Chapters 12 and 13 Flashcards
Definition of generalised eigenspace
Generalised eigenspace of a subalgebra
Root space decomposition
Define a Cartan subalgebra
Equivalent definition of a Cartan subalgebra
Define an abstract root system
Theorem about Killing forms and root systems
Lie’s theorem
If g is semisimple, what can you say about it’s Cartan subalgebra?
It’s Cartan subalgebra is abelian
If g is simple what can you say about it’s root system
If g is simple if and only if it’s root system is irreducible
Define a root system that is reduced, reducible, irreducible and equivalent to another
Existence theorem
Proposition about abstract root system
Explanation of the 3 dynkin diagrams with 2 nodes
All possible dynkin diagrams of irreducible reduced root system
Define roots and root spaces
Define a Weyl reflection and Weyl group
Simple root
A root is simple if it is not a sum of two positive roots
The Dynkin diagram of an irreducible reduced root system is
Connected
