Universal covering map of SL(2,R)
Send a homotopy class of paths starting at I to its endpoint. Path multiplication gives the covering-group homomorphism. Kernel: ⟨z²⟩ ≅ Z, where z lifts a π rotation.
Send a homotopy class of paths starting at I to its endpoint. Path multiplication gives the covering-group homomorphism.
Definition sourcesThe group map
Reference properties
| Property | Value | Evidence |
|---|---|---|
| Domain | universal-cover-sl-2-r | Sources |
| Codomain | sl-2-r | Sources |
| Specified homomorphism | Send a homotopy class of paths starting at I to its endpoint. Path multiplication gives the covering-group homomorphism. | Sources |
| Kernel | ⟨z²⟩ ≅ Z, where z lifts a π rotation | Sources |
| Image | Sources | |
| Injective | No | Sources |
| Surjective | Yes | Sources |
| Covering degree | Infinitely many sheets | Sources |
Global and representation consequences
The center of the domain is ⟨z⟩. Its subgroup ⟨z²⟩ is the kernel of this projection; composing with SL(2,R) → PSL(2,R) enlarges the kernel to the full center. [sources]
Choose R(θ) as the standard planar rotation matrix inside SL(2,R). The path θ:0→π ends at −I and defines z; the path θ:0→2π is one fundamental loop in SL(2,R) and defines z². [sources]
Related objects and maps
- Domain group
- Codomain group
Claims and sources
Each assertion has a stable identifier within this record and a source locator or derivation. Cite the dataset edition when referring to a corrected or historical claim.
Definition
Send a homotopy class of paths starting at I to its endpoint. Path multiplication gives the covering-group homomorphism.
- Lie Groups and Lie AlgebrasSection 3.2: covering Lie groupsOpen source
- K-theory and limit-of-discrete-series for the universal cover of SL₂(R)Section 1, page 2: universal cover and its infinite cyclic centerOpen source
map:universal-sl-2-r-to-sl-2-r#definitionDomain
universal-cover-sl-2-r
- Lie Groups and Lie AlgebrasSection 3.2: covering Lie groupsOpen source
- K-theory and limit-of-discrete-series for the universal cover of SL₂(R)Section 1, page 2: universal cover and its infinite cyclic centerOpen source
map:universal-sl-2-r-to-sl-2-r#domainCodomain
sl-2-r
- Lie Groups and Lie AlgebrasSection 3.2: covering Lie groupsOpen source
- K-theory and limit-of-discrete-series for the universal cover of SL₂(R)Section 1, page 2: universal cover and its infinite cyclic centerOpen source
map:universal-sl-2-r-to-sl-2-r#codomainSpecified homomorphism
Send a homotopy class of paths starting at I to its endpoint. Path multiplication gives the covering-group homomorphism.
- Lie Groups and Lie AlgebrasSection 3.2: covering Lie groupsOpen source
- K-theory and limit-of-discrete-series for the universal cover of SL₂(R)Section 1, page 2: universal cover and its infinite cyclic centerOpen source
map:universal-sl-2-r-to-sl-2-r#formulaKernel
⟨z²⟩ ≅ Z, where z lifts a π rotation
- Lie Groups and Lie AlgebrasSection 3.2: covering Lie groupsOpen source
- K-theory and limit-of-discrete-series for the universal cover of SL₂(R)Section 1, page 2: universal cover and its infinite cyclic centerOpen source
map:universal-sl-2-r-to-sl-2-r#kernelImage
SL(2,R)
- Lie Groups and Lie AlgebrasSection 3.2: covering Lie groupsOpen source
- K-theory and limit-of-discrete-series for the universal cover of SL₂(R)Section 1, page 2: universal cover and its infinite cyclic centerOpen source
map:universal-sl-2-r-to-sl-2-r#imageInjective
No
- Lie Groups and Lie AlgebrasSection 3.2: covering Lie groupsOpen source
- K-theory and limit-of-discrete-series for the universal cover of SL₂(R)Section 1, page 2: universal cover and its infinite cyclic centerOpen source
map:universal-sl-2-r-to-sl-2-r#injectiveSurjective
Yes
- Lie Groups and Lie AlgebrasSection 3.2: covering Lie groupsOpen source
- K-theory and limit-of-discrete-series for the universal cover of SL₂(R)Section 1, page 2: universal cover and its infinite cyclic centerOpen source
map:universal-sl-2-r-to-sl-2-r#surjectiveCovering degree
Infinitely many sheets
- Lie Groups and Lie AlgebrasSection 3.2: covering Lie groupsOpen source
- K-theory and limit-of-discrete-series for the universal cover of SL₂(R)Section 1, page 2: universal cover and its infinite cyclic centerOpen source
map:universal-sl-2-r-to-sl-2-r#covering-degreeGlobal and representation consequences
The center of the domain is ⟨z⟩. Its subgroup ⟨z²⟩ is the kernel of this projection; composing with SL(2,R) → PSL(2,R) enlarges the kernel to the full center.
- Lie Groups and Lie AlgebrasSection 3.2: covering Lie groupsOpen source
- K-theory and limit-of-discrete-series for the universal cover of SL₂(R)Section 1, page 2: universal cover and its infinite cyclic centerOpen source
- Lecture 17: Classification of Connected Compact and Complex Reductive GroupsSection 43.1, Proposition 43.2 and Corollary 43.5: weight lattices and descent to central quotientsOpen source
map:universal-sl-2-r-to-sl-2-r#meaning-1Global and representation consequences
Choose R(θ) as the standard planar rotation matrix inside SL(2,R). The path θ:0→π ends at −I and defines z; the path θ:0→2π is one fundamental loop in SL(2,R) and defines z².
- Lecture 17: Classification of Connected Compact and Complex Reductive GroupsSection 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)Open source
- Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source
Premises: SL(2,R) retracts to SO(2) · π₁(SO(2)) is generated by θ:0→2π · R(π)=−I
map:universal-sl-2-r-to-sl-2-r#meaning-2Domain group
group:universal-cover-sl-2-r
- Lie Groups and Lie AlgebrasSection 3.2: covering Lie groupsOpen source
- K-theory and limit-of-discrete-series for the universal cover of SL₂(R)Section 1, page 2: universal cover and its infinite cyclic centerOpen source
map:universal-sl-2-r-to-sl-2-r#relation-1Codomain group
group:sl-2-r
- Lie Groups and Lie AlgebrasSection 3.2: covering Lie groupsOpen source
- K-theory and limit-of-discrete-series for the universal cover of SL₂(R)Section 1, page 2: universal cover and its infinite cyclic centerOpen source
map:universal-sl-2-r-to-sl-2-r#relation-2