Lie theory · Group

$\mathrm{Sp}(4, \mathbb R)$ Real symplectic group on a four-dimensional space

Connected spin double cover of SO₀(2,3); not its universal cover. The real 4 × 4 matrices A satisfying AᵀJA = J for a fixed nondegenerate alternating matrix J.

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Reference coverageNoncompact Lie groupSimple algebraGlobal form

Definition

The real 4 × 4 matrices A satisfying AᵀJA = J for a fixed nondegenerate alternating matrix J.

Definition sources

Real dimension: $10$
Center: $\mathrm{C}_{2}$
Fundamental group π₁: Z

Reference properties

Property	Value	Evidence
Real dimension	$10$	Sources
Real rank	$2$	Sources
Center	$\mathrm{C}_{2}$	Sources
Fundamental group π₁	Z	Sources
Compact	No	Sources
Connected	Yes	Sources
Abelian	No	Sources
Simple real Lie algebra	Yes	Sources
Global form	Connected spin double cover of SO₀(2,3); not its universal cover	Sources
Real Lie algebra	$\mathfrak{sp}(4, \mathbb R) \cong \mathfrak{so}(2,3)$	Sources
Maximal compact subgroup	$\mathrm{U}(2)$	Sources

Global structure and topology

Cartan decomposition retracts Sp(4,R) onto U(2), giving fundamental group Z. Its center is {±I}. Its orthogonal five-dimensional action is a double cover of SO₀(2,3), whose fundamental group is Z × C₂. This real group is different from compact Sp(2). [sources]

Representation category and descent

The real defining 4 preserves its symplectic form. The primitive part of Λ²R⁴ has dimension 5, carries an invariant quadratic form of signature (2,3) up to overall sign, and gives the double-cover map. [sources]

Symplectic notation

This site writes compact Sp(n) for quaternionic n × n unitary matrices, also called USp(2n). It writes Sp(2n,R) for real matrices of size 2n preserving an alternating form. The two conventions count different dimensions. [sources]

Names and aliases

Spin₀(2,3) · Spin₀(3,2)

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

Related objects and maps

$\mathfrak{sp}(4, \mathbb R) \cong \mathfrak{so}(2,3)$
sp(4, R) ≅ so(2,3) — real Lie algebra
Real Lie algebra
$\mathfrak{sp}(4, \mathbb R) \cong \mathfrak{so}(2,3)$
sp(4, R) ≅ so(2,3) real form
Real-form context
$\mathrm{Sp}(4, \mathbb R) \longrightarrow \mathrm{SO}_0(2,3)$
Primitive-exterior-square spin cover in signature (2,3)
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

The real 4 × 4 matrices A satisfying AᵀJA = J for a fixed nondegenerate alternating matrix J.

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:sp-4-r#definition

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Real dimension

10

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:sp-4-r#dimension

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Real rank

2

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

group:sp-4-r#real-rank

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Center

C₂

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:sp-4-r#center

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Fundamental group π₁

Z

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: U(2) · 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:sp-4-r#pi1

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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:sp-4-r#compact

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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:sp-4-r#connected

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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:sp-4-r#abelian

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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:sp-4-r#simple-algebra

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Global form

Connected spin double cover of SO₀(2,3); not its universal cover

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 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
Sporadic isogenies to orthogonal groupsSection 2.8: real symplectic group → SO(3,2); source notation Sp₂(R) uses symplectic rankOpen source

group:sp-4-r#global-form

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Real Lie algebra

sp(4, R) ≅ so(2,3)

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:sp-4-r#lie-algebra

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Maximal compact subgroup

U(2)

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:sp-4-r#maximal-compact

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Global structure and topology

Cartan decomposition retracts Sp(4,R) onto U(2), giving fundamental group Z. Its center is {±I}. Its orthogonal five-dimensional action is a double cover of SO₀(2,3), whose fundamental group is Z × C₂. This real group is different from compact Sp(2).

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
Sporadic isogenies to orthogonal groupsSection 2.8: real symplectic group → SO(3,2); source notation Sp₂(R) uses symplectic rankOpen source

group:sp-4-r#topology-1

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Representation category and descent

The real defining 4 preserves its symplectic form. The primitive part of Λ²R⁴ has dimension 5, carries an invariant quadratic form of signature (2,3) up to overall sign, and gives the double-cover map.

Sporadic isogenies to orthogonal groupsSection 2.8, page 7: five-dimensional orthogonal construction for the real rank-two symplectic groupOpen source

group:sp-4-r#representations-1

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Symplectic notation

This site writes compact Sp(n) for quaternionic n × n unitary matrices, also called USp(2n). It writes Sp(2n,R) for real matrices of size 2n preserving an alternating form. The two conventions count different dimensions.

Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source

group:sp-4-r#notation-1

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Real Lie algebra

algebra:sp-4-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:sp-4-r#relation-1

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Real-form context

real-form:sp-4-r

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

group:sp-4-r#relation-2

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Explicit homomorphism

map:sp-4-r-to-so-2-3

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
Sporadic isogenies to orthogonal groupsSection 2.8: real symplectic group → SO(3,2); source notation Sp₂(R) uses symplectic rankOpen source

group:sp-4-r#relation-3

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