Lie theory · Group

$\mathrm{SO}_0(1,3)$ Proper orthochronous Lorentz group in four dimensions

SL(2,C)/{±I}. The identity component of the real determinant-one matrices preserving a quadratic form with signature (1,3).

Compare this group

Reference coverageNoncompact Lie groupSimple algebraGlobal form

Definition

The identity component of the real determinant-one matrices preserving a quadratic form with signature (1,3).

Definition sources

Real dimension: $6$
Center: $1$
Fundamental group π₁: $\mathrm{C}_{2}$

Reference properties

Property	Value	Evidence
Real dimension	$6$	Sources
Real rank	$1$	Sources
Center	$1$	Sources
Fundamental group π₁	$\mathrm{C}_{2}$	Sources
Compact	No	Sources
Connected	Yes	Sources
Abelian	No	Sources
Simple real Lie algebra	Yes	Sources
Global form	$\mathrm{SL}(2,\mathbb C)/\{\pm I\}$	Sources
Real Lie algebra	$\mathfrak{sl}(2,\mathbb C)_{\mathbb R}$	Sources
Maximal compact subgroup	$\mathrm{SO}(3)$	Sources

Global structure and topology

Its maximal compact subgroup is SO(3), and its manifold is SO(3) × R³. The simply connected double cover is SL(2,C), acting on Hermitian 2 × 2 matrices by X ↦ AXA†. [sources]

Representation category and descent

Finite-dimensional complex irreducibles of the SL(2,C) cover have labels (jL,jR). They descend to SO₀(1,3) exactly when jL + jR is an integer. The two Weyl spinors therefore belong to the cover, while the vector (1/2,1/2) descends. [sources]

Names and aliases

SO⁺(1,3) · SO₀(3,1) · Proper orthochronous Lorentz group

Aliases identify the stated object under this record’s conventions. Type names and broader search terms are kept separate.

Related objects and maps

$\mathfrak{sl}(2,\mathbb C)_{\mathbb R}$
sl(2, C) viewed over R — real Lie algebra
Real Lie algebra
$\mathfrak{sl}(2,\mathbb C)_{\mathbb R}$
sl(2, C) viewed over R real form
Real-form context
$\mathrm{SL}(2, \mathbb C) \longrightarrow \mathrm{SO}_0(1,3)$
Hermitian-matrix Lorentz spin cover
Explicit homomorphism
$\mathrm{SL}(2,\mathbb C)$
Lorentz spin group
Specified connected covering 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

The identity component of the real determinant-one matrices preserving a quadratic form with signature (1,3).

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:so-1-3-identity#definition

Link to claim Suggest a correction

Real dimension

6

Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
Lecture 6: Maximal Root, Exponents, Coxeter Numbers, Dual RepresentationsSection 32.4, printed pages 175–176: Coxeter and dual Coxeter numbersOpen source

group:so-1-3-identity#dimension

Link to claim Suggest a correction

Real rank

1

Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source

group:so-1-3-identity#real-rank

Link to claim Suggest a correction

Center

1

Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source

group:so-1-3-identity#center

Link to claim Suggest a correction

Fundamental group π₁

C₂

Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source
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

Derivation: Cartan decomposition is a deformation retraction onto the stated maximal compact subgroup; compute its fundamental group.

Premises: Maximal compact subgroup: SO(3) · π₁(SO(2))=Z, π₁(SO(n))=C₂ for n≥3, π₁(U(2))=Z · For S(U(2)×U(2)), the determinant map has simply connected fiber SU(2)×SU(2), giving π₁=Z

group:so-1-3-identity#pi1

Link to claim Suggest a correction

Compact

No

Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source

group:so-1-3-identity#compact

Link to claim Suggest a correction

Connected

Yes

Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source

group:so-1-3-identity#connected

Link to claim Suggest a correction

Abelian

No

Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source

group:so-1-3-identity#abelian

Link to claim Suggest a correction

Simple real Lie algebra

Yes

Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source

group:so-1-3-identity#simple-algebra

Link to claim Suggest a correction

Global form

SL(2,C)/{±I}

Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source
Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source
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
Sporadic isogenies to orthogonal groupsSection 2.4: SL(2,C) covering SO(3,1)Open source

group:so-1-3-identity#global-form

Link to claim Suggest a correction

Real Lie algebra

sl(2, C) viewed over R

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:so-1-3-identity#lie-algebra

Link to claim Suggest a correction

Maximal compact subgroup

SO(3)

Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source
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:so-1-3-identity#maximal-compact

Link to claim Suggest a correction

Global structure and topology

Its maximal compact subgroup is SO(3), and its manifold is SO(3) × R³. The simply connected double cover is SL(2,C), acting on Hermitian 2 × 2 matrices by X ↦ AXA†.

Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source
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
Sporadic isogenies to orthogonal groupsSection 2.4: SL(2,C) covering SO(3,1)Open source

group:so-1-3-identity#topology-1

Link to claim Suggest a correction

Representation category and descent

Finite-dimensional complex irreducibles of the SL(2,C) cover have labels (jL,jR). They descend to SO₀(1,3) exactly when jL + jR is an integer. The two Weyl spinors therefore belong to the cover, while the vector (1/2,1/2) descends.

Conrad and Landesman · Compact Lie groupsLectures 27–28: highest-weight classification and fundamental representationsOpen 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
Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsTheorem 6.105: complex simple algebras viewed as realOpen source

Derivation: The complexified Lorentz algebra is sl(2,C) ⊕ sl(2,C). The covering element −I acts with product sign (−1)^(2jL+2jR).

Premises: algebra:sl-2-c-as-real · SL(2,C) → SO₀(1,3) has kernel {±I}

group:so-1-3-identity#representations-1

Link to claim Suggest a correction

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:so-1-3-identity#relation-1

Link to claim Suggest a correction

Real-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:so-1-3-identity#relation-2

Link to claim Suggest a correction

Explicit homomorphism

map:sl-2-c-to-lorentz

Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source
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
Sporadic isogenies to orthogonal groupsSection 2.4: SL(2,C) covering SO(3,1)Open source

group:so-1-3-identity#relation-3

Link to claim Suggest a correction

Specified connected covering group

group:sl-2-c

Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source
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
Sporadic isogenies to orthogonal groupsSection 2.4: SL(2,C) covering SO(3,1)Open source

group:so-1-3-identity#relation-4

Link to claim Suggest a correction