Lie theory · Group
Rotation group in five dimensions
Sp(2)/{±I}. The real orthogonal 5 × 5 matrices of determinant 1.
Reference coverageCompact Lie groupSimple algebraGlobal form
Reference properties
Global structure and topology
The exceptional isomorphism Spin(5) ≅ Sp(2) realizes the universal double cover. The compact symplectic group in this statement acts on quaternionic dimension two. [sources]
Representation category and descent
Use finite-dimensional continuous complex representations. An irreducible representation of the simply connected compact cover descends precisely when the specified covering kernel acts trivially; the same Lie algebra does not remove that global condition. [sources]
Names and aliases
PSp(2)
Related objects and maps
- Real Lie algebra
- Real-form context
- Explicit homomorphism
- 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 real orthogonal 5 × 5 matrices of determinant 1.
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
- Conrad and Landesman · Compact Lie groupsLectures 26–28 and Appendix V: compact semisimple groups and highest weightsOpen source
- Conrad and Landesman · Compact Lie groupsLecture 16, printed page 73: SU(n) and Sp(n) are simply connectedOpen source
group:so-5#definitionReal 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:so-5#dimensionCompact rank
2
- Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
group:so-5#rankCenter
1
- Math 210C. Centers of simply connected semisimple groupsPages 2–4: centers of simply connected groups by root-system typeOpen source
- Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source
group:so-5#centerFundamental group π₁
C₂
- Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source
- Conrad and Landesman · Compact Lie groupsLecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3Open source
group:so-5#pi1Compact
Yes
- Conrad and Landesman · Compact Lie groupsLectures 26–28 and Appendix V: compact semisimple groups and highest weightsOpen source
- Conrad and Landesman · Compact Lie groupsLecture 16, printed page 73: SU(n) and Sp(n) are simply connectedOpen source
group:so-5#compactConnected
Yes
- Conrad and Landesman · Compact Lie groupsLectures 26–28 and Appendix V: compact semisimple groups and highest weightsOpen source
- Conrad and Landesman · Compact Lie groupsLecture 16, printed page 73: SU(n) and Sp(n) are simply connectedOpen source
group:so-5#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:so-5#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:so-5#simple-algebraGlobal form
Sp(2)/{±I}
- Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source
- Math 210C. Centers of simply connected semisimple groupsPage 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphismsOpen source
- Conrad and Landesman · Compact Lie groupsLecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3Open source
- Sporadic isogenies to orthogonal groupsSection 2.6: compact Sp(2) → SO(5)Open source
group:so-5#global-formReal Lie algebra
sp(2) ≅ so(5)
- 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-5#lie-algebraGlobal structure and topology
The exceptional isomorphism Spin(5) ≅ Sp(2) realizes the universal double cover. The compact symplectic group in this statement acts on quaternionic dimension two.
- Math 210C. Centers of simply connected semisimple groupsPage 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphismsOpen source
- Conrad and Landesman · Compact Lie groupsLecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3Open source
- Sporadic isogenies to orthogonal groupsSection 2.6: compact Sp(2) → SO(5)Open source
group:so-5#topology-1Representation category and descent
Use finite-dimensional continuous complex representations. An irreducible representation of the simply connected compact cover descends precisely when the specified covering kernel acts trivially; the same Lie algebra does not remove that global condition.
- 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
group:so-5#representations-1Real Lie algebra
algebra:sp-2-compact-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-5#relation-1Real-form context
real-form:sp-2-compact-r
- Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
group:so-5#relation-2Explicit homomorphism
map:sp-2-to-so-5
- Math 210C. Centers of simply connected semisimple groupsPage 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphismsOpen source
- Conrad and Landesman · Compact Lie groupsLecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3Open source
- Sporadic isogenies to orthogonal groupsSection 2.6: compact Sp(2) → SO(5)Open source
group:so-5#relation-3Specified connected covering group
group:sp-2
- Math 210C. Centers of simply connected semisimple groupsPage 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphismsOpen source
- Conrad and Landesman · Compact Lie groupsLecture 16, including (16.6), and Appendix P: Spin(n) as the simply connected double cover for n ≥ 3Open source
- Sporadic isogenies to orthogonal groupsSection 2.6: compact Sp(2) → SO(5)Open source
group:so-5#relation-4