Buildings, geometries and algebraic groups study group X

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.

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