Lorentz spin group
As a real Lie group, SL(2,ℂ) is the double cover of the identity component SO⁺(3,1) of the Lorentz group.
Reference properties
Representations
Finite-dimensional irreducible complex representations of the real Lie group carry labels (jL,jR). Particle-state representations require the Poincaré group and a different category. [sources]
Topology and global form
Polar decomposition gives SL(2,ℂ) ≃ SU(2) × ℝ³ as a manifold. It is simply connected and noncompact. [sources]
In physics
Weyl and Dirac fields, Lorentz covariance, and the distinction between fields and states. [sources]
The Lorentz algebra is a realification
SL(2,C) is being used as a real six-dimensional Lie group. Its real Lie algebra complexifies to sl(2,C) ⊕ sl(2,C), rather than to a single A₁ factor. Finite-dimensional complex irreducibles consequently carry two nonnegative half-integer labels (jL,jR), with dimension (2jL+1)(2jR+1). [sources]
Names and aliases
Spin+(3,1) · Lorentz spin cover
Related objects and maps
- Related group
- Related group
- Actual real Lie algebra
- Real-form context
- Explicit homomorphism
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
2 × 2 complex matrices of determinant 1
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
group:sl-2-c#definitionReal dimension
6
- Lie Groups and Lie AlgebrasSections 6.1–6.3: sl(2,C) has complex dimension 3 and real dimension 6Open source
group:sl-2-c#dimensionCenter
ℤ₂
- Sporadic isogenies to orthogonal groupsSection 2.4: the SL(2,C) spin covering kernel {±I}Open source
group:sl-2-c#centerFundamental group
0
- 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
group:sl-2-c#pi1Compact
No
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
group:sl-2-c#compactConnected
Yes
- 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
group:sl-2-c#connectedReal rank
1
- Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsTheorem 6.105: complex simple algebras as real algebras; the split diagonal subgroup of SL(2,C) has real rank 1Open source
group:sl-2-c#rankRepresentations
Finite-dimensional irreducible complex representations of the real Lie group carry labels (jL,jR). Particle-state representations require the Poincaré group and a different category.
- Conrad and Landesman · Compact Lie groupsLectures 27–28: highest-weight classification and fundamental representationsOpen source
- Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsTheorem 6.105: sl(2,C) viewed as a real simple algebraOpen source
- Lie Groups and Lie AlgebrasSection 9.4: real forms and complexificationOpen source
group:sl-2-c#representations-1Topology and global form
Polar decomposition gives SL(2,ℂ) ≃ SU(2) × ℝ³ as a manifold. It is simply connected and noncompact.
- 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
group:sl-2-c#topology-1In physics
Weyl and Dirac fields, Lorentz covariance, and the distinction between fields and states.
- Zuber · Invariances in Physics and Group TheoryApplications to the indicated symmetry and representation categoryOpen source
group:sl-2-c#physics-1The Lorentz algebra is a realification
SL(2,C) is being used as a real six-dimensional Lie group. Its real Lie algebra complexifies to sl(2,C) ⊕ sl(2,C), rather than to a single A₁ factor. Finite-dimensional complex irreducibles consequently carry two nonnegative half-integer labels (jL,jR), with dimension (2jL+1)(2jR+1).
- Conrad and Landesman · Compact Lie groupsLectures 27–28: highest-weight classification and fundamental representationsOpen source
- Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsTheorem 6.105: sl(2,C) viewed as a real simple algebraOpen source
- Lie Groups and Lie AlgebrasSection 9.4: real forms and complexificationOpen source
Premises: algebra:sl-2-c-as-real · algebra:a-1-plus-a-1-c · SU(2) highest weight 2j has dimension 2j+1 · SL(2,C) is simply connected by polar decomposition
group:sl-2-c#real-algebra-1Related group
group:su-2
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
group:sl-2-c#relation-1Related group
group:spin-4
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
group:sl-2-c#relation-2Actual real Lie algebra
algebra:sl-2-c-as-real
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
group:sl-2-c#relation-3Real-form context
real-form:sl-2-c-as-real
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
group:sl-2-c#relation-4Explicit homomorphism
map:sl-2-c-to-lorentz
- Sporadic isogenies to orthogonal groupsSection 2.4: SL(2,C) orthogonal covering of signature (3,1)Open source
group:sl-2-c#relation-5