Indefinite special unitary group of signature (2,2)
Connected spin double cover of SO₀(2,4); not its universal cover. The complex 4 × 4 matrices A with det A = 1 and A†HA = H, where H = diag(1,1,−1,−1).
The complex 4 × 4 matrices A with det A = 1 and A†HA = H, where H = diag(1,1,−1,−1).
Definition sourcesReference properties
| Property | Value | Evidence |
|---|---|---|
| Real dimension | Sources | |
| Real rank | Sources | |
| Center | 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(2,4); not its universal cover | Sources |
| Real Lie algebra | Sources | |
| Maximal compact subgroup | S(U(2) × U(2)) | Sources |
Global structure and topology
The maximal compact subgroup is S(U(2) × U(2)), whose determinant winding gives fundamental group Z. The scalar center is C₄. The orthogonal action on a real form of Λ²C⁴ gives a double cover of SO₀(2,4) with kernel {±I}; SU(2,2) itself is not simply connected. [sources]
Representation category and descent
The defining complex 4 is a chiral spinor for the connected orthogonal cover. Its exterior square has an invariant real structure and gives the real vector 6 of signature (2,4). Because −I acts nontrivially on the 4, that 4 does not descend to SO₀(2,4). [sources]
Names and aliases
Spin₀(2,4) · Spin₀(4,2)
Related objects and maps
- 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
The complex 4 × 4 matrices A with det A = 1 and A†HA = H, where H = diag(1,1,−1,−1).
- 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:su-2-2#definitionReal dimension
15
- 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:su-2-2#dimensionReal rank
2
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
group:su-2-2#real-rankCenter
C₄
- 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:su-2-2#centerFundamental 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
Premises: Maximal compact subgroup: S(U(2) × 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:su-2-2#pi1Compact
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:su-2-2#compactConnected
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:su-2-2#connectedAbelian
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:su-2-2#abelianSimple 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:su-2-2#simple-algebraGlobal form
Connected spin double cover of SO₀(2,4); 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 2.11, pages 9–10: SU(2,2) → SO(4,2)Open source
group:su-2-2#global-formReal Lie algebra
su(2,2) ≅ so(2,4)
- 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:su-2-2#lie-algebraMaximal compact subgroup
S(U(2) × 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:su-2-2#maximal-compactGlobal structure and topology
The maximal compact subgroup is S(U(2) × U(2)), whose determinant winding gives fundamental group Z. The scalar center is C₄. The orthogonal action on a real form of Λ²C⁴ gives a double cover of SO₀(2,4) with kernel {±I}; SU(2,2) itself is not simply connected.
- 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 2.11, pages 9–10: SU(2,2) → SO(4,2)Open source
group:su-2-2#topology-1Representation category and descent
The defining complex 4 is a chiral spinor for the connected orthogonal cover. Its exterior square has an invariant real structure and gives the real vector 6 of signature (2,4). Because −I acts nontrivially on the 4, that 4 does not descend to SO₀(2,4).
- Sporadic isogenies to orthogonal groupsSection 2.11: exterior-square construction and double-cover kernelOpen source
group:su-2-2#representations-1Real Lie algebra
algebra:su-2-2-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:su-2-2#relation-1Real-form context
real-form:su-2-2-r
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
group:su-2-2#relation-2Explicit homomorphism
map:su-2-2-to-so-2-4
- 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 2.11, pages 9–10: SU(2,2) → SO(4,2)Open source
group:su-2-2#relation-3