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 and a positive definite bilinear form on , we can define the reflection

Then a *root system* is a subset of such that

- and ,
- If then the only scalar multiples of in are ,
- for all ,
- for all .

A root system is called *irreducible* if it cannot be written as a disjoint union of and such that .

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

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