# Monday, October 24

:::{.remark}
Classifying root systems: $\Delta \subseteq \Phi \subseteq \EE$ a base yields a decomposition 

- $\EE = \bigoplus_{i=1}^t \EE_i$,
- $\Phi= \bigoplus_{i=1}^t \Phi_i$,
- $\Delta = \bigoplus_{i=1}^t \Delta_i$,
where are orthogonal direct sums with respect to $(\wait, \wait)$.
Note that the sub-bases $\Delta_i$ biject with connected components of the Coxeter graph $\Gamma$ of $\Delta$.
We saw $\inp{\alpha_i}{\alpha_j} \inp {\alpha_j}{\alpha_i} \in \ts{0,1,2,3}$ is the number of edges between nodes $i$ and $j$ in $\Gamma$, using that the first term is $4\cos^2(\theta)\in [0, 3] \intersect \ZZ$.
It suffices to classify irreducible root systems, corresponding to connected Coxeter graphs.
Recall arrows point from long to short roots.
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:::{.theorem title="?"}
If $\Phi$ is an irreducible root system of rank $\ell$, then its Dynkin diagram is one of the following:

- The four infinite families, corresponding to classical types:

![](figures/2022-10-24_09-19-12.png)

- Exceptional classes

![](figures/2022-10-24_09-19-36.png)

Types ADE are called **simply laced** since they have no multiple edges.

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:::{.remark}
Idea: classify possible connected Coxeter graphs, ignoring relative root lengths.
If \( \alpha, \beta \) are simple roots, note that for any $c$,
\[
\inner{ \alpha }{ \beta } \inner{ \beta }{ \alpha } = {2(c \alpha, \beta)\over (\beta, \beta)} {2( \beta, c \alpha) \over (c \alpha, c \alpha)} \in \ts{0,1,2,3}
,\]
so $\alpha \mapsto c\alpha$ leaves this number invariant and we can assume all simple roots are unit vectors.
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:::{.definition title="?"}
Let $\EE$ be a finite dimensional Euclidean space, then a subset $A =\ts{\tl \eps n }\subseteq \EE$ of linearly independent unit vectors satisfying

- $(\eps_i, \eps_j) \leq 0$ for all $i,j$,
- $4(\eps_i, \eps_j)^2 = 4\cos^2(\theta) \in \ts{0,1,2,3}$ for all $i\neq j$ where $\theta$ is the angle between $\eps_i$ and $\eps_j$

is called **admissible**.
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:::{.example title="?"}
Any base for a root system where each vector is normalized is admissible.
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:::{.remark}
To such an $A$ we associate a graph $\Gamma$ as before with vertices $1,\cdots, n$ where $i,j$ are joined by $4(\eps_i, \eps_j)^2$ edges.
We'll determine all connected graphs $\Gamma$ that can occur, since these include all connected Coxeter graphs.
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## Proof of classification

:::{.proof title="Sketch"}
An easy 10 steps:

1. If some $\eps_i$ are discarded, the remaining ones still form an admissible set in $\EE$ whose graph is obtained from $\Gamma$ by omitting the corresponding discarded vertices.
2. The number of pairs of vertices in $\Gamma$ connected by at least one edge is strictly less than $n$.
  
  Proof: Set $\eps \da\sum_{i=1}^n \eps_i$, which is nonzero by linear independence.
  Then $0 < (\eps, \eps) = n + \sum_{i<j} 2 (\eps_i, \eps_j)$, and if $i< j$ are joined, so $(\eps_i, \eps_j)\neq 0$, then $4(\eps_i, \eps_j)^2 = 1,2,3$ and so $2(\eps_i, \eps_j)\leq -1$.
  However, since the sum is positive, there are $\leq n-1$ such pairs.

3. $\Gamma$ contains no cycles.

  Proof: A cycle is subgraph that corresponds to an admissible subset $A' \subseteq A$ with $m$ vertices and $m$ edges corresponding to $m$ connected pairs.
  This contradicts (2).

4. No more than 3 edges can be incident to any vertex of $\Gamma$.

  Let $\eps\in A$ be a vertex, and suppose $\tl \eta k$ are the vertices connected to $\eps$:

\begin{tikzpicture}
\fontsize{45pt}{1em} 
\node (node_one) at (0,0) { \import{/home/zack/SparkleShare/github.com/Notes/Class_Notes/2022/Fall/LieAlgebras/sections/figures}{2022-10-24_09-40.pdf_tex} };
\end{tikzpicture}

  By (3), no two $\eta_i, \eta_j$ are connected, so $( \eta_i, \eta_j) = 0$ for all $i,j$.
  Apply Gram-Schmidt to $\ts{\tl \eta k , \eps}$ only involves modifying $\eps$ since $\ts{\tl \eta k}$ are already orthonormal.
  Call the new vector $\eta_0$ and let $\ts{\tlz \eta k}$ be the resulting orthonormal set.
  One can then write $\eps = \sum_{i=0}^k (\eps, \eta_i)\eta_i$, which implies $(\eps, \eta_0)\neq 0$ by linear independence.
  Then $1 = (\eps, \eps) = \sum_{i=0}^k (\eps, \eta_i)^2$, but $(\eps, \eta_0)^2 > 0$ so $\sum_{i=1}^k (\eps, \eta_i)^2 < 1$ and thus $\sum_{i=1}^k 4 (\eps, \eta_i)^2 < 4$.
  But this sum is the number of edges incident to $\eps$ in $\Gamma$.

5. The only connected graph which contains a triple edge is the Coxeter graph of $G_2$ by (4), since the triple edge forces each vertex to already have 3 incident edges.

6. Let $\ts{\eps_1,\cdots, \eps_k} \subseteq A$ have a simple chain $\cdot \to \cdot \to \cdots \to \cdot$ as a subgraph.
  If $A' \da \ts{A\smts{\tl \eps k}} \union\ts{\eps}$ where $\eps \da \sum_{i=1}^k \eps_i$, then $A'$ is admissible.
  The corresponding graph $\Gamma'$ is obtained by shrinking the chain to a point, where any edge that was incident to any vertex in the chain is now incident to $\eps$, with the same multiplicity.

  Proof: number the vertices in the chain $1,\cdots, k$.
  Linear independence of $A'$ is clear.
  Note $4(\eps_i, \eps_{i+1})^2 = 1\implies 2(\eps_i, \eps_{i+1}) \implies (\eps, \eps) = k + 2 \sum_{i< j} (\eps_i, \eps_j) = k + (-1)(k-1) = 1$.
  Any $\eta\in A\smts{\tl \eps k}$ is connected to at most one of $\tl \eps k$ since this would otherwise form a cycle, so $(\eta, \eps) = (\eta, \eps_i)$ for a single $i$.
  So $4(\eta, \eps)^2 = 4(\eta, \eps_i)^2 \in \ts{0,1,2,3}$ and $(\eta, \eps) = (\eta, \eps_i) \leq 0$, which verifies all of the admissibility criteria.

7. $\Gamma$ contains no graphs of the following forms:

<!-- Xournal file: /home/zack/SparkleShare/github.com/Notes/Class_Notes/2022/Fall/LieAlgebras/sections/figures/2022-10-24_09-59.xoj -->

  ![](figures/2022-10-24_10-01-11.png)

  Proof: collapsing the chain in the middle produces a vertex with 4 incident edges.

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