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\begin{document}
\title{Cartan matrices, Dynkin diagrams and classification}
\author{Your name here}
\date{\today}
\maketitle
%\thispagestyle{empty}
{\Large 18.099 - 18.06 CI.}
{Due on Monday, May 10 in class.}
\vspace{1cm}
{\it Write a paper proving the statements and working through the examples
formulated below. Add your own examples, asides and discussions
whenever necessary. }
Let $V$ be a Euclidean space, that is
a finite dimensional real linear space with a symmetric
positive definite inner product $\la, \ra$.
Recall that for a root system $\Delta$ in
$V$, there exists a simple root system
$\Pi \subset \Delta$ (not unique), which is a basis in $V$,
and a positive root system $\Delta^+$
such that any positive root is a linear combination of simple roots with
nonnegative integer coefficients.
Below we will assume that the root system $\Delta$ is reduced (that is,
for any $\alpha \in \Delta$, $2\alpha \notin \Delta$).
For a given pair $(\Pi, \Delta)$ with a fixed enumeration of simple roots
$\{ \alpha_1, \alpha_2, \ldots, \alpha_l \}$, where $l = \dim(V)$,
we defined in \cite{5}
the Cartan matrix $A$ by setting
$$ A_{ij} = \frac{2\langle \alpha_i, \alpha_j \rangle}{\langle \alpha_i,
\alpha_i \rangle}.$$
An abstract Cartan matrix is an $l\times l$ matrix with the following
properties:
\begin{enumerate}
\item{all entries $A_{ij}$ are integers;}
\item{$A_{ii}=2$ for all $i$;}
\item{$A_{ij} \leq 0$ for all $i \neq j$;}
\item{$A_{ij}=0$ if and only if $A_{ji}=0$;}
\item{there exists a diagonal matrix $D$ with positive entries such
that $DAD^{-1}$ is symmetric positive definite.}
\end{enumerate}
A Cartan matrix is irreducible if it is not isomorphic (conjugate by
a product of permutation matrices)
to a block diagonal matrix with more than one block.
We know that a Cartan matrix determines the simple root system uniquely
up to isomorphism (\cite{5}).
Recall that two root systems $\Delta$ and $\Delta'$ in
$V$ are isomorphic if there exists a linear automorphism of $V$ that
maps $\Delta$ onto $\Delta'$ preserving the numbers $n(\alpha, \beta)$
for all $\alpha, \beta \in \Delta$.
The goal of this paper is to show that any abstract Cartan matrix
determines a unique reduced root system up to isomorphism.
This provides a tool for classification of abstract root systems.
The proof requires several steps.
\begin{theorem} \label{W}
Let $\Pi = \{\alpha_1, \alpha_2, \ldots , \alpha_l \}
\subset \Delta$ be a set of simple roots in a reduced root system
$\Delta$. For any root $\alpha \in \Delta$, there exists an element
$\alpha_j \in \Pi$ such that $\alpha = w(\alpha_j)$, where
$w$ is a composition of reflections with respect to simple roots in $\Pi$.
\end{theorem}
Hint: First assume that $\alpha = \sum_{i=1}^l n_i \alpha_i$
is a positive root and proceed by induction in the number $\sum_{i=1}^l n_i$
(the level of a root).
Use Theorem 5 in \cite{4} to find an
element $\alpha_k \in \Pi$ such that the reflection with respect to
$\alpha_k$ maps $\alpha$ to a positive root with a smaller
level. This is the induction step.
Then extend the result for the negative roots, using that
the reflection with respect to $\alpha_i \in \Pi$ maps $\alpha_i$ to
$-\alpha_i$.
\begin{theorem} The set $\{\alpha_1, \alpha_2, \ldots, \alpha_l \}$
of simple roots determines the set of all roots in a reduced root system.
\end{theorem}
Hint: use Theorem \ref{W}.
The last Theorem together with Theorem 11 in \cite{5} shows that
an abstract Cartan matrix corresponds to at most one
reduced root system, up to isomorphism.
Using the properties of abstract Cartan matrices,
it is possible to further show that \emph{every}
abstract Cartan matrix determines a
reduced abstract root system. Consequently, it is possible to classify
all reduced root systems based on the properties of Cartan matrices.
Here is an example in dimension $3$.
\begin{example} Find all $3\times3$ abstract Cartan matrices up to
isomorphism and
construct the corresponding root systems. Using examples in
\cite{2,3,4,5}, identify the type of a root system whenever possible.
\end{example}
Hint: Start with the block diagonal Cartan matrices with more than
one block. The properties (1)-(4) of an abstract Cartan matrix
together with the conditions on the numbers
$n(\alpha, \beta)= \frac{2\langle \beta, \alpha \rangle}{\langle
\alpha, \alpha \rangle}$ for
simple roots, discussed at the end of \cite{2}, provide
sufficient information to classify all such Cartan
matrices. For the irreducible $3\times 3$ Cartan matrices, use that
all upper left determinants of a positive definite matrix are positive.
A similar argument in higher dimensions eventually leads
to a complete classification
of the abstract (reduced) root systems in Euclidean spaces.
{\it Continue this paper with a review of literature (without proofs) on
abstract root systems, in particular define the Dynkin diagram of a root
system and formulate the complete classification of abstract irreducible
root systems. Suggested sources: \cite{6}, \S 5.12-5.15,
\cite{7}, \S 5.8-5.9, \cite{8}, \S 4.5. Any one of these sourses will be
sufficient.}
\begin{thebibliography}{2}
\bibitem[2]{2} Your classmate, {\it Abstract root systems},
preprint, MIT, 2004.
\bibitem[3]{3} Your classmate, {\it Simple and positive roots},
preprint, MIT, 2004.
\bibitem[4]{4} Your classmate, {\it Properties of simple roots},
preprint, MIT, 2004.
\bibitem[5]{5} Your classmate, {\it Cartan matrix of a root system},
preprint, MIT, 2004.
\bibitem[6]{6} J.-P. Serre, {\it Complex semisimple Lie algebras},
New York, Springer-Verlag, 1987
\bibitem[7]{7} W.A. De Graaf, {\it Lie algebras: theory and algorithms},
Amsterdam, New York, Elsevier, 2000
\bibitem[8]{8} N. Jacobson, {\it Lie algebras}, New York, Dover, 1979
\end{thebibliography}
\end{document}