# 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](descriptions.md)

- [Root Systems and Root Data](descriptions-2.md)

- [Coxeter and Reflection Groups](groups.md)

- [Lie Algebras and Groups of Lie Type](descriptions-3.md)

- [Highest Weight Representations](descriptions-4.md)

- [Universal Enveloping Algebras and Quantum Groups](descriptions-5.md)