Classification of root systems
29 novembro 2017, 11:00 • Gustavo Granja
Definition of base of a root system. Simple roots, positive and negative roots, height. The partial order on roots. Examples. Lemma: If \Delta is a base of \Phi then for all \alpha,\beta \in \Phi, (\alpha,\beta)<=0 and \alpha-\beta is not a root (so \alpha and \beta are not comparable). Regular and singular elements in E. Weyl chambers. The base \Delta(x) associated to a regular element. This depends only on the Weyl chamber associated to x.
Proposition. The function x \mapsto \Delta(x) induces a bijective correspondence between Weyl chambers and bases of \Phi.
Examples. The Weyl group acts compatibly on Weyl chambers and bases.
Theorem: 1. The Weyl group acts simply transitively on bases and Weyl chambers
2. If \alpha is in \Phi then there is \sigma in \mathcal W sending \alpha to \Delta (i.e. every root is part of a base)
3. \mathcal W is generated by simple reflections (reflections determined by simple roots
4. The closure of a Weyl chamber is a fundamental domain for the action of \mathcal W on E.
Definition of Cartan matrix associated to \Delta. Examples.
Proposition: The Cartan matrix determined \Phi up to isomorphism.
The Coxeter graph and the Dynkin diagram. The Dynkin diagram is just a visual depiction of the Cartan matrix. Examples.
A root system is irreducible if it can not be partitioned into non-empty subsets. This is equivalent to saying that \Delta can not be so partitioned or that the Coxeter graph is connected. Any root system can be written uniquely as a cartesian product of irreducible root systems. If L is semisimple with simple factors L_i then the root system of L is the cartesian product of the root systems of the L_i
Prop: Let (E,\Phi) be an irreducible root system with base \Delta. Then
1. There's a unique maximal root and all its coefficients are positive.
2. The Weyl group orbit of any root spans E
3. At most two root lengths occur in \Phi (these are called long and short roots). Moreover roots of the same length form a single orbit of \mathcal W
4. The maximal root is long.
Theorem: If \Phi is an irreducible root system then it is either A_l for l>=1, B_l for l>=2, C_l for l>=3, D_l for l>=4 or G_2, F_4, E_6, E_7, or E_8.
Flavor of the proof: proof that the Coxeter graph of an irreducible root system can not contain a loop.
The root systems above all arise from semi-simple Lie algebras. A_l from sl(l+1), B_l from so(2l+1), C_l form Sp(2l) and D_l from so(2l). The other come from more complicated Lie algebras called exceptional. G_2 is the (complexification of the) Lie algebra of derivations of the octonions and F_4 the complexification of the Lie algebra of the group of isometries of the octonionic projective plane.
Proposition: If \Phi is a root system, Aut(\Phi) is the semi-direct product of the group \Gamma = \{ f \in Aut(\Phi) \colon f(\Delta)=\Delta} of graph automorphisms with the Weyl group.
If L is semisimple over a field of char 0, Int(L) denotes the subgroup of GL(L) generated by \{ exp(ad(x)): x \in L nilpotent\}. This is a group of automorphisms of L because the binomial formula for the iterated application of a derivation shows more generally that if \delta is a nilpotent derivation of an algebra then exp(\delta) is an automorphism. Over the complex numbers, exp is the matrix exponential so exp(ad x) is just Ad(exp x).
Theorem: All Cartan subalgebras of a semisimple Lie algebra over an algebraically closed field are conjugate by Int(L) (and hence the root system associated to such an algebra is well defined up to isomorphism).