lecture-notes
Lecture 17: Classification of Connected Compact and Complex Reductive Groups
Pavel Etingof
Use in this reference: Linked MIT OpenCourseWare notes; global-form and representation criteria are restated.
Relevant material: Sections 43.1–43.2, printed pages 230–235.
Dependent assertions
| Record | Assertion | Source locator |
|---|---|---|
| SO(3) | Representations | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SO(3) | In physics | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SU(3) | Representations | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SU(3) | A₂ Dynkin-label convention | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SU(3) | A₂ Dynkin-label convention | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SU(3) | Explicit homomorphism | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| PSU(3) | Center | Section 43.1: the compact adjoint form has trivial center |
| PSU(3) | Representations | Section 43.1, Proposition 43.2: adjoint-group weights are the root lattice |
| PSU(3) | A₂ Dynkin-label convention | Section 43.1, Proposition 43.2: adjoint-group weights are the root lattice |
| PSU(3) | Explicit homomorphism | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SL(2,ℂ) | Fundamental group | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SL(2,ℂ) | Connected | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SL(2,ℂ) | Topology and global form | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| Aᵣ | Representation convention | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| Bᵣ | Representation convention | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| Cᵣ | Representation convention | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| Dᵣ | Representation convention | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| E₆ | Representation convention | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| E₇ | Representation convention | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| E₈ | Representation convention | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| F₄ | Representation convention | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| G₂ | Representation convention | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| sl(2, C) | Using the algebra record | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| sl(3, C) | Using the algebra record | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| sl(4, C) | Using the algebra record | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| sl(5, C) | Using the algebra record | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| so(5, C) ≅ sp(4, C) | Using the algebra record | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| sp(6, C) | Using the algebra record | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| so(8, C) | Using the algebra record | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| so(10, C) | Using the algebra record | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| e₆(C) | Using the algebra record | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| e₇(C) | Using the algebra record | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| e₈(C) | Using the algebra record | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| f₄(C) | Using the algebra record | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| g₂(C) | Using the algebra record | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| su(2) | From algebra to group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| su(3) | From algebra to group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| su(4) ≅ so(6) | From algebra to group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| su(5) | From algebra to group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| sp(2) ≅ so(5) | From algebra to group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| sp(3) | From algebra to group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| so(8) | From algebra to group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| so(10) | From algebra to group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| e₆ compact | From algebra to group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| e₇ compact | From algebra to group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| e₈ compact | From algebra to group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| f₄ compact | From algebra to group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| g₂ compact | From algebra to group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| su(2) ⊕ su(2) | From algebra to group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| u(1) | From algebra to group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| u(2) | From algebra to group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| sl(2, R) | From algebra to group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| sl(3, R) | From algebra to group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| su(2,2) ≅ so(2,4) | From algebra to group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| sp(4, R) ≅ so(2,3) | From algebra to group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| sl(2, C) viewed over R | From algebra to group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| PSU(4) | Representation category and descent | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SU(5) | Representation category and descent | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| U(2) | Representation category and descent | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SO(4) | Representation category and descent | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SO(5) | Representation category and descent | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SO(6) | Representation category and descent | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| Sp(3) | Representation category and descent | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SO(8) | Representation category and descent | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| PSO(8) | Representation category and descent | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SO(10) | Representation category and descent | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| E₆ compact, simply connected | Representation category and descent | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| E₇ compact, simply connected | Representation category and descent | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| E₈ compact | Representation category and descent | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| E₆ compact, adjoint | Representation category and descent | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| E₇ compact, adjoint | Representation category and descent | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SL(2, R) | Fundamental group π₁ | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SL(2, R) | Global form | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SL(2, R) | Maximal compact subgroup | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SL(2, R) | Global structure and topology | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SL(2, R) | Explicit homomorphism | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SL(2, R) | Explicit homomorphism | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SL(2, R) | Specified connected covering group | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| PSL(2, R) | Fundamental group π₁ | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| PSL(2, R) | Global form | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| PSL(2, R) | Maximal compact subgroup | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| PSL(2, R) | Global structure and topology | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| PSL(2, R) | Representation category and descent | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| PSL(2, R) | Explicit homomorphism | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| PSL(2, R) | Specified connected covering group | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SL(3, R) | Fundamental group π₁ | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SL(3, R) | Global form | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SL(3, R) | Maximal compact subgroup | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SL(3, R) | Global structure and topology | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SU(2,2) | Fundamental group π₁ | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SU(2,2) | Maximal compact subgroup | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| Sp(4, R) | Fundamental group π₁ | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| Sp(4, R) | Maximal compact subgroup | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SO₀(1,3) | Fundamental group π₁ | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SO₀(1,3) | Global form | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SO₀(1,3) | Maximal compact subgroup | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SO₀(1,3) | Global structure and topology | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SO₀(1,3) | Representation category and descent | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SO₀(1,3) | Explicit homomorphism | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SO₀(1,3) | Specified connected covering group | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SO₀(2,4) | Fundamental group π₁ | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SO₀(2,4) | Maximal compact subgroup | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SO₀(2,3) | Fundamental group π₁ | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SO₀(2,3) | Maximal compact subgroup | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SU(2) → SO(3) | Global and representation consequences | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SU(3) → PSU(3) | Definition | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SU(3) → PSU(3) | Domain | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SU(3) → PSU(3) | Codomain | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SU(3) → PSU(3) | Specified homomorphism | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SU(3) → PSU(3) | Kernel | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SU(3) → PSU(3) | Image | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SU(3) → PSU(3) | Injective | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SU(3) → PSU(3) | Surjective | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SU(3) → PSU(3) | Covering degree | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SU(3) → PSU(3) | Global and representation consequences | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients · Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SU(3) → PSU(3) | Domain group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SU(3) → PSU(3) | Codomain group | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SU(2) ↪ SU(3) | Global and representation consequences | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SU(2) → SU(3), spin 1 | Global and representation consequences | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| U(1) ↪ SU(2) | Global and representation consequences | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| U(1) ≅ SO(2) ↪ SO(3) | Global and representation consequences | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| Spin(4) → SO(4) | Global and representation consequences | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SU(4) → SO(6) | Global and representation consequences | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SU(4) → PSU(4) | Global and representation consequences | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| Sp(2) → SO(5) | Global and representation consequences | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| Spin(8) → SO(8) | Global and representation consequences | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| Spin(10) → SO(10) | Global and representation consequences | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SL(2, C) → SO₀(1,3) | Global and representation consequences | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| SL(2, C) → SO₀(1,3) | Global and representation consequences | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SL(2, R) → PSL(2, R) | Global and representation consequences | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| Universal SL(2,R) cover → SL(2,R) | Global and representation consequences | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| Universal SL(2,R) cover → SL(2,R) | Global and representation consequences | Section 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C) |
| SU(2,2) → SO₀(2,4) | Global and representation consequences | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |
| Sp(4, R) → SO₀(2,3) | Global and representation consequences | Section 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotients |