This module covers the foundations of Lie group and Lie algebra theory, including the exponential map, adjoint representations, root systems, and the classification of semisimple Lie algebras.
Lie Groups
Basic Definitions
- Lie group
- Lie subgroup
- Connected Lie group
- Simply connected Lie group
- Compact Lie group
- Abelian Lie group
- Closed subgroup
- Discrete subgroup
Group Structure
- Center of a Lie group
- Commutator subgroup
- Normal Lie subgroup
- Quotient Lie group
- Product Lie group
- Covering Lie group
- Universal covering group
Translations
- Left translation
- Right translation
- Left-invariant vector field
- Right-invariant vector field
- Bi-invariant metric
Lie Algebras
Basic Definitions
- Lie algebra
- Lie algebra of a Lie group
- Lie bracket
- Lie subalgebra
- Ideal (Lie algebra)
- Lie algebra homomorphism
- Lie algebra isomorphism
Structure
- Center of a Lie algebra
- Derived subalgebra
- Solvable Lie algebra
- Nilpotent Lie algebra
- Simple Lie algebra
- Semisimple Lie algebra
- Abelian Lie algebra
Derivations
Exponential Map and Maurer-Cartan
- Exponential map
- One-parameter subgroup
- Logarithm map
- Baker-Campbell-Hausdorff formula
- Left Maurer-Cartan form
- Right Maurer-Cartan form
- Maurer-Cartan equation
Representations
Adjoint Representations
General Representations
- Representation of a Lie group
- Representation of a Lie algebra
- Irreducible representation (group)
- Irreducible representation (algebra)
- Completely reducible representation
Weight Theory
Structure Theory
Killing Form and Cartan Theory
Classification
Classical Lie Groups
Actions and Homogeneous Spaces
Key Theorems
- Closed subgroup theorem
- Lie's third theorem
- Ado's theorem
- Lie correspondence
- Cartan's criterion for solvability
- Cartan's criterion for semisimplicity
- Weyl's complete reducibility theorem
- Levi decomposition
- Peter-Weyl theorem
- Maximal torus theorem