Section
Lie Groups and Lie Algebras
Core theory of Lie groups, Lie algebras, representations, and structure theory at graduate level.
Core idea
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
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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
- Example: (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
Noncompact and unitary representation theory
Continuous and unitary representations
- Cyclic vector and cyclic unitary representation
- Intertwining operator between unitary representations
- Invariant closed subspace of a unitary representation
- Irreducible unitary representation
- Projective unitary representation
- Schur's lemma for unitary representations
- Strongly continuous unitary representation
- Unitary equivalence of representations
- Weak and strong continuity for unitary representations
- Weakly continuous unitary representation
Smooth vectors and derived representations
- Analytic vector of a Lie-group representation
- Derived representation on smooth vectors
- Distribution vectors of a Lie-group representation
- Dixmier–Malliavin factorization theorem
- Essential skew-adjointness of derived representation operators
- Gårding density theorem
- Gårding subspace
- Infinitesimal generator of a unitary representation
- Nelson Laplacian
- Nelson's analytic vector theorem
- Poincaré–Birkhoff–Witt theorem
- Smooth vector of a Lie-group representation
- Stone's theorem for one-parameter unitary groups
- Universal enveloping algebra action on smooth vectors
- Universal enveloping algebra of a Lie algebra
Admissibility and Harish-Chandra modules
- (g,K)-module
- Admissible representation of a real reductive group
- Casselman–Wallach globalization theorem
- Casselman–Wallach representation
- Harish–Chandra admissibility theorem
- Harish–Chandra module
- Harish–Chandra module of K-finite vectors
- Infinitesimal equivalence of admissible representations
- K-finite vector
- K-type
Real reductive structure
- Cartan decomposition of a real reductive Lie algebra
- Cartan involution of a real Lie algebra
- Furstenberg boundary of a semisimple Lie group
- Global Cartan decomposition of a real reductive group
- Iwasawa decomposition
- Langlands decomposition of a parabolic subgroup
- Maximal compact subgroup of a real reductive group
- Minimal parabolic subgroup
- Parabolic subgroup of a real reductive group
- Real rank of a reductive group
- Real reductive Lie group
- Restricted root system
- Restricted Weyl group
- Riemannian symmetric space of noncompact type
Noncompact representation series
- Casselman subrepresentation theorem
- Complementary series representation
- Discrete series representation
- Harish–Chandra's discrete-series criterion
- Knapp–Stein intertwining operator
- Langlands classification for real reductive groups
- Langlands quotient
- Limit of discrete series representation
- Principal series representation
- Spherical principal series representation
- Square-integrable representation modulo the center
- Standard module of a real reductive group
Infinitesimal characters and distribution characters
- Casimir element and Casimir operator
- Center of the universal enveloping algebra
- Global character of an admissible representation
- Harish–Chandra isomorphism
- Harish–Chandra regularity theorem
- Infinitesimal character
Induction, direct integrals, and Plancherel theory
Spherical representation theory
Orbit method
- Induced representation attached to a polarization
- Kirillov correspondence for nilpotent Lie groups
- Orbit method
- Polarization at a coadjoint functional