lecture-notes
Lie Groups and Lie Algebras
Pavel Etingof
Use in this reference: Linked MIT OpenCourseWare lecture notes. Original summaries only; website content is not offered as a reproduction of these notes.
Relevant material: MIT 18.755, Spring 2024. Cite section numbers and printed pages.
Dependent assertions
| Record | Assertion | Source locator |
|---|---|---|
| SU(2) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(2) | Real dimension | Sections 6.1–6.3: su(2) consists of traceless skew-Hermitian 2 × 2 matrices; real dimension 2²−1=3 |
| SU(2) | Fundamental group | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SU(2) | Compact | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(2) | Representations | Section 11.4: irreducible sl(2,C) modules of highest weight n and dimension n+1 |
| SU(2) | Topology and global form | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SU(2) | In physics | Section 11.4: irreducible sl(2,C) modules of highest weight n and dimension n+1 |
| SU(2) | Related group | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(2) | Related group | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(2) | Related group | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(2) | Actual real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(2) | Explicit homomorphism | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SU(2) | Specified homogeneous quotient | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SO(3) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(3) | Real dimension | Sections 6.1–6.3: real skew-symmetric 3 × 3 matrices have dimension 3·2/2=3 |
| SO(3) | Fundamental group | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SO(3) | Compact | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(3) | Topology and global form | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SO(3) | Related group | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(3) | Related group | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(3) | Related group | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(3) | Related group | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(3) | Actual real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(3) | Explicit homomorphism | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SO(3) | Specified homogeneous quotient | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SU(3) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(3) | Real dimension | Sections 6.1–6.3: su(3) has real dimension 3²−1=8 |
| SU(3) | Compact | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(3) | Representations | Section 26.5: Weyl dimension formula, specialized to the three positive A₂ roots · Section 27.2: SL(n,C) fundamental representations as exterior powers |
| SU(3) | A₂ Dynkin-label convention | Section 26.5: Weyl dimension formula, specialized to the three positive A₂ roots · Section 27.2: SL(n,C) fundamental representations as exterior powers |
| SU(3) | A₂ Dynkin-label convention | Section 26.5: Weyl dimension formula, specialized to the three positive A₂ roots · Section 27.2: SL(n,C) fundamental representations as exterior powers |
| SU(3) | Related group | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(3) | Related group | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(3) | Actual real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| PSU(3) | Real dimension | Sections 6.1–6.3: dim su(3)=8; a discrete central quotient preserves the Lie algebra |
| PSU(3) | Representations | Section 27.2: the two A₂ fundamental modules have scalar charges 1 and 2 |
| PSU(3) | A₂ Dynkin-label convention | Section 27.2: the two A₂ fundamental modules have scalar charges 1 and 2 |
| PSU(3) | Actual real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| U(1) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| U(1) | Real dimension | Section 6: U(1) has Lie algebra iR of real dimension 1 |
| U(1) | Center | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| U(1) | Fundamental group | Section 3.2: R → R/Z as a covering of the circle |
| U(1) | Compact | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| U(1) | Connected | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| U(1) | Rank | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| U(1) | Topology and global form | Section 3.2: covering groups of the circle |
| U(1) | Related group | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| U(1) | Related group | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| U(1) | Actual real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| U(1) | Explicit homomorphism | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) | Explicit homomorphism | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| Spin(4) | Fundamental group | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| Spin(4) | Topology and global form | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| Spin(4) | Actual real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(2,ℂ) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(2,ℂ) | Real dimension | Sections 6.1–6.3: sl(2,C) has complex dimension 3 and real dimension 6 |
| SL(2,ℂ) | Compact | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(2,ℂ) | Representations | Section 9.4: real forms and complexification |
| SL(2,ℂ) | The Lorentz algebra is a realification | Section 9.4: real forms and complexification |
| SL(2,ℂ) | Related group | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(2,ℂ) | Related group | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(2,ℂ) | Actual real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(2, C) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(2, C) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(2, C) | Complex dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(3, C) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(3, C) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(3, C) | Complex dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(4, C) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(4, C) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(4, C) | Complex dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(5, C) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(5, C) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(5, C) | Complex dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| so(5, C) ≅ sp(4, C) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| so(5, C) ≅ sp(4, C) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| so(5, C) ≅ sp(4, C) | Complex dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sp(6, C) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sp(6, C) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sp(6, C) | Complex dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| so(8, C) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| so(8, C) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| so(8, C) | Complex dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| so(10, C) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| so(10, C) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| so(10, C) | Complex dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₆(C) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₆(C) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₆(C) | Complex dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₇(C) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₇(C) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₇(C) | Complex dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₈(C) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₈(C) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₈(C) | Complex dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| f₄(C) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| f₄(C) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| f₄(C) | Complex dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| g₂(C) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| g₂(C) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| g₂(C) | Complex dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(2, C) ⊕ sl(2, C) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(2, C) ⊕ sl(2, C) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(2, C) ⊕ sl(2, C) | Complex dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(2, C) ⊕ sl(2, C) | Using the algebra record | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| C (zero bracket) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| C (zero bracket) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| C (zero bracket) | Complex dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| C (zero bracket) | Using the algebra record | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(2, C) ⊕ C | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(2, C) ⊕ C | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(2, C) ⊕ C | Complex dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(2, C) ⊕ C | Using the algebra record | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(2) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(2) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(2) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(3) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(3) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(3) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(4) ≅ so(6) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(4) ≅ so(6) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(4) ≅ so(6) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(5) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(5) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(5) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sp(2) ≅ so(5) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sp(2) ≅ so(5) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sp(2) ≅ so(5) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sp(3) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sp(3) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sp(3) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| so(8) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| so(8) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| so(8) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| so(10) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| so(10) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| so(10) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₆ compact | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₆ compact | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₆ compact | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₇ compact | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₇ compact | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₇ compact | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₈ compact | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₈ compact | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₈ compact | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| f₄ compact | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| f₄ compact | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| f₄ compact | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| g₂ compact | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| g₂ compact | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| g₂ compact | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(2) ⊕ su(2) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(2) ⊕ su(2) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(2) ⊕ su(2) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| u(1) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| u(1) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| u(1) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| u(1) | From algebra to group | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| u(2) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| u(2) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| u(2) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| u(2) | From algebra to group | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(2, R) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(2, R) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(2, R) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(3, R) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(3, R) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(3, R) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(2,2) ≅ so(2,4) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(2,2) ≅ so(2,4) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(2,2) ≅ so(2,4) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sp(4, R) ≅ so(2,3) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sp(4, R) ≅ so(2,3) | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sp(4, R) ≅ so(2,3) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(2, C) viewed over R | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(2, C) viewed over R | Scalar field | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(2, C) viewed over R | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(2, C) viewed over R | From algebra to group | Section 9.4: real forms and complexification |
| su(2) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(2) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(3) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(3) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(4) ≅ so(6) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(4) ≅ so(6) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(5) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(5) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sp(2) ≅ so(5) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sp(2) ≅ so(5) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sp(3) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sp(3) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| so(8) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| so(8) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| so(10) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| so(10) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₆ compact | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₆ compact | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₇ compact | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₇ compact | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₈ compact | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| e₈ compact | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| f₄ compact | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| f₄ compact | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| g₂ compact | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| g₂ compact | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(2) ⊕ su(2) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(2) ⊕ su(2) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(2, R) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(2, R) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(3, R) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(3, R) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(2,2) ≅ so(2,4) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| su(2,2) ≅ so(2,4) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sp(4, R) ≅ so(2,3) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sp(4, R) ≅ so(2,3) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(2, C) viewed over R | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| sl(2, C) viewed over R | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(4) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(4) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(4) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(4) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(4) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| PSU(4) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| PSU(4) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| PSU(4) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| PSU(4) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| PSU(4) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(5) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(5) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(5) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(5) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(5) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| U(2) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| U(2) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| U(2) | Center | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| U(2) | Fundamental group π₁ | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| U(2) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| U(2) | Global form | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| U(2) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| U(2) | Global structure and topology | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| U(2) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(4) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(4) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(4) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(4) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(4) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(5) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(5) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(5) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(5) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(5) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(6) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(6) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(6) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(6) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(6) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(2) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(2) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(2) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(2) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(2) | Symplectic notation | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(2) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(3) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(3) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(3) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(3) | Global form | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(3) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(3) | Global structure and topology | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(3) | Symplectic notation | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(3) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Spin(8) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Spin(8) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Spin(8) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Spin(8) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Spin(8) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(8) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(8) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(8) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(8) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(8) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| PSO(8) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| PSO(8) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| PSO(8) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| PSO(8) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| PSO(8) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Spin(10) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Spin(10) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Spin(10) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Spin(10) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Spin(10) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(10) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(10) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(10) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(10) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(10) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| G₂ compact | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| G₂ compact | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| G₂ compact | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| G₂ compact | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| F₄ compact | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| F₄ compact | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| F₄ compact | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| F₄ compact | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₆ compact, simply connected | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₆ compact, simply connected | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₆ compact, simply connected | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₆ compact, simply connected | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₆ compact, simply connected | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₇ compact, simply connected | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₇ compact, simply connected | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₇ compact, simply connected | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₇ compact, simply connected | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₇ compact, simply connected | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₈ compact | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₈ compact | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₈ compact | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₈ compact | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₈ compact | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₆ compact, adjoint | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₆ compact, adjoint | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₆ compact, adjoint | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₆ compact, adjoint | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₆ compact, adjoint | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₇ compact, adjoint | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₇ compact, adjoint | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₇ compact, adjoint | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₇ compact, adjoint | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| E₇ compact, adjoint | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(2, R) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(2, R) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(2, R) | Center | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(2, R) | Compact | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(2, R) | Connected | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(2, R) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(2, R) | Global form | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(2, R) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(2, R) | Maximal compact subgroup | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(2, R) | Global structure and topology | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(2, R) | Representation category and descent | Remark 36.7, printed page 196: finite-dimensional representations of the universal cover factor through SL(2,R) |
| SL(2, R) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(2, R) | Explicit homomorphism | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(2, R) | Explicit homomorphism | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(2, R) | Specified connected covering group | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| PSL(2, R) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| PSL(2, R) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| PSL(2, R) | Center | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| PSL(2, R) | Compact | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| PSL(2, R) | Connected | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| PSL(2, R) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| PSL(2, R) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| PSL(2, R) | Maximal compact subgroup | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| PSL(2, R) | Representation category and descent | Remark 36.7: finite-dimensional representations and the universal cover |
| PSL(2, R) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Universal cover of SL(2, R) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Universal cover of SL(2, R) | Fundamental group π₁ | Section 3.2: universal covering Lie groups |
| Universal cover of SL(2, R) | Compact | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Universal cover of SL(2, R) | Connected | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Universal cover of SL(2, R) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Universal cover of SL(2, R) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Universal cover of SL(2, R) | Representation category and descent | Remark 36.7, printed page 196; Exercise 11.20: finite-dimensional representations and nonlinearity of the universal cover |
| Universal cover of SL(2, R) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(3, R) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(3, R) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(3, R) | Center | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(3, R) | Compact | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(3, R) | Connected | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(3, R) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(3, R) | Global form | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(3, R) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(3, R) | Maximal compact subgroup | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(3, R) | Global structure and topology | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SL(3, R) | Representation category and descent | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 9: Lie-group and Lie-algebra representations; connected integration |
| SL(3, R) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(2,2) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(2,2) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(2,2) | Center | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(2,2) | Compact | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(2,2) | Connected | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(2,2) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(2,2) | Global form | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(2,2) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(2,2) | Maximal compact subgroup | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(2,2) | Global structure and topology | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(2,2) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(2,2) | Explicit homomorphism | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(4, R) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(4, R) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(4, R) | Center | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(4, R) | Compact | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(4, R) | Connected | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(4, R) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(4, R) | Global form | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(4, R) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(4, R) | Maximal compact subgroup | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(4, R) | Global structure and topology | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(4, R) | Symplectic notation | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(4, R) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(4, R) | Explicit homomorphism | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(1,3) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(1,3) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(1,3) | Center | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(1,3) | Compact | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(1,3) | Connected | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(1,3) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(1,3) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(1,3) | Maximal compact subgroup | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(1,3) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(2,4) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(2,4) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(2,4) | Center | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(2,4) | Compact | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(2,4) | Connected | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(2,4) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(2,4) | Global form | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(2,4) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(2,4) | Maximal compact subgroup | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(2,4) | Global structure and topology | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(2,4) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(2,4) | Explicit homomorphism | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(2,4) | Specified connected covering group | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(2,3) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(2,3) | Real dimension | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(2,3) | Center | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(2,3) | Compact | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(2,3) | Connected | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(2,3) | Abelian | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(2,3) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(2,3) | Maximal compact subgroup | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO₀(2,3) | Real Lie algebra | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(2) → SO(3) | Global and representation consequences | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ↪ SU(2) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ↪ SU(2) | Domain | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ↪ SU(2) | Codomain | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ↪ SU(2) | Specified homomorphism | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ↪ SU(2) | Kernel | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ↪ SU(2) | Image | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ↪ SU(2) | Injective | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ↪ SU(2) | Surjective | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ↪ SU(2) | Global and representation consequences | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ↪ SU(2) | Global and representation consequences | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ↪ SU(2) | Domain group | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ↪ SU(2) | Codomain group | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ↪ SU(2) | Associated homogeneous quotient | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ≅ SO(2) ↪ SO(3) | Definition | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ≅ SO(2) ↪ SO(3) | Domain | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ≅ SO(2) ↪ SO(3) | Codomain | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ≅ SO(2) ↪ SO(3) | Specified homomorphism | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ≅ SO(2) ↪ SO(3) | Kernel | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ≅ SO(2) ↪ SO(3) | Image | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ≅ SO(2) ↪ SO(3) | Injective | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ≅ SO(2) ↪ SO(3) | Surjective | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ≅ SO(2) ↪ SO(3) | Global and representation consequences | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ≅ SO(2) ↪ SO(3) | Global and representation consequences | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ≅ SO(2) ↪ SO(3) | Domain group | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ≅ SO(2) ↪ SO(3) | Codomain group | Sections 6.1–6.3: classical matrix groups and their Lie algebras · Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| U(1) ≅ SO(2) ↪ SO(3) | Associated homogeneous quotient | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SL(2, C) → SO₀(1,3) | Global and representation consequences | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Universal SL(2,R) cover → SL(2,R) | Definition | Section 3.2: covering Lie groups |
| Universal SL(2,R) cover → SL(2,R) | Domain | Section 3.2: covering Lie groups |
| Universal SL(2,R) cover → SL(2,R) | Codomain | Section 3.2: covering Lie groups |
| Universal SL(2,R) cover → SL(2,R) | Specified homomorphism | Section 3.2: covering Lie groups |
| Universal SL(2,R) cover → SL(2,R) | Kernel | Section 3.2: covering Lie groups |
| Universal SL(2,R) cover → SL(2,R) | Image | Section 3.2: covering Lie groups |
| Universal SL(2,R) cover → SL(2,R) | Injective | Section 3.2: covering Lie groups |
| Universal SL(2,R) cover → SL(2,R) | Surjective | Section 3.2: covering Lie groups |
| Universal SL(2,R) cover → SL(2,R) | Covering degree | Section 3.2: covering Lie groups |
| Universal SL(2,R) cover → SL(2,R) | Global and representation consequences | Section 3.2: covering Lie groups |
| Universal SL(2,R) cover → SL(2,R) | Domain group | Section 3.2: covering Lie groups |
| Universal SL(2,R) cover → SL(2,R) | Codomain group | Section 3.2: covering Lie groups |
| Sp(4, R) → SO₀(2,3) | Definition | Section 31.1, printed page 165: the primitive exterior-square symplectic module |
| Sp(4, R) → SO₀(2,3) | Domain | Section 31.1, printed page 165: the primitive exterior-square symplectic module |
| Sp(4, R) → SO₀(2,3) | Codomain | Section 31.1, printed page 165: the primitive exterior-square symplectic module |
| Sp(4, R) → SO₀(2,3) | Specified homomorphism | Section 31.1, printed page 165: the primitive exterior-square symplectic module |
| Sp(4, R) → SO₀(2,3) | Kernel | Section 31.1, printed page 165: the primitive exterior-square symplectic module |
| Sp(4, R) → SO₀(2,3) | Image | Section 31.1, printed page 165: the primitive exterior-square symplectic module |
| Sp(4, R) → SO₀(2,3) | Injective | Section 31.1, printed page 165: the primitive exterior-square symplectic module |
| Sp(4, R) → SO₀(2,3) | Surjective | Section 31.1, printed page 165: the primitive exterior-square symplectic module |
| Sp(4, R) → SO₀(2,3) | Covering degree | Section 31.1, printed page 165: the primitive exterior-square symplectic module |
| Sp(4, R) → SO₀(2,3) | Global and representation consequences | Section 31.1, printed page 165: the primitive exterior-square symplectic module |
| Sp(4, R) → SO₀(2,3) | Global and representation consequences | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| Sp(4, R) → SO₀(2,3) | Domain group | Section 31.1, printed page 165: the primitive exterior-square symplectic module |
| Sp(4, R) → SO₀(2,3) | Codomain group | Section 31.1, printed page 165: the primitive exterior-square symplectic module |
| SU(2)/U(1) ≅ S² | Definition | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SU(2)/U(1) ≅ S² | Real dimension | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SU(2)/U(1) ≅ S² | Manifold | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SU(2)/U(1) ≅ S² | Fiber angle period | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SU(2)/U(1) ≅ S² | Boundary π₂(S²) → π₁(U(1)) | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SU(2)/U(1) ≅ S² | The exact sequence with generators | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SU(2)/U(1) ≅ S² | The exact sequence with generators | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SU(2)/U(1) ≅ S² | Total group | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SU(2)/U(1) ≅ S² | Embedded torus | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SU(2)/U(1) ≅ S² | Normalized subgroup map | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SU(2)/U(1) ≅ S² | Same S², different fibration | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SO(3)/SO(2) ≅ S² | Definition | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SO(3)/SO(2) ≅ S² | Real dimension | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SO(3)/SO(2) ≅ S² | Manifold | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SO(3)/SO(2) ≅ S² | Physical rotation angle period | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(3)/SO(2) ≅ S² | Inclusion π₁(SO(2)) → π₁(SO(3)) | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SO(3)/SO(2) ≅ S² | Boundary π₂(S²) → π₁(SO(2)) | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SO(3)/SO(2) ≅ S² | The exact sequence with generators | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SO(3)/SO(2) ≅ S² | The exact sequence with generators | Sections 6.1–6.3: classical matrix groups and their Lie algebras |
| SO(3)/SO(2) ≅ S² | Total group | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SO(3)/SO(2) ≅ S² | Circle realized as SO(2) | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SO(3)/SO(2) ≅ S² | Normalized rotational subgroup map | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |
| SO(3)/SO(2) ≅ S² | Same S², different fibration | Section 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³ |