lecture-notes
Sporadic isogenies to orthogonal groups
Paul Garrett
Use in this reference: Author’s lecture notes linked at an accessible mirror. No redistribution license is inferred; matrix constructions are explained independently.
Relevant material: May 7, 2015 notes, Sections 2.1–2.11. The linked file is an accessible mirror of the author’s notes.
Dependent assertions
| Record | Assertion | Source locator |
|---|---|---|
| SU(2) | Explicit homomorphism | Section 2.1: SU(2) → SO(3), construction and kernel |
| SO(3) | Representations | Section 2.1: kernel {±I} of SU(2) → SO(3) |
| SO(3) | In physics | Section 2.1: kernel {±I} of SU(2) → SO(3) |
| SO(3) | Explicit homomorphism | Section 2.1: SU(2) → SO(3), construction and kernel |
| Spin(4) | Definition | Section 2.3: SU(2) × SU(2) → SO(4) on quaternions |
| Spin(4) | Topology and global form | Section 2.3: quaternionic product model and diagonal kernel |
| Spin(4) | Related group | Section 2.3: SU(2) × SU(2) → SO(4) on quaternions |
| Spin(4) | Related group | Section 2.3: SU(2) × SU(2) → SO(4) on quaternions |
| Spin(4) | Explicit homomorphism | Section 2.3: quaternionic SO(4) covering construction |
| SL(2,ℂ) | Center | Section 2.4: the SL(2,C) spin covering kernel {±I} |
| SL(2,ℂ) | Explicit homomorphism | Section 2.4: SL(2,C) orthogonal covering of signature (3,1) |
| SU(4) | Representation category and descent | Section 2.9: SU(4) acting on the real form of its exterior square |
| SO(4) | Global form | Section 2.3: quaternionic construction of Spin(4) → SO(4) |
| SO(4) | Global structure and topology | Section 2.3: quaternionic construction of Spin(4) → SO(4) |
| SO(4) | Explicit homomorphism | Section 2.3: quaternionic construction of Spin(4) → SO(4) |
| SO(4) | Specified connected covering group | Section 2.3: quaternionic construction of Spin(4) → SO(4) |
| SO(5) | Global form | Section 2.6: compact Sp(2) → SO(5) |
| SO(5) | Global structure and topology | Section 2.6: compact Sp(2) → SO(5) |
| SO(5) | Explicit homomorphism | Section 2.6: compact Sp(2) → SO(5) |
| SO(5) | Specified connected covering group | Section 2.6: compact Sp(2) → SO(5) |
| Sp(2) | Representation category and descent | Section 2.6: five-dimensional orthogonal action of compact Sp(2) |
| PSL(2, R) | Global form | Section 2.2: the SL(2,R) orthogonal action of signature (2,1) |
| PSL(2, R) | Global structure and topology | Section 2.2: the SL(2,R) orthogonal action of signature (2,1) |
| PSL(2, R) | Explicit homomorphism | Section 2.2: the SL(2,R) orthogonal action of signature (2,1) |
| PSL(2, R) | Specified connected covering group | Section 2.2: the SL(2,R) orthogonal action of signature (2,1) |
| SU(2,2) | Global form | Section 2.11, pages 9–10: SU(2,2) → SO(4,2) |
| SU(2,2) | Global structure and topology | Section 2.11, pages 9–10: SU(2,2) → SO(4,2) |
| SU(2,2) | Representation category and descent | Section 2.11: exterior-square construction and double-cover kernel |
| SU(2,2) | Explicit homomorphism | Section 2.11, pages 9–10: SU(2,2) → SO(4,2) |
| Sp(4, R) | Global form | Section 2.8: real symplectic group → SO(3,2); source notation Sp₂(R) uses symplectic rank |
| Sp(4, R) | Global structure and topology | Section 2.8: real symplectic group → SO(3,2); source notation Sp₂(R) uses symplectic rank |
| Sp(4, R) | Representation category and descent | Section 2.8, page 7: five-dimensional orthogonal construction for the real rank-two symplectic group |
| Sp(4, R) | Explicit homomorphism | Section 2.8: real symplectic group → SO(3,2); source notation Sp₂(R) uses symplectic rank |
| SO₀(1,3) | Global form | Section 2.4: SL(2,C) covering SO(3,1) |
| SO₀(1,3) | Global structure and topology | Section 2.4: SL(2,C) covering SO(3,1) |
| SO₀(1,3) | Explicit homomorphism | Section 2.4: SL(2,C) covering SO(3,1) |
| SO₀(1,3) | Specified connected covering group | Section 2.4: SL(2,C) covering SO(3,1) |
| SO₀(2,4) | Global form | Section 2.11: SU(2,2) double cover |
| SO₀(2,4) | Global structure and topology | Section 2.11: SU(2,2) double cover |
| SO₀(2,4) | Representation category and descent | Section 2.11: exterior-square orthogonal action and its kernel |
| SO₀(2,4) | Explicit homomorphism | Section 2.11: SU(2,2) double cover |
| SO₀(2,4) | Specified connected covering group | Section 2.11: SU(2,2) double cover |
| SO₀(2,3) | Global form | Section 2.8: real symplectic double cover of SO(3,2) |
| SO₀(2,3) | Global structure and topology | Section 2.8: real symplectic double cover of SO(3,2) |
| SO₀(2,3) | Representation category and descent | Section 2.8: four-dimensional defining module and five-dimensional orthogonal action |
| SO₀(2,3) | Explicit homomorphism | Section 2.8: real symplectic double cover of SO(3,2) |
| SO₀(2,3) | Specified connected covering group | Section 2.8: real symplectic double cover of SO(3,2) |
| SU(2) → SO(3) | Definition | Section 2.1: SU(2) → SO(3), construction and kernel |
| SU(2) → SO(3) | Domain | Section 2.1: SU(2) → SO(3), construction and kernel |
| SU(2) → SO(3) | Codomain | Section 2.1: SU(2) → SO(3), construction and kernel |
| SU(2) → SO(3) | Specified homomorphism | Section 2.1: SU(2) → SO(3), construction and kernel |
| SU(2) → SO(3) | Kernel | Section 2.1: SU(2) → SO(3), construction and kernel |
| SU(2) → SO(3) | Image | Section 2.1: SU(2) → SO(3), construction and kernel |
| SU(2) → SO(3) | Injective | Section 2.1: SU(2) → SO(3), construction and kernel |
| SU(2) → SO(3) | Surjective | Section 2.1: SU(2) → SO(3), construction and kernel |
| SU(2) → SO(3) | Covering degree | Section 2.1: SU(2) → SO(3), construction and kernel |
| SU(2) → SO(3) | Global and representation consequences | Section 2.1: SU(2) → SO(3), construction and kernel |
| SU(2) → SO(3) | Global and representation consequences | Section 2.1: adjoint rotation action |
| SU(2) → SO(3) | Domain group | Section 2.1: SU(2) → SO(3), construction and kernel |
| SU(2) → SO(3) | Codomain group | Section 2.1: SU(2) → SO(3), construction and kernel |
| Spin(4) → SO(4) | Definition | Section 2.3: quaternionic SO(4) covering construction |
| Spin(4) → SO(4) | Domain | Section 2.3: quaternionic SO(4) covering construction |
| Spin(4) → SO(4) | Codomain | Section 2.3: quaternionic SO(4) covering construction |
| Spin(4) → SO(4) | Specified homomorphism | Section 2.3: quaternionic SO(4) covering construction |
| Spin(4) → SO(4) | Kernel | Section 2.3: quaternionic SO(4) covering construction |
| Spin(4) → SO(4) | Image | Section 2.3: quaternionic SO(4) covering construction |
| Spin(4) → SO(4) | Injective | Section 2.3: quaternionic SO(4) covering construction |
| Spin(4) → SO(4) | Surjective | Section 2.3: quaternionic SO(4) covering construction |
| Spin(4) → SO(4) | Covering degree | Section 2.3: quaternionic SO(4) covering construction |
| Spin(4) → SO(4) | Global and representation consequences | Section 2.3: quaternionic SO(4) covering construction |
| Spin(4) → SO(4) | Domain group | Section 2.3: quaternionic SO(4) covering construction |
| Spin(4) → SO(4) | Codomain group | Section 2.3: quaternionic SO(4) covering construction |
| SU(4) → SO(6) | Definition | Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴ |
| SU(4) → SO(6) | Domain | Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴ |
| SU(4) → SO(6) | Codomain | Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴ |
| SU(4) → SO(6) | Specified homomorphism | Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴ |
| SU(4) → SO(6) | Kernel | Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴ |
| SU(4) → SO(6) | Image | Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴ |
| SU(4) → SO(6) | Injective | Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴ |
| SU(4) → SO(6) | Surjective | Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴ |
| SU(4) → SO(6) | Covering degree | Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴ |
| SU(4) → SO(6) | Global and representation consequences | Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴ |
| SU(4) → SO(6) | Domain group | Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴ |
| SU(4) → SO(6) | Codomain group | Section 2.9, pages 7–8: SU(4) → SO(6) via Λ²C⁴ |
| Sp(2) → SO(5) | Definition | Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices |
| Sp(2) → SO(5) | Domain | Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices |
| Sp(2) → SO(5) | Codomain | Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices |
| Sp(2) → SO(5) | Specified homomorphism | Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices |
| Sp(2) → SO(5) | Kernel | Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices |
| Sp(2) → SO(5) | Image | Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices |
| Sp(2) → SO(5) | Injective | Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices |
| Sp(2) → SO(5) | Surjective | Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices |
| Sp(2) → SO(5) | Covering degree | Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices |
| Sp(2) → SO(5) | Global and representation consequences | Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices |
| Sp(2) → SO(5) | Domain group | Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices |
| Sp(2) → SO(5) | Codomain group | Section 2.6: compact symplectic group acting on traceless quaternionic Hermitian matrices |
| SL(2, C) → SO₀(1,3) | Definition | Section 2.4: SL(2,C) orthogonal covering of signature (3,1) |
| SL(2, C) → SO₀(1,3) | Domain | Section 2.4: SL(2,C) orthogonal covering of signature (3,1) |
| SL(2, C) → SO₀(1,3) | Codomain | Section 2.4: SL(2,C) orthogonal covering of signature (3,1) |
| SL(2, C) → SO₀(1,3) | Specified homomorphism | Section 2.4: SL(2,C) orthogonal covering of signature (3,1) |
| SL(2, C) → SO₀(1,3) | Kernel | Section 2.4: SL(2,C) orthogonal covering of signature (3,1) |
| SL(2, C) → SO₀(1,3) | Image | Section 2.4: SL(2,C) orthogonal covering of signature (3,1) |
| SL(2, C) → SO₀(1,3) | Injective | Section 2.4: SL(2,C) orthogonal covering of signature (3,1) |
| SL(2, C) → SO₀(1,3) | Surjective | Section 2.4: SL(2,C) orthogonal covering of signature (3,1) |
| SL(2, C) → SO₀(1,3) | Covering degree | Section 2.4: SL(2,C) orthogonal covering of signature (3,1) |
| SL(2, C) → SO₀(1,3) | Global and representation consequences | Section 2.4: SL(2,C) orthogonal covering of signature (3,1) |
| SL(2, C) → SO₀(1,3) | Domain group | Section 2.4: SL(2,C) orthogonal covering of signature (3,1) |
| SL(2, C) → SO₀(1,3) | Codomain group | Section 2.4: SL(2,C) orthogonal covering of signature (3,1) |
| SL(2, R) → PSL(2, R) | Definition | Section 2.2: SL(2,R) covering the identity component of SO(2,1) |
| SL(2, R) → PSL(2, R) | Domain | Section 2.2: SL(2,R) covering the identity component of SO(2,1) |
| SL(2, R) → PSL(2, R) | Codomain | Section 2.2: SL(2,R) covering the identity component of SO(2,1) |
| SL(2, R) → PSL(2, R) | Specified homomorphism | Section 2.2: SL(2,R) covering the identity component of SO(2,1) |
| SL(2, R) → PSL(2, R) | Kernel | Section 2.2: SL(2,R) covering the identity component of SO(2,1) |
| SL(2, R) → PSL(2, R) | Image | Section 2.2: SL(2,R) covering the identity component of SO(2,1) |
| SL(2, R) → PSL(2, R) | Injective | Section 2.2: SL(2,R) covering the identity component of SO(2,1) |
| SL(2, R) → PSL(2, R) | Surjective | Section 2.2: SL(2,R) covering the identity component of SO(2,1) |
| SL(2, R) → PSL(2, R) | Covering degree | Section 2.2: SL(2,R) covering the identity component of SO(2,1) |
| SL(2, R) → PSL(2, R) | Global and representation consequences | Section 2.2: SL(2,R) covering the identity component of SO(2,1) |
| SL(2, R) → PSL(2, R) | Domain group | Section 2.2: SL(2,R) covering the identity component of SO(2,1) |
| SL(2, R) → PSL(2, R) | Codomain group | Section 2.2: SL(2,R) covering the identity component of SO(2,1) |
| SU(2,2) → SO₀(2,4) | Definition | Section 2.11, pages 9–10: SU(2,2) → SO(4,2) |
| SU(2,2) → SO₀(2,4) | Domain | Section 2.11, pages 9–10: SU(2,2) → SO(4,2) |
| SU(2,2) → SO₀(2,4) | Codomain | Section 2.11, pages 9–10: SU(2,2) → SO(4,2) |
| SU(2,2) → SO₀(2,4) | Specified homomorphism | Section 2.11, pages 9–10: SU(2,2) → SO(4,2) |
| SU(2,2) → SO₀(2,4) | Kernel | Section 2.11, pages 9–10: SU(2,2) → SO(4,2) |
| SU(2,2) → SO₀(2,4) | Image | Section 2.11, pages 9–10: SU(2,2) → SO(4,2) |
| SU(2,2) → SO₀(2,4) | Injective | Section 2.11, pages 9–10: SU(2,2) → SO(4,2) |
| SU(2,2) → SO₀(2,4) | Surjective | Section 2.11, pages 9–10: SU(2,2) → SO(4,2) |
| SU(2,2) → SO₀(2,4) | Covering degree | Section 2.11, pages 9–10: SU(2,2) → SO(4,2) |
| SU(2,2) → SO₀(2,4) | Global and representation consequences | Section 2.11, pages 9–10: SU(2,2) → SO(4,2) |
| SU(2,2) → SO₀(2,4) | Domain group | Section 2.11, pages 9–10: SU(2,2) → SO(4,2) |
| SU(2,2) → SO₀(2,4) | Codomain group | Section 2.11, pages 9–10: SU(2,2) → SO(4,2) |
| Sp(4, R) → SO₀(2,3) | Definition | Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R) |
| Sp(4, R) → SO₀(2,3) | Domain | Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R) |
| Sp(4, R) → SO₀(2,3) | Codomain | Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R) |
| Sp(4, R) → SO₀(2,3) | Specified homomorphism | Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R) |
| Sp(4, R) → SO₀(2,3) | Kernel | Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R) |
| Sp(4, R) → SO₀(2,3) | Image | Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R) |
| Sp(4, R) → SO₀(2,3) | Injective | Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R) |
| Sp(4, R) → SO₀(2,3) | Surjective | Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R) |
| Sp(4, R) → SO₀(2,3) | Covering degree | Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R) |
| Sp(4, R) → SO₀(2,3) | Global and representation consequences | Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R) |
| Sp(4, R) → SO₀(2,3) | Domain group | Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R) |
| Sp(4, R) → SO₀(2,3) | Codomain group | Section 2.8, page 7: the real symplectic five-dimensional orthogonal action; source notation Sp₂(R) |