# Friday January 10th

## Root Space Decomposition

Let $\lieg$ be a finite dimensional semisimple lie algebra over $\CC$.
Recall that this means it has no proper solvable ideals.
A more useful characterization is that the Killing form $\kappa: \lieg\tensor \lieg \to \lieg$ is a *non-degenerate* symmetric (associative) bilinear form.
The running example we'll use is $\lieg = \liesl(n, \CC)$, the trace zero $n\times n$ matrices.
Let $\lieh$ be a maximal toral subalgebra, where $x\in\lieg$ is *toral* if $x$ is semisimple, i.e. $\ad x$ is semisimple (i.e. diagonalizable).

:::{.example title="?"}
$\lieh$ is the diagonal matrices in $\liesl(n, \CC)$.
:::

:::{.remark}
$\lieh$ is abelian, so $\ad \lieh$ consists of commuting semisimple elements, which (theorem from linear algebra) can be simultaneously diagonalized.
:::

:::{.definition title="Root Space Decomposition"}
This leads to the root space decomposition,

\[
\lieg = \lieh \oplus \bigoplus_{\alpha\in \Phi} \lieg_\alpha
.\]

where 
\[
\lieg_\alpha = \theset{x\in \lieg \suchthat [h x] = \alpha(h) x ~\forall h\in \lieh}
\]
where $\alpha \in \lieh\dual$ is a linear functional.
:::

Here $\lieh = C_\lieg(\lieh)$, so $[h x] = 0$ corresponds to zero eigenvalues, and (fact) it turns out that $\lieh$ is its own centralizer.

:::{.definition title="Root System"}
We then obtain a set of roots of $\lieh, \lieg$ given by 
\[
\Phi = \theset{\alpha\in\lieh\dual \suchthat \alpha\neq 0, \lieg_\alpha \neq \theset{0}}
\]
:::

:::{.example title="?"}
$\lieg_\alpha = \CC E_{ij}$ for some $i\neq j$, the matrix with a 1 in the $i,j$ position and zero elsewhere.
:::


:::{.remark}
The restriction $\restrictionof{\kappa}{\lieh}$ is nondegenerate, so we can identify $\lieh, \lieh\dual$ via $\kappa$ (can always do this with vector spaces with a nondegenerate bilinear form), where $\kappa$ maps to another bilinear form $(\wait, \wait)$.
We thus get a correspondence
\[
\lieh\dual \ni \lambda \iff t_\lambda \in \lieh \\
\lambda(h) = \kappa(t_\lambda, h) \quad\text{where } (\lambda, \mu) = \kappa(t_\lambda, t_\mu)
.\]

:::

## Facts About $\Phi$ and Root Spaces


:::{.definition title="Abstract Root System"}
Let $\alpha, \beta \in \Phi$ be roots.

1. $\Phi$ spans $\lieh\dual$ and does not contain zero.

2. If $\alpha \in \Phi$ then $-\alpha \in \Phi$, but no other scalar multiple of $\alpha$ is in $\Phi$.

    - Note: see \cref{rmk:aside1}.


 3. $(\beta, \alpha\dual) \in \ZZ$

4. $s_\alpha(\beta) \definedas \beta - (\beta, \alpha\dual)\alpha \in \Phi$.

    - Note: see \cref{rmk:aside2}
:::

:::{.remark} 
\label{rmk:aside1}
An aside:

- $\dim \lieg_\alpha = 1$.

- If $0 \neq x_\alpha \in \lieg_\alpha$ then there exists a unique $y_\alpha \in \lieg_{-\alpha}$ such that $x_\alpha, y_\alpha, h_\alpha \definedas [x_\alpha, y_\alpha]$ spans a 3-dimensional subalgebra in $\liesl_2$, given by 
\[
  x_\alpha = 
  \begin{bmatrix}
  0 & 1 \\
  0 & 0
  \end{bmatrix}, 
  \quad
  y_\alpha = 
  \begin{bmatrix}
  0 & 0 \\
  1 & 0
  \end{bmatrix},
  \quad
  h_\alpha =\begin{bmatrix}
  1 & 0 \\
  0 & -1
  \end{bmatrix}
  .\]

- Under the correspondence $\lieh \iff \lieh\dual$ induced by $\kappa$, 
  \[
  h_\alpha \iff \alpha\dual \definedas \frac{2\alpha}{(\alpha, \alpha)}
  \]
  Thus for all $\lambda \in\lieh\dual$, 
  \[
  \lambda(h_\alpha) = (\lambda, \alpha\dual) = \frac{2(\lambda, \alpha)}{(\alpha, \alpha)}
  .\]

- If $\alpha + \beta \neq 0$, then $\kappa(g_\alpha, g_\beta) = 0$.

:::

:::{.remark}
\label{rmk:aside2}
If $\alpha + \beta \in \Phi$, then $[\lieg_\alpha \lieg_\beta] = \lieg_{\alpha+\beta}$.

:::{.example title="?"}
Example: If $\alpha = E_{ij}, \beta = E_{jk}$ where $k\neq i$, then $[E_{ij}, E_{jk}]= E_{ik}$.
:::

- $\lieg$ is generated as an algebra by the root spaces $\lieg_\alpha$
- Root strings: If $\beta \neq \pm\alpha$, then the roots of the form $\alpha + k\beta$ for $k\in \ZZ$ form an unbroken string 
\[
\alpha - r\beta, \cdots, \alpha-\beta, \alpha,\alpha+\beta,\cdots,\alpha + \ell \beta
\]
consisting of at most 4 roots where $r-s = (\alpha, \beta\dual)$.

:::


:::{.example title="?"}
The circled roots below form the root string for $\beta$:

![Image](figures/2020-01-10-09:34.png)\

In general, a subset $\Phi$ of a real euclidean space $E$ satisfying conditions (1) through (4) is an *(abstract) root system*.
Note that when $\Phi$ comes from a $\lieg$, we define $E\definedas \RR \Phi$.
:::

### The Root System


:::{.definition title="Simple System"}
There exists a subset $\Delta \subseteq \Phi$ such that 

- $\Delta$ is a $\CC\dash$basis for $\lieg\dual$
- $\beta\in\Phi$ implies that $\beta = \sum_{\alpha \in \Delta} c_\alpha \alpha$ with either 
  - All $c_\alpha \in \ZZ_{\geq 0} \iff \beta \in \Phi^+$ or $\beta < 0$.
  - All $c_\alpha \in \ZZ_{\leq 0} \iff \beta \in \Phi^-$ or $\beta > 0$.

$\Delta$ is called a **simple system**.
:::

:::{.definition title="Positive Roots, Height"}

If $\Delta = \theset{a_1, \cdots, a_\ell}$ then $\Phi^+$ are the *positive roots*, and if $\Phi^+ \ni \beta = \sum_{\alpha \in \Delta} c_\alpha \alpha$, then the *height* of $\beta$ is defined as 

\[
\height(\beta) \da 
\sum c_\alpha \in \ZZ_{> 0}
\]

:::

:::{.definition title="Root Lattice, Dual Root System"}
Note that $\ZZ \Phi \definedas \Lambda_r$ is a lattice, and is referred to as the *root lattice*, and $\Lambda_r \subset E = \RR \Phi$.

We also have 
\[
\Phi^+ = \theset{\beta\dual \suchthat \beta \in \Phi}
,\]
the *dual root system*, is a root system with simple system $\Delta\dual$.
:::

:::{.proposition title="Important subalgebras of a Lie algebra"}
\[  
\lien = \lien^+ &\da \sum_{\beta > 0} \lieg_\beta
&& \text{Upper triangular with zero diagonal,} \\
\lien^- &\da \sum_{\beta > 0} \lieg_{-\beta}
&& \text{Lower triangular with zero diagonal,} \\
\lieb &\da \lieh + \lien
&& \text{Upper triangular (the "Borel" subalgebra),} \\
\lieb^- &\da \lieh + \lien^-
&& \text{Lower triangular.} 
.\]
:::


:::{.definition title="Triangular/Cartan Decomposition"}
\[
\lieg = \lien^- \oplus \lieh \oplus \lien
\]
:::

:::{.fact}
If $\beta \in \Phi^+\setminus \Delta$, and if $\alpha \in \Delta$ such that $(\beta, \alpha\dual) > 0$, then $\beta - (\beta,\alpha\dual)\alpha \in \Phi^+$ has height strictly less than the height of $\beta$.
:::

:::{.remark}
By root strings, $\beta-\alpha\in\Phi^+$ is  positive root of height one less than $\beta$, yielding a way to induct on heights (useful technique).
:::


### Weyl Groups

For $\alpha \in \Phi$, define

\[
S_\alpha : \lieh\dual &\to \lieh\dual \\
\lambda &\mapsto \lambda - (\lambda, \alpha\dual)\alpha
.\]

This is reflection in the hyperplane in $E$ perpendicular to $\alpha$:

![Reflection through a hyperplane](figures/2020-01-10-09:51.png){width=350px}

Note that $s_\alpha^2 = \id$.

:::{.definition title="Weyl Group"}
Define $W$ as the subgroup of $\gl(E)$ generated by all $s_\alpha$ for $\alpha \in \Phi$, this is the *Weyl group* of $\lieg$ or $\Phi$, which is finite and $W = \generators{s_\alpha \suchthat \alpha\in\Delta}$ is generated by simple reflections.


:::
By (4), $W$ leaves $\Phi$ invariant.
In fact $W$ is a finite Coxeter group with generators $S = \theset{s_\alpha \suchthat \alpha\in \Delta}$ and defining relations $(s_\alpha s_\beta)^{m(\alpha, \beta)} = 1$ for $\alpha,\beta \in \Delta$ where $m(\alpha, \beta) \in \theset{2,3,4,6}$ when $\alpha \neq \beta$ and $m(\alpha, \alpha) = 1$.

:::{.definition title="Crystallographic group"}
If this finiteness on numerical conditions are met, then $W$ is referred to as a *Crystallographic group*.
:::