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Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple Groups
Anthony W. Knapp
Use in this reference: Linked author-hosted chapter excerpt. No reproduction or redistribution permission is assumed.
Relevant material: Chapter VI, including Theorem 6.46 and Theorem 6.105 (printed page 362).
Dependent assertions
| Record | Assertion | Source locator |
|---|---|---|
| SL(2,ℂ) | Real rank | Theorem 6.105: complex simple algebras as real algebras; the split diagonal subgroup of SL(2,C) has real rank 1 |
| SL(2,ℂ) | Representations | Theorem 6.105: sl(2,C) viewed as a real simple algebra |
| SL(2,ℂ) | The Lorentz algebra is a realification | Theorem 6.105: sl(2,C) viewed as a real simple algebra |
| Aᵣ | Classical real forms | Theorem 6.105, printed page 362: nonredundant real-form classification and coincidences |
| Bᵣ | Classical real forms | Theorem 6.105, printed page 362: nonredundant real-form classification and coincidences |
| Cᵣ | Classical real forms | Theorem 6.105, printed page 362: nonredundant real-form classification and coincidences |
| Dᵣ | Classical real forms | Theorem 6.105, printed page 362: nonredundant real-form classification and coincidences |
| su(2) | Simple over R | Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R |
| su(3) | Simple over R | Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R |
| su(4) ≅ so(6) | Simple over R | Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R |
| su(5) | Simple over R | Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R |
| sp(2) ≅ so(5) | Simple over R | Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R |
| sp(3) | Simple over R | Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R |
| so(8) | Simple over R | Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R |
| so(10) | Simple over R | Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R |
| e₆ compact | Simple over R | Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R |
| e₇ compact | Simple over R | Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R |
| e₈ compact | Simple over R | Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R |
| f₄ compact | Simple over R | Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R |
| g₂ compact | Simple over R | Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R |
| su(2) ⊕ su(2) | Simple over R | Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R |
| u(1) | Simple over R | Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R |
| u(2) | Simple over R | Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R |
| sl(2, R) | Simple over R | Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R |
| sl(3, R) | Simple over R | Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R |
| su(2,2) ≅ so(2,4) | Simple over R | Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R |
| sp(4, R) ≅ so(2,3) | Simple over R | Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R |
| sl(2, C) viewed over R | Definition | Theorem 6.105, printed page 362: complex simple Lie algebras viewed as simple real algebras |
| sl(2, C) viewed over R | Simple over R | Theorem 6.105, printed page 362: simple real algebras, including complex simple algebras viewed over R |
| sl(2, C) viewed over R | From algebra to group | Theorem 6.105: complex simple algebras considered as real algebras |
| sl(2, C) viewed over R | Definition | Theorem 6.105, printed page 362: complex simple Lie algebras viewed over R |
| sl(2, C) viewed over R | One algebra, several integrations | Theorem 6.105: real simple classification |
| SL(2, R) | Center | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SL(2, R) | Fundamental group π₁ | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SL(2, R) | Compact | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SL(2, R) | Connected | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SL(2, R) | Global form | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SL(2, R) | Maximal compact subgroup | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| PSL(2, R) | Center | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| PSL(2, R) | Fundamental group π₁ | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| PSL(2, R) | Compact | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| PSL(2, R) | Connected | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| PSL(2, R) | Global form | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| PSL(2, R) | Maximal compact subgroup | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| Universal cover of SL(2, R) | Compact | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| Universal cover of SL(2, R) | Connected | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| Universal cover of SL(2, R) | Global form | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 · Theorem 6.46: Iwasawa decomposition |
| Universal cover of SL(2, R) | Global structure and topology | Theorem 6.46: Iwasawa decomposition |
| Universal cover of SL(2, R) | Explicit homomorphism | Theorem 6.46: Iwasawa decomposition |
| SL(3, R) | Center | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SL(3, R) | Fundamental group π₁ | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SL(3, R) | Compact | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SL(3, R) | Connected | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SL(3, R) | Global form | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SL(3, R) | Maximal compact subgroup | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SU(2,2) | Center | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SU(2,2) | Fundamental group π₁ | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SU(2,2) | Compact | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SU(2,2) | Connected | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SU(2,2) | Global form | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 · Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SU(2,2) | Maximal compact subgroup | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SU(2,2) | Global structure and topology | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SU(2,2) | Explicit homomorphism | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| Sp(4, R) | Center | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| Sp(4, R) | Fundamental group π₁ | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| Sp(4, R) | Compact | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| Sp(4, R) | Connected | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| Sp(4, R) | Global form | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 · Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| Sp(4, R) | Maximal compact subgroup | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| Sp(4, R) | Global structure and topology | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| Sp(4, R) | Explicit homomorphism | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(1,3) | Center | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(1,3) | Fundamental group π₁ | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(1,3) | Compact | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(1,3) | Connected | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(1,3) | Global form | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 · Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(1,3) | Maximal compact subgroup | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(1,3) | Global structure and topology | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(1,3) | Representation category and descent | Theorem 6.105: complex simple algebras viewed as real |
| SO₀(1,3) | Explicit homomorphism | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(1,3) | Specified connected covering group | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(2,4) | Center | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(2,4) | Fundamental group π₁ | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(2,4) | Compact | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(2,4) | Connected | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(2,4) | Global form | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 · Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(2,4) | Maximal compact subgroup | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(2,4) | Global structure and topology | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(2,4) | Explicit homomorphism | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(2,4) | Specified connected covering group | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(2,3) | Center | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(2,3) | Fundamental group π₁ | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(2,3) | Compact | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(2,3) | Connected | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(2,3) | Global form | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 · Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(2,3) | Maximal compact subgroup | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(2,3) | Global structure and topology | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(2,3) | Explicit homomorphism | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| SO₀(2,3) | Specified connected covering group | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |
| Sp(4, R) → SO₀(2,3) | Global and representation consequences | Chapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316 |