Root systems

24 novembro 2017, 10:00 • Gustavo Granja

The Killing form takes rational values on the \Q span of the roots. \dim_\Q \Q\cdot \Phi =\dim_k \mathfrak h and the restriction of the Killing form to \Q\cdot \Phi is positive definite. Example: Taking into account this Euclidean structure the roots of sl(3) form a regular hexagon. Definition of root system and Weyl group. Isomorphisms of root systems and the dual root system. Lemma: If a linear f \in GL(E) preserves the set of roots then it is an isomorphism of the root system and it sends a generating reflection s_\alpha in the Weyl group to s_{f(\alpha)}. In particular the Weyl group is a normal subgroup of the group of automorphisms of the root system. The axioms of a root system put strong restriction on angle and relative lengths of the roots. Table of all the possible angles and relative lengths between two roots. Proposition: If \alpha and \beta are non proportional roots then if (\alpha,\beta)>0 then \alpha-\beta is a root and if (\alpha,\beta)<0 then \alpha+\beta is a root. Proposition: The \alpha string through \beta has no holes and length at most 4.