Lie algebras – Examples sheet 3
1. Let (ai,j)1≤i,j ≤lbe the Cartan matrix of an abstract root system. Prove
directly from the definition that
(i) ai,i = 2;
(ii) ai,j = 0 if and only if aj,i = 0;
(iii) if i!=jthen ai,j ≤0.
2. Prove that the only abstract root systems of rank two are
A1A1A2B2G2
!2 0
0 2 " ! 2−1
−1 2 " ! 2−2
−1 2 " ! 2−1
−3 2 "
3. Let Φ⊂Ebe an abstract root system with base ∆. For α∈∆, show
that sα∈Wstabilises Φ+\ {α}. Deduce that sα(ρ) = ρ−αfor all α∈∆,
where
ρ=1
2#
β∈Φ+
β.
4. Let Φ⊂Ebe an abstract root system.
(a) Show that
Φ∨=$2α
(α,α)%
%
%
%
α∈Φ&
is also an abstract root system in E, known as the dual root system.
(b) Show that the Weyl group of Φ∨is isomorphic to the Weyl group of
Φ.
(c) Show that Φ∨is irreducible if and only if Φis irreducible, and that
the double dual of Φis isomorphic to Φ.
(d) Show that the dual root system to Al, Bl, Clor Dlis Al, Cl, Blor Dl
respectively.
5. Let Φ⊂Ebe an abstract root system.
(a) Let Φ$⊂Φbe a subset such that if α1,...,αn∈Φ$and α='aiαi∈
Φfor certain coefficients ai∈Z, then α∈Φ$. Show that Φ$is a root system
in the subspace E$< E that it spans. Such subsystems of the root system
Φare called closed subsystems.
(b) Verify that the set of long roots in the root system of type G2is
1
