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.