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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