Introduction to Lie Theory

A number of structures from Lie theory and the theory of Coxeter groups can be handled by Magma. Specifically, facilities are provided for:

1. Coxeter matrices, Coxeter graphs, Cartan matrices, Dynkin diagrams, and Cartan’s naming system for Coxeter groups;

2. Finite root systems and finite root data;

3. Coxeter groups in three different formats: as finitely presented groups, as permutation groups, and as reflection groups;

4. Complex reflection groups;

5. Lie algebras, given as structure constant algebras, matrix algebras, or finitely generated algebras;

6. Groups of Lie type (connected reductive algebraic groups);

7. Representations of Lie algebras and groups of Lie type;

8. Universal enveloping algebras and Quantum groups.

• Descriptions of Coxeter Groups
• Root Systems and Root Data
• Coxeter and Reflection Groups
• Lie Algebras and Groups of Lie Type
• Highest Weight Representations
• Universal Enveloping Algebras and Quantum Groups