This week Sükrü led the discussion which focussed on root systems, algebraic groups and the Curtis-Tits Theorem.

Recall from an earlier week that given ${\alpha\in V=\mathbb{R}^n}$ and a positive definite bilinear form ${(.,.)}$ on ${V}$, we can define the reflection

$\displaystyle \sigma_{\alpha}:\beta\mapsto \beta-\frac{2(\alpha,\beta)}{(\alpha,\alpha)}\alpha$

Then a root system is a subset ${\Phi}$ of ${V}$ such that

1. ${\langle \Phi\rangle=V}$ and ${0\notin \Phi}$,
2. If ${\alpha\in\Phi}$ then the only scalar multiples of ${\alpha}$ in ${\Phi}$ are ${\pm \alpha}$,
3. ${\sigma_{\alpha}(\Phi)=\Phi}$ for all ${\alpha\in\Phi}$,
4. ${2(\alpha,\beta)/(\alpha,\alpha)\in\mathbb{Z}}$ for all ${\alpha,\beta\in\Phi}$.

A root system is called irreducible if it cannot be written as a disjoint union of ${\Phi_1}$ and ${\Phi_2}$ such that ${\langle \Phi_1\rangle \perp \langle \Phi_2\rangle}$.

A fundamental system is then a subset ${\Pi}$ of ${\Phi}$ which is a basis for ${V}$ and such that each element of ${\Phi}$ can be written as a linear combination of the elements of ${\Pi}$ such that the coefficients are either all nonnegative integers or all nonpositive integers.