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
- 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
Lie Algebras
Basic Definitions
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
- 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
Examples
- SU(2) and su(2)
- SO(3) and so(3)
- SL(2,C) and sl(2,C)
- Heisenberg algebra
- Upper triangular (solvable)
- Strictly upper triangular (nilpotent)
- S² as homogeneous space
Uncategorized
- Adjoint representation: discrete kernel iff discrete center
- Bi-invariant differential form
- Cartan subalgebras are self-normalizing
- Coadjoint representation of a Lie algebra
- Bi-invariant metrics on compact Lie groups
- Structure of compact connected Lie groups
- Structure of connected abelian Lie groups
- Connected subgroup determined by its Lie algebra
- Derived series of a Lie algebra
- Derived subalgebra is an ideal
- Differential of a Lie group homomorphism
- Direct sum of Lie algebras
- Dual (contragredient) representation
- Effective action
- Example: the torus $T^n$
- Example: $U(1)$ (the circle group)
- Exponential map is a local diffeomorphism
- Exponentials and one-parameter subgroups
- Free action
- Fundamental representation
- General linear Lie algebra
- Heisenberg group
- Highest-weight theorem
- Kernel of Ad and the center
- Kernel of ad and the center
- Ad-invariance of the Killing form
- Killing form nondegeneracy criterion
- Left-invariant differential form
- Left-invariant vector fields form the Lie algebra
- Lie algebra automorphism
- Compact Lie algebra is reductive
- Lie algebra of a product
- Lie algebra of a subgroup lemma
- Lorentz group
- Lower central series
- Maurer–Cartan equation lemma
- Nilpotent implies solvable
- One-parameter subgroups as integral curves
- Orbit space
- Orthogonal Lie algebra
- Poincaré group
- Proper action
- Quotient Lie algebra
- Right-invariant differential form
- Root space decomposition
- Schur orthogonality for compact Lie groups
- Semisimple Lie algebra as a direct sum of simple ideals
- Center of a simple Lie algebra is trivial
- Simply connected Lie groups are determined by their Lie algebras
- Special linear Lie algebra
- Special unitary Lie algebra
- Subrepresentation of a Lie algebra
- Symplectic Lie algebra
- Tensor product of representations
- Equivalent characterizations of nilpotency for Lie algebras
- Equivalent characterizations of semisimplicity for Lie algebras
- Equivalent characterizations of solvability for Lie algebras
- Transitive Lie group action
- Unitary Lie algebra
- Existence of universal covering groups
- Weights in the dual Cartan