lecture-notes
Conrad and Landesman · Compact Lie groups
Brian Conrad · Aaron Landesman
Use in this reference: Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted.
Relevant material: Numbered lectures and appendices; page numbers refer to the printed pagination.
Dependent assertions
| Record | Assertion | Source locator |
|---|---|---|
| SU(2) | Fundamental group | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SU(2) | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SU(2) | Rank | Lecture 25 and Appendix T: root systems and classical root computations |
| SO(3) | Fundamental group | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO(3) | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SO(3) | Rank | Lecture 25 and Appendix T: root systems and classical root computations |
| SO(3) | Explicit homomorphism | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SU(3) | Fundamental group | Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SU(3) | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SU(3) | Rank | Lecture 25 and Appendix T: root systems and classical root computations |
| SU(3) | Topology and global form | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SU(3) | Explicit homomorphism | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| PSU(3) | Definition | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| PSU(3) | Fundamental group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| PSU(3) | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| PSU(3) | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| PSU(3) | Rank | Lecture 25 and Appendix T: root systems and classical root computations |
| PSU(3) | Topology and global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| PSU(3) | Related group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| PSU(3) | Explicit homomorphism | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| U(1) | Representations | Lecture 27: characters of a torus; U(1) character lattice Z |
| U(1) | In physics | Lecture 27: characters of a torus; U(1) character lattice Z |
| U(1) | Explicit homomorphism | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(4) | Fundamental group | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(4) | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Spin(4) | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Spin(4) | Representations | Lecture 27: representations of a product group |
| SL(2,ℂ) | Representations | Lectures 27–28: highest-weight classification and fundamental representations |
| SL(2,ℂ) | The Lorentz algebra is a realification | Lectures 27–28: highest-weight classification and fundamental representations |
| Aᵣ | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| Aᵣ | Rank | Lecture 25 and Appendix T: root systems and classical root computations |
| Aᵣ | Complex algebra dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| Aᵣ | Compact simply connected group | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Aᵣ | Algebra, real form, global group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| Aᵣ | Algebra, real form, global group | Appendix V: finite fundamental groups; Remark T.2.1, printed page 262: outer automorphisms, central kernels, and D₄ triality |
| Aᵣ | Representation convention | Lectures 27–28: highest-weight classification and fundamental representations |
| Aᵣ | Rank conventions and coincidences | Appendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ triality |
| Aᵣ | Concrete algebra record | Lecture 25 and Appendix T: root systems and classical root computations |
| Aᵣ | Concrete algebra record | Lecture 25 and Appendix T: root systems and classical root computations |
| Aᵣ | Concrete algebra record | Lecture 25 and Appendix T: root systems and classical root computations |
| Aᵣ | Concrete algebra record | Lecture 25 and Appendix T: root systems and classical root computations |
| Bᵣ | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| Bᵣ | Rank | Lecture 25 and Appendix T: root systems and classical root computations |
| Bᵣ | Complex algebra dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| Bᵣ | Compact simply connected group | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Bᵣ | Algebra, real form, global group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| Bᵣ | Algebra, real form, global group | Appendix V: finite fundamental groups; Remark T.2.1, printed page 262: outer automorphisms, central kernels, and D₄ triality |
| Bᵣ | Representation convention | Lectures 27–28: highest-weight classification and fundamental representations |
| Bᵣ | Rank conventions and coincidences | Appendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ triality |
| Bᵣ | Rank conventions and coincidences | Appendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ triality |
| Bᵣ | Concrete algebra record | Lecture 25 and Appendix T: root systems and classical root computations |
| Cᵣ | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| Cᵣ | Rank | Lecture 25 and Appendix T: root systems and classical root computations |
| Cᵣ | Complex algebra dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| Cᵣ | Compact simply connected group | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Cᵣ | Algebra, real form, global group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| Cᵣ | Algebra, real form, global group | Appendix V: finite fundamental groups; Remark T.2.1, printed page 262: outer automorphisms, central kernels, and D₄ triality |
| Cᵣ | Representation convention | Lectures 27–28: highest-weight classification and fundamental representations |
| Cᵣ | Rank conventions and coincidences | Appendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ triality |
| Cᵣ | Rank conventions and coincidences | Appendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ triality |
| Cᵣ | Concrete algebra record | Lecture 25 and Appendix T: root systems and classical root computations |
| Dᵣ | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| Dᵣ | Rank | Lecture 25 and Appendix T: root systems and classical root computations |
| Dᵣ | Complex algebra dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| Dᵣ | Compact simply connected group | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Dᵣ | Algebra, real form, global group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| Dᵣ | Algebra, real form, global group | Appendix V: finite fundamental groups; Remark T.2.1, printed page 262: outer automorphisms, central kernels, and D₄ triality |
| Dᵣ | Representation convention | Lectures 27–28: highest-weight classification and fundamental representations |
| Dᵣ | Rank conventions and coincidences | Appendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ triality |
| Dᵣ | Rank conventions and coincidences | Appendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ triality |
| Dᵣ | Rank conventions and coincidences | Appendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ triality |
| Dᵣ | Rank conventions and coincidences | Appendix T, Sections T.1–T.4: classical root systems; Remark T.2.1, printed page 262: D₄ triality |
| Dᵣ | Concrete algebra record | Lecture 25 and Appendix T: root systems and classical root computations |
| Dᵣ | Concrete algebra record | Lecture 25 and Appendix T: root systems and classical root computations |
| E₆ | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| E₆ | Rank | Lecture 25 and Appendix T: root systems and classical root computations |
| E₆ | Complex algebra dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| E₆ | Compact simply connected group | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| E₆ | Algebra, real form, global group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| E₆ | Algebra, real form, global group | Appendix V: finite fundamental groups; Remark T.2.1, printed page 262: outer automorphisms, central kernels, and D₄ triality |
| E₆ | Representation convention | Lectures 27–28: highest-weight classification and fundamental representations |
| E₆ | Concrete algebra record | Lecture 25 and Appendix T: root systems and classical root computations |
| E₇ | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| E₇ | Rank | Lecture 25 and Appendix T: root systems and classical root computations |
| E₇ | Complex algebra dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| E₇ | Compact simply connected group | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| E₇ | Algebra, real form, global group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| E₇ | Algebra, real form, global group | Appendix V: finite fundamental groups; Remark T.2.1, printed page 262: outer automorphisms, central kernels, and D₄ triality |
| E₇ | Representation convention | Lectures 27–28: highest-weight classification and fundamental representations |
| E₇ | Concrete algebra record | Lecture 25 and Appendix T: root systems and classical root computations |
| E₈ | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| E₈ | Rank | Lecture 25 and Appendix T: root systems and classical root computations |
| E₈ | Complex algebra dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| E₈ | Compact simply connected group | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| E₈ | Algebra, real form, global group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| E₈ | Algebra, real form, global group | Appendix V: finite fundamental groups; Remark T.2.1, printed page 262: outer automorphisms, central kernels, and D₄ triality |
| E₈ | Representation convention | Lectures 27–28: highest-weight classification and fundamental representations |
| E₈ | Concrete algebra record | Lecture 25 and Appendix T: root systems and classical root computations |
| F₄ | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| F₄ | Rank | Lecture 25 and Appendix T: root systems and classical root computations |
| F₄ | Complex algebra dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| F₄ | Compact simply connected group | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| F₄ | Algebra, real form, global group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| F₄ | Algebra, real form, global group | Appendix V: finite fundamental groups; Remark T.2.1, printed page 262: outer automorphisms, central kernels, and D₄ triality |
| F₄ | Representation convention | Lectures 27–28: highest-weight classification and fundamental representations |
| F₄ | Concrete algebra record | Lecture 25 and Appendix T: root systems and classical root computations |
| G₂ | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| G₂ | Rank | Lecture 25 and Appendix T: root systems and classical root computations |
| G₂ | Complex algebra dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| G₂ | Compact simply connected group | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| G₂ | Algebra, real form, global group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| G₂ | Algebra, real form, global group | Appendix V: finite fundamental groups; Remark T.2.1, printed page 262: outer automorphisms, central kernels, and D₄ triality |
| G₂ | Representation convention | Lectures 27–28: highest-weight classification and fundamental representations |
| G₂ | Concrete algebra record | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, C) | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, C) | Complex dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, C) | Complex rank | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, C) | Root-system type | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, C) | Simple over C | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, C) | Using the algebra record | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(2, C) | Using the algebra record | Lectures 27–28: highest-weight classification and fundamental representations |
| sl(2, C) | Simple-type classification | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(3, C) | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(3, C) | Complex dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(3, C) | Complex rank | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(3, C) | Root-system type | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(3, C) | Simple over C | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(3, C) | Using the algebra record | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(3, C) | Using the algebra record | Lectures 27–28: highest-weight classification and fundamental representations |
| sl(3, C) | Simple-type classification | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(4, C) | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(4, C) | Complex dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(4, C) | Complex rank | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(4, C) | Root-system type | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(4, C) | Simple over C | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(4, C) | Using the algebra record | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(4, C) | Using the algebra record | Lectures 27–28: highest-weight classification and fundamental representations |
| sl(4, C) | Simple-type classification | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(5, C) | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(5, C) | Complex dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(5, C) | Complex rank | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(5, C) | Root-system type | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(5, C) | Simple over C | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(5, C) | Using the algebra record | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(5, C) | Using the algebra record | Lectures 27–28: highest-weight classification and fundamental representations |
| sl(5, C) | Simple-type classification | Lecture 25 and Appendix T: root systems and classical root computations |
| so(5, C) ≅ sp(4, C) | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| so(5, C) ≅ sp(4, C) | Complex dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| so(5, C) ≅ sp(4, C) | Complex rank | Lecture 25 and Appendix T: root systems and classical root computations |
| so(5, C) ≅ sp(4, C) | Root-system type | Lecture 25 and Appendix T: root systems and classical root computations |
| so(5, C) ≅ sp(4, C) | Simple over C | Lecture 25 and Appendix T: root systems and classical root computations |
| so(5, C) ≅ sp(4, C) | Using the algebra record | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| so(5, C) ≅ sp(4, C) | Using the algebra record | Lectures 27–28: highest-weight classification and fundamental representations |
| so(5, C) ≅ sp(4, C) | Simple-type classification | Lecture 25 and Appendix T: root systems and classical root computations |
| sp(6, C) | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| sp(6, C) | Complex dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| sp(6, C) | Complex rank | Lecture 25 and Appendix T: root systems and classical root computations |
| sp(6, C) | Root-system type | Lecture 25 and Appendix T: root systems and classical root computations |
| sp(6, C) | Simple over C | Lecture 25 and Appendix T: root systems and classical root computations |
| sp(6, C) | Using the algebra record | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sp(6, C) | Using the algebra record | Lectures 27–28: highest-weight classification and fundamental representations |
| sp(6, C) | Simple-type classification | Lecture 25 and Appendix T: root systems and classical root computations |
| so(8, C) | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| so(8, C) | Complex dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| so(8, C) | Complex rank | Lecture 25 and Appendix T: root systems and classical root computations |
| so(8, C) | Root-system type | Lecture 25 and Appendix T: root systems and classical root computations |
| so(8, C) | Simple over C | Lecture 25 and Appendix T: root systems and classical root computations |
| so(8, C) | Using the algebra record | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| so(8, C) | Using the algebra record | Lectures 27–28: highest-weight classification and fundamental representations |
| so(8, C) | Simple-type classification | Lecture 25 and Appendix T: root systems and classical root computations |
| so(10, C) | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| so(10, C) | Complex dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| so(10, C) | Complex rank | Lecture 25 and Appendix T: root systems and classical root computations |
| so(10, C) | Root-system type | Lecture 25 and Appendix T: root systems and classical root computations |
| so(10, C) | Simple over C | Lecture 25 and Appendix T: root systems and classical root computations |
| so(10, C) | Using the algebra record | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| so(10, C) | Using the algebra record | Lectures 27–28: highest-weight classification and fundamental representations |
| so(10, C) | Simple-type classification | Lecture 25 and Appendix T: root systems and classical root computations |
| e₆(C) | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| e₆(C) | Complex dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| e₆(C) | Complex rank | Lecture 25 and Appendix T: root systems and classical root computations |
| e₆(C) | Root-system type | Lecture 25 and Appendix T: root systems and classical root computations |
| e₆(C) | Simple over C | Lecture 25 and Appendix T: root systems and classical root computations |
| e₆(C) | Using the algebra record | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| e₆(C) | Using the algebra record | Lectures 27–28: highest-weight classification and fundamental representations |
| e₆(C) | Simple-type classification | Lecture 25 and Appendix T: root systems and classical root computations |
| e₇(C) | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| e₇(C) | Complex dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| e₇(C) | Complex rank | Lecture 25 and Appendix T: root systems and classical root computations |
| e₇(C) | Root-system type | Lecture 25 and Appendix T: root systems and classical root computations |
| e₇(C) | Simple over C | Lecture 25 and Appendix T: root systems and classical root computations |
| e₇(C) | Using the algebra record | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| e₇(C) | Using the algebra record | Lectures 27–28: highest-weight classification and fundamental representations |
| e₇(C) | Simple-type classification | Lecture 25 and Appendix T: root systems and classical root computations |
| e₈(C) | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| e₈(C) | Complex dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| e₈(C) | Complex rank | Lecture 25 and Appendix T: root systems and classical root computations |
| e₈(C) | Root-system type | Lecture 25 and Appendix T: root systems and classical root computations |
| e₈(C) | Simple over C | Lecture 25 and Appendix T: root systems and classical root computations |
| e₈(C) | Using the algebra record | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| e₈(C) | Using the algebra record | Lectures 27–28: highest-weight classification and fundamental representations |
| e₈(C) | Simple-type classification | Lecture 25 and Appendix T: root systems and classical root computations |
| f₄(C) | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| f₄(C) | Complex dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| f₄(C) | Complex rank | Lecture 25 and Appendix T: root systems and classical root computations |
| f₄(C) | Root-system type | Lecture 25 and Appendix T: root systems and classical root computations |
| f₄(C) | Simple over C | Lecture 25 and Appendix T: root systems and classical root computations |
| f₄(C) | Using the algebra record | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| f₄(C) | Using the algebra record | Lectures 27–28: highest-weight classification and fundamental representations |
| f₄(C) | Simple-type classification | Lecture 25 and Appendix T: root systems and classical root computations |
| g₂(C) | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| g₂(C) | Complex dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| g₂(C) | Complex rank | Lecture 25 and Appendix T: root systems and classical root computations |
| g₂(C) | Root-system type | Lecture 25 and Appendix T: root systems and classical root computations |
| g₂(C) | Simple over C | Lecture 25 and Appendix T: root systems and classical root computations |
| g₂(C) | Using the algebra record | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| g₂(C) | Using the algebra record | Lectures 27–28: highest-weight classification and fundamental representations |
| g₂(C) | Simple-type classification | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, C) ⊕ sl(2, C) | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, C) ⊕ sl(2, C) | Complex dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, C) ⊕ sl(2, C) | Complex rank | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, C) ⊕ sl(2, C) | Root-system type | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, C) ⊕ sl(2, C) | Simple over C | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, C) ⊕ sl(2, C) | Using the algebra record | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| C (zero bracket) | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| C (zero bracket) | Complex dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| C (zero bracket) | Complex rank | Lecture 25 and Appendix T: root systems and classical root computations |
| C (zero bracket) | Root-system type | Lecture 25 and Appendix T: root systems and classical root computations |
| C (zero bracket) | Simple over C | Lecture 25 and Appendix T: root systems and classical root computations |
| C (zero bracket) | Using the algebra record | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(2, C) ⊕ C | Definition | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, C) ⊕ C | Complex dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, C) ⊕ C | Complex rank | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, C) ⊕ C | Root-system type | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, C) ⊕ C | Simple over C | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, C) ⊕ C | Using the algebra record | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(2) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| su(2) | Absolute rank (of complexification) | Lecture 25 and Appendix T: root systems and classical root computations |
| su(2) | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(2) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(2) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(3) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| su(3) | Absolute rank (of complexification) | Lecture 25 and Appendix T: root systems and classical root computations |
| su(3) | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(3) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(3) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(4) ≅ so(6) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| su(4) ≅ so(6) | Absolute rank (of complexification) | Lecture 25 and Appendix T: root systems and classical root computations |
| su(4) ≅ so(6) | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(4) ≅ so(6) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(4) ≅ so(6) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(4) ≅ so(6) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(5) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| su(5) | Absolute rank (of complexification) | Lecture 25 and Appendix T: root systems and classical root computations |
| su(5) | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(5) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sp(2) ≅ so(5) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| sp(2) ≅ so(5) | Absolute rank (of complexification) | Lecture 25 and Appendix T: root systems and classical root computations |
| sp(2) ≅ so(5) | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sp(2) ≅ so(5) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sp(2) ≅ so(5) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sp(3) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| sp(3) | Absolute rank (of complexification) | Lecture 25 and Appendix T: root systems and classical root computations |
| sp(3) | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sp(3) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| so(8) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| so(8) | Absolute rank (of complexification) | Lecture 25 and Appendix T: root systems and classical root computations |
| so(8) | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| so(8) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| so(8) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| so(8) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| so(10) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| so(10) | Absolute rank (of complexification) | Lecture 25 and Appendix T: root systems and classical root computations |
| so(10) | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| so(10) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| so(10) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| e₆ compact | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| e₆ compact | Absolute rank (of complexification) | Lecture 25 and Appendix T: root systems and classical root computations |
| e₆ compact | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| e₆ compact | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| e₆ compact | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| e₇ compact | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| e₇ compact | Absolute rank (of complexification) | Lecture 25 and Appendix T: root systems and classical root computations |
| e₇ compact | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| e₇ compact | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| e₇ compact | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| e₈ compact | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| e₈ compact | Absolute rank (of complexification) | Lecture 25 and Appendix T: root systems and classical root computations |
| e₈ compact | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| e₈ compact | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| f₄ compact | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| f₄ compact | Absolute rank (of complexification) | Lecture 25 and Appendix T: root systems and classical root computations |
| f₄ compact | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| f₄ compact | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| g₂ compact | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| g₂ compact | Absolute rank (of complexification) | Lecture 25 and Appendix T: root systems and classical root computations |
| g₂ compact | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| g₂ compact | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(2) ⊕ su(2) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| su(2) ⊕ su(2) | Absolute rank (of complexification) | Lecture 25 and Appendix T: root systems and classical root computations |
| su(2) ⊕ su(2) | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(2) ⊕ su(2) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(2) ⊕ su(2) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| u(1) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| u(1) | Absolute rank (of complexification) | Lecture 25 and Appendix T: root systems and classical root computations |
| u(1) | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| u(1) | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| u(1) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| u(2) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| u(2) | Absolute rank (of complexification) | Lecture 25 and Appendix T: root systems and classical root computations |
| u(2) | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| u(2) | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| u(2) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(2, R) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, R) | Absolute rank (of complexification) | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, R) | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(2, R) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(2, R) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(2, R) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(3, R) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(3, R) | Absolute rank (of complexification) | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(3, R) | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(3, R) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(2,2) ≅ so(2,4) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| su(2,2) ≅ so(2,4) | Absolute rank (of complexification) | Lecture 25 and Appendix T: root systems and classical root computations |
| su(2,2) ≅ so(2,4) | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(2,2) ≅ so(2,4) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(2,2) ≅ so(2,4) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sp(4, R) ≅ so(2,3) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| sp(4, R) ≅ so(2,3) | Absolute rank (of complexification) | Lecture 25 and Appendix T: root systems and classical root computations |
| sp(4, R) ≅ so(2,3) | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sp(4, R) ≅ so(2,3) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sp(4, R) ≅ so(2,3) | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(2, C) viewed over R | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, C) viewed over R | Absolute rank (of complexification) | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, C) viewed over R | From algebra to group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(2, C) viewed over R | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(2, C) viewed over R | Curated integrating group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(2) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| su(2) | One algebra, several integrations | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(2) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(2) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(3) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| su(3) | One algebra, several integrations | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(3) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(3) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(4) ≅ so(6) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| su(4) ≅ so(6) | One algebra, several integrations | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(4) ≅ so(6) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(4) ≅ so(6) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(4) ≅ so(6) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(5) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| su(5) | One algebra, several integrations | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(5) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sp(2) ≅ so(5) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| sp(2) ≅ so(5) | One algebra, several integrations | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sp(2) ≅ so(5) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sp(2) ≅ so(5) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sp(3) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| sp(3) | One algebra, several integrations | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sp(3) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| so(8) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| so(8) | One algebra, several integrations | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| so(8) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| so(8) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| so(8) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| so(10) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| so(10) | One algebra, several integrations | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| so(10) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| so(10) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| e₆ compact | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| e₆ compact | One algebra, several integrations | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| e₆ compact | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| e₆ compact | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| e₇ compact | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| e₇ compact | One algebra, several integrations | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| e₇ compact | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| e₇ compact | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| e₈ compact | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| e₈ compact | One algebra, several integrations | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| e₈ compact | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| f₄ compact | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| f₄ compact | One algebra, several integrations | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| f₄ compact | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| g₂ compact | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| g₂ compact | One algebra, several integrations | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| g₂ compact | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(2) ⊕ su(2) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| su(2) ⊕ su(2) | One algebra, several integrations | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(2) ⊕ su(2) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(2) ⊕ su(2) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(2, R) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, R) | One algebra, several integrations | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(2, R) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(2, R) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(2, R) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(3, R) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(3, R) | One algebra, several integrations | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(3, R) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(2,2) ≅ so(2,4) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| su(2,2) ≅ so(2,4) | One algebra, several integrations | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(2,2) ≅ so(2,4) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| su(2,2) ≅ so(2,4) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sp(4, R) ≅ so(2,3) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| sp(4, R) ≅ so(2,3) | One algebra, several integrations | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sp(4, R) ≅ so(2,3) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sp(4, R) ≅ so(2,3) | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(2, C) viewed over R | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| sl(2, C) viewed over R | One algebra, several integrations | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(2, C) viewed over R | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| sl(2, C) viewed over R | Curated global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(4) | Definition | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SU(4) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| SU(4) | Compact rank | Lecture 25 and Appendix T: root systems and classical root computations |
| SU(4) | Center | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(4) | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SU(4) | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SU(4) | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SU(4) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| SU(4) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| SU(4) | Global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(4) | Global structure and topology | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(4) | Representation category and descent | Lectures 27–28: highest-weight classification and fundamental representations |
| SU(4) | Explicit homomorphism | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(4) | Explicit homomorphism | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| PSU(4) | Definition | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| PSU(4) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| PSU(4) | Compact rank | Lecture 25 and Appendix T: root systems and classical root computations |
| PSU(4) | Center | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| PSU(4) | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| PSU(4) | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| PSU(4) | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| PSU(4) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| PSU(4) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| PSU(4) | Global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| PSU(4) | Global structure and topology | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| PSU(4) | Representation category and descent | Lectures 27–28: highest-weight classification and fundamental representations |
| PSU(4) | Explicit homomorphism | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| PSU(4) | Specified connected covering group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(5) | Definition | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SU(5) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| SU(5) | Compact rank | Lecture 25 and Appendix T: root systems and classical root computations |
| SU(5) | Center | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(5) | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SU(5) | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SU(5) | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SU(5) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| SU(5) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| SU(5) | Global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(5) | Global structure and topology | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(5) | Representation category and descent | Lectures 27–28: highest-weight classification and fundamental representations |
| U(2) | Definition | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| U(2) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| U(2) | Compact rank | Lecture 25 and Appendix T: root systems and classical root computations |
| U(2) | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| U(2) | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| U(2) | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| U(2) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| U(2) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| U(2) | Global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| U(2) | Global structure and topology | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| U(2) | Representation category and descent | Lectures 27–28: highest-weight classification and fundamental representations |
| SO(4) | Definition | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SO(4) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| SO(4) | Compact rank | Lecture 25 and Appendix T: root systems and classical root computations |
| SO(4) | Center | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SO(4) | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO(4) | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SO(4) | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SO(4) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| SO(4) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| SO(4) | Global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO(4) | Global structure and topology | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO(4) | Representation category and descent | Lectures 27–28: highest-weight classification and fundamental representations |
| SO(4) | Explicit homomorphism | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO(4) | Specified connected covering group | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO(5) | Definition | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SO(5) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| SO(5) | Compact rank | Lecture 25 and Appendix T: root systems and classical root computations |
| SO(5) | Center | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SO(5) | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO(5) | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SO(5) | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SO(5) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| SO(5) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| SO(5) | Global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO(5) | Global structure and topology | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO(5) | Representation category and descent | Lectures 27–28: highest-weight classification and fundamental representations |
| SO(5) | Explicit homomorphism | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO(5) | Specified connected covering group | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO(6) | Definition | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SO(6) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| SO(6) | Compact rank | Lecture 25 and Appendix T: root systems and classical root computations |
| SO(6) | Center | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SO(6) | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO(6) | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SO(6) | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SO(6) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| SO(6) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| SO(6) | Global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO(6) | Global structure and topology | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO(6) | Representation category and descent | Lectures 27–28: highest-weight classification and fundamental representations |
| SO(6) | Explicit homomorphism | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO(6) | Specified connected covering group | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Sp(2) | Definition | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Sp(2) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| Sp(2) | Compact rank | Lecture 25 and Appendix T: root systems and classical root computations |
| Sp(2) | Center | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| Sp(2) | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Sp(2) | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Sp(2) | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Sp(2) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| Sp(2) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| Sp(2) | Global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Sp(2) | Global structure and topology | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Sp(2) | Representation category and descent | Lectures 27–28: highest-weight classification and fundamental representations |
| Sp(2) | Explicit homomorphism | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Sp(3) | Definition | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Sp(3) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| Sp(3) | Compact rank | Lecture 25 and Appendix T: root systems and classical root computations |
| Sp(3) | Center | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| Sp(3) | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Sp(3) | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Sp(3) | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Sp(3) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| Sp(3) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| Sp(3) | Global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Sp(3) | Global structure and topology | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Sp(3) | Representation category and descent | Lectures 27–28: highest-weight classification and fundamental representations |
| Spin(8) | Definition | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Spin(8) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| Spin(8) | Compact rank | Lecture 25 and Appendix T: root systems and classical root computations |
| Spin(8) | Center | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| Spin(8) | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(8) | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Spin(8) | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Spin(8) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| Spin(8) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| Spin(8) | Global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 · Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| Spin(8) | Global structure and topology | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 · Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| Spin(8) | Representation category and descent | Appendix P: spin representations; Remark T.2.1, printed page 262: D₄ triality and central kernels |
| Spin(8) | Explicit homomorphism | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 · Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SO(8) | Definition | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SO(8) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| SO(8) | Compact rank | Lecture 25 and Appendix T: root systems and classical root computations |
| SO(8) | Center | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SO(8) | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO(8) | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SO(8) | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SO(8) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| SO(8) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| SO(8) | Global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 · Appendix T.2, printed pages 261–262, especially Remark T.2.1: center, intermediate quotients, and triality |
| SO(8) | Global structure and topology | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 · Appendix T.2, printed pages 261–262, especially Remark T.2.1: center, intermediate quotients, and triality |
| SO(8) | Representation category and descent | Lectures 27–28: highest-weight classification and fundamental representations |
| SO(8) | Explicit homomorphism | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 · Appendix T.2, printed pages 261–262, especially Remark T.2.1: center, intermediate quotients, and triality |
| SO(8) | Specified connected covering group | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 · Appendix T.2, printed pages 261–262, especially Remark T.2.1: center, intermediate quotients, and triality |
| PSO(8) | Definition | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| PSO(8) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| PSO(8) | Compact rank | Lecture 25 and Appendix T: root systems and classical root computations |
| PSO(8) | Center | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| PSO(8) | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| PSO(8) | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| PSO(8) | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| PSO(8) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| PSO(8) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| PSO(8) | Global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| PSO(8) | Global structure and topology | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| PSO(8) | Representation category and descent | Appendix T.2, printed pages 261–262, especially Remark T.2.1: D₄ center, central kernels, and triality |
| PSO(8) | Specified connected covering group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| Spin(10) | Definition | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Spin(10) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| Spin(10) | Compact rank | Lecture 25 and Appendix T: root systems and classical root computations |
| Spin(10) | Center | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| Spin(10) | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(10) | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Spin(10) | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| Spin(10) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| Spin(10) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| Spin(10) | Global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(10) | Global structure and topology | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(10) | Representation category and descent | Appendix P: Clifford modules and half-spin representations; dim S± = 2^(r−1) in dimension 2r |
| Spin(10) | Explicit homomorphism | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO(10) | Definition | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SO(10) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| SO(10) | Compact rank | Lecture 25 and Appendix T: root systems and classical root computations |
| SO(10) | Center | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SO(10) | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO(10) | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SO(10) | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| SO(10) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| SO(10) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| SO(10) | Global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO(10) | Global structure and topology | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO(10) | Representation category and descent | Lectures 27–28: highest-weight classification and fundamental representations |
| SO(10) | Explicit homomorphism | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO(10) | Specified connected covering group | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| G₂ compact | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| G₂ compact | Compact rank | Lecture 25 and Appendix T: root systems and classical root computations |
| G₂ compact | Center | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| G₂ compact | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| G₂ compact | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| G₂ compact | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| G₂ compact | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| G₂ compact | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| G₂ compact | Global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| G₂ compact | Global structure and topology | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| F₄ compact | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| F₄ compact | Compact rank | Lecture 25 and Appendix T: root systems and classical root computations |
| F₄ compact | Center | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| F₄ compact | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| F₄ compact | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| F₄ compact | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| F₄ compact | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| F₄ compact | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| F₄ compact | Global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| F₄ compact | Global structure and topology | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| E₆ compact, simply connected | Definition | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| E₆ compact, simply connected | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| E₆ compact, simply connected | Compact rank | Lecture 25 and Appendix T: root systems and classical root computations |
| E₆ compact, simply connected | Center | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| E₆ compact, simply connected | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| E₆ compact, simply connected | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| E₆ compact, simply connected | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| E₆ compact, simply connected | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| E₆ compact, simply connected | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| E₆ compact, simply connected | Global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| E₆ compact, simply connected | Global structure and topology | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| E₆ compact, simply connected | Representation category and descent | Lectures 27–28: highest-weight classification and fundamental representations |
| E₇ compact, simply connected | Definition | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| E₇ compact, simply connected | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| E₇ compact, simply connected | Compact rank | Lecture 25 and Appendix T: root systems and classical root computations |
| E₇ compact, simply connected | Center | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| E₇ compact, simply connected | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| E₇ compact, simply connected | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| E₇ compact, simply connected | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| E₇ compact, simply connected | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| E₇ compact, simply connected | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| E₇ compact, simply connected | Global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| E₇ compact, simply connected | Global structure and topology | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| E₇ compact, simply connected | Representation category and descent | Lectures 27–28: highest-weight classification and fundamental representations |
| E₈ compact | Definition | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| E₈ compact | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| E₈ compact | Compact rank | Lecture 25 and Appendix T: root systems and classical root computations |
| E₈ compact | Center | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| E₈ compact | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| E₈ compact | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| E₈ compact | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| E₈ compact | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| E₈ compact | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| E₈ compact | Global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| E₈ compact | Global structure and topology | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| E₈ compact | Representation category and descent | Lectures 27–28: highest-weight classification and fundamental representations |
| E₆ compact, adjoint | Definition | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| E₆ compact, adjoint | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| E₆ compact, adjoint | Compact rank | Lecture 25 and Appendix T: root systems and classical root computations |
| E₆ compact, adjoint | Center | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| E₆ compact, adjoint | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| E₆ compact, adjoint | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| E₆ compact, adjoint | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| E₆ compact, adjoint | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| E₆ compact, adjoint | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| E₆ compact, adjoint | Global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| E₆ compact, adjoint | Global structure and topology | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| E₆ compact, adjoint | Specified connected covering group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| E₇ compact, adjoint | Definition | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| E₇ compact, adjoint | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| E₇ compact, adjoint | Compact rank | Lecture 25 and Appendix T: root systems and classical root computations |
| E₇ compact, adjoint | Center | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| E₇ compact, adjoint | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| E₇ compact, adjoint | Compact | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| E₇ compact, adjoint | Connected | Lectures 26–28 and Appendix V: compact semisimple groups and highest weights · Lecture 16, printed page 73: SU(n) and Sp(n) are simply connected |
| E₇ compact, adjoint | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| E₇ compact, adjoint | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| E₇ compact, adjoint | Global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients · Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| E₇ compact, adjoint | Global structure and topology | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| E₇ compact, adjoint | Specified connected covering group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SL(2, R) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| SL(2, R) | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SL(2, R) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| SL(2, R) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| SL(2, R) | Representation category and descent | Lectures 27–28: highest-weight classification and fundamental representations |
| PSL(2, R) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| PSL(2, R) | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| PSL(2, R) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| PSL(2, R) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| PSL(2, R) | Representation category and descent | Lectures 27–28: highest-weight classification and fundamental representations |
| Universal cover of SL(2, R) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| Universal cover of SL(2, R) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| Universal cover of SL(2, R) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| Universal cover of SL(2, R) | Global form | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| Universal cover of SL(2, R) | Global structure and topology | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| Universal cover of SL(2, R) | Explicit homomorphism | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SL(3, R) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| SL(3, R) | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SL(3, R) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| SL(3, R) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| SL(3, R) | Global form | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SL(3, R) | Global structure and topology | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SL(3, R) | Representation category and descent | Lectures 27–28: highest-weight classification and fundamental representations |
| SU(2,2) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| SU(2,2) | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(2,2) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| SU(2,2) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| Sp(4, R) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| Sp(4, R) | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| Sp(4, R) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| Sp(4, R) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| SO₀(1,3) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| SO₀(1,3) | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SO₀(1,3) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| SO₀(1,3) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| SO₀(1,3) | Representation category and descent | Lectures 27–28: highest-weight classification and fundamental representations |
| SO₀(2,4) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| SO₀(2,4) | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SO₀(2,4) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| SO₀(2,4) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| SO₀(2,4) | Global form | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO₀(2,4) | Global structure and topology | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO₀(2,4) | Explicit homomorphism | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO₀(2,4) | Specified connected covering group | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO₀(2,3) | Real dimension | Lecture 25 and Appendix T: root systems and classical root computations |
| SO₀(2,3) | Fundamental group π₁ | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SO₀(2,3) | Abelian | Lecture 25 and Appendix T: root systems and classical root computations |
| SO₀(2,3) | Simple real Lie algebra | Lecture 25 and Appendix T: root systems and classical root computations |
| SO₀(2,3) | Global form | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO₀(2,3) | Global structure and topology | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO₀(2,3) | Explicit homomorphism | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO₀(2,3) | Specified connected covering group | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SU(3) → PSU(3) | Definition | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(3) → PSU(3) | Domain | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(3) → PSU(3) | Codomain | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(3) → PSU(3) | Specified homomorphism | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(3) → PSU(3) | Kernel | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(3) → PSU(3) | Image | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(3) → PSU(3) | Injective | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(3) → PSU(3) | Surjective | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(3) → PSU(3) | Covering degree | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(3) → PSU(3) | Global and representation consequences | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(3) → PSU(3) | Domain group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(3) → PSU(3) | Codomain group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(2) ↪ SU(3) | Global and representation consequences | Lectures 27–28: highest-weight classification and fundamental representations |
| SU(2) → SU(3), spin 1 | Global and representation consequences | Lectures 27–28: highest-weight classification and fundamental representations |
| U(1) ≅ SO(2) ↪ SO(3) | Definition | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| U(1) ≅ SO(2) ↪ SO(3) | Domain | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| U(1) ≅ SO(2) ↪ SO(3) | Codomain | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| U(1) ≅ SO(2) ↪ SO(3) | Specified homomorphism | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| U(1) ≅ SO(2) ↪ SO(3) | Kernel | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| U(1) ≅ SO(2) ↪ SO(3) | Image | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| U(1) ≅ SO(2) ↪ SO(3) | Injective | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| U(1) ≅ SO(2) ↪ SO(3) | Surjective | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| U(1) ≅ SO(2) ↪ SO(3) | Global and representation consequences | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| U(1) ≅ SO(2) ↪ SO(3) | Global and representation consequences | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| U(1) ≅ SO(2) ↪ SO(3) | Domain group | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| U(1) ≅ SO(2) ↪ SO(3) | Codomain group | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SU(4) → PSU(4) | Definition | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(4) → PSU(4) | Domain | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(4) → PSU(4) | Codomain | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(4) → PSU(4) | Specified homomorphism | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(4) → PSU(4) | Kernel | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(4) → PSU(4) | Image | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(4) → PSU(4) | Injective | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(4) → PSU(4) | Surjective | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(4) → PSU(4) | Covering degree | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(4) → PSU(4) | Global and representation consequences | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(4) → PSU(4) | Domain group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| SU(4) → PSU(4) | Codomain group | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| Spin(8) → SO(8) | Definition | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(8) → SO(8) | Domain | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(8) → SO(8) | Codomain | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(8) → SO(8) | Specified homomorphism | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(8) → SO(8) | Kernel | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(8) → SO(8) | Image | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(8) → SO(8) | Injective | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(8) → SO(8) | Surjective | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(8) → SO(8) | Covering degree | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(8) → SO(8) | Global and representation consequences | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(8) → SO(8) | Global and representation consequences | Remark T.2.1, printed page 262: D₄ triality and quotient groups |
| Spin(8) → SO(8) | Domain group | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(8) → SO(8) | Codomain group | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(10) → SO(10) | Definition | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(10) → SO(10) | Domain | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(10) → SO(10) | Codomain | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(10) → SO(10) | Specified homomorphism | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(10) → SO(10) | Kernel | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(10) → SO(10) | Image | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(10) → SO(10) | Injective | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(10) → SO(10) | Surjective | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(10) → SO(10) | Covering degree | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(10) → SO(10) | Global and representation consequences | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(10) → SO(10) | Domain group | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Spin(10) → SO(10) | Codomain group | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| Universal SL(2,R) cover → SL(2,R) | Global and representation consequences | Appendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotients |
| Sp(4, R) → SO₀(2,3) | Global and representation consequences | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |
| SO(3)/SO(2) ≅ S² | Inclusion π₁(SO(2)) → π₁(SO(3)) | Lecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3 |