mathbb{C}}
\newcommand{\znums}{\mathbb{Z}}
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\newtheorem{theorem}[proposition]{Theorem} |
Content:
\paragraph{Summary.} A root system is an key notion which is needed
for the classification and the representation theory of reflection groups
and of semi-simple Lie algebras.
\paragraph{Definitions.}
Let $E$ be a Euclidean vector space with inner product
$(\cdot,\cdot)$. A root system is a finite spanning set $R\subset E$
such that for every $u\in R$, the orthogonal reflection
$$v\mapsto v-2\frac{(u,v)}{(v,v)},\quad v\in E$$
preserves $R$.
A root system is called \emph{crystallographic} if
$2\frac{(u,v)}{(u,u)}$ is an integer for all $u,v\in R$.
A root system is called {\em reduced} if for all $u\in R$, we have
$ku\in R$ for $k=\pm 1$ only.
We call a root system {\em indecomposable} if there is no proper
decomposition $R=R'\cup R''$ such that every vector in $R'$ is orthogonal to
every vector in $R''$.
\paragraph{Classification.}
Root systems arise in the classification of semi-simple Lie algebras
in the following manner: If $\fr g$ is a semi-simple complex Lie
algebra, then one can choose a maximal self-normalizing subalgebra of
$\fr g$ (alternatively, this is the commutant of an element with
commutant of minimal dimension), called a Cartan subalgebra,
traditionally denote $\fr h$. These act on $\fr g$ by the adjoint
action by diagonalizable linear maps. Since these maps all commute,
they are all simultaneously diagonalizable. The simultaneous
eigenspaces of this action are called {\em root spaces}, and the
decomposition of $\fr g$ into $\fr h$ and the root spaces is called a
{\em root decompositon} of $\fr g$. It turns out that all root spaces
are all one dimensional. Now, for each eigenspace, we have a map
$\lambda:\fr h\to\mathbb{C}$, given by $Hv=\lambda(H)v$ for $v$ an
element of that eigenspace. The set $R\subset\fr h^*$ of these
$\lambda$ is called the {\em root system} of the algebra $\fr g$. The
Cartan subalgebra $\fr h$ has a natural inner product (the Killing
form), which in turn induces an inner product on $\fr h^*$. With
respect to this inner product, the root system $R$ is an abstract root
system, in the sense defined up above.
Conversely, given any reduced, crystallographic root system $R$, there is a unique
semi-simple complex Lie algebra $\fr g$ such that $R$ is its root system.
Thus to classify complex semi-simple Lie algebras, we need only classify
indecomposable roots systems, a somewhat easier task. We only
need to classify indecomposable root systems, since all other root
systems are built out of these. The Lie algebra corresponding to a
root system is semi-simple if and only if the associated root system is
indecomposable.
By convention $e_1,\ldots,e_n$ are orthonormal vectors, and the
subscript on the name of the root system is the dimension of the space
it is contained in, also called the {\em rank} of the system, and the
indices $i$ and $j$ will run from $1$ to $n$. There are four infinite
series of indecomposable root systems :
\begin{itemize}
\item $A_n=\{e_i-e_j,\delta+e_i\}_{i\neq j}$, where
$\delta=\sum_{k=1}^ne_k$. This system corresponds to $\fr{sl}_2\cnums$.
\item $B_n=\{\pm e_i\pm e_j\}_{i<j}\cup\{e_i\}$. This system
corresponds to $\fr{so}_{2n+1}\cnums$.
\item $C_n=\{\pm e_i\pm e_j\}_{i<j}\cup\{2e_i\}$. This system
corresponds to $\fr{sp}_{2n}\cnums$.
\item $D_n=\{\pm e_i\pm e_j\}_{i<j}$. This sytem corresponds to
$\fr{so}_{2n}\cnums$.
\end{itemize}
There are also five exceptional root systems $G_2,F_4,E_6,E_7,E_8$,
with five corresponding exceptional algebras, generally denoted by the
same letter in lower-case Fraktur ($\fr g_2$, etc.). |
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