Compact symplectic group of quaternionic rank two
Simply connected compact form of B₂ = C₂. Quaternionic unitary 2 × 2 matrices. Equivalently, this is the compact subgroup of U(4) preserving a complex symplectic form.
Quaternionic unitary 2 × 2 matrices. Equivalently, this is the compact subgroup of U(4) preserving a complex symplectic form.
Definition sourcesReference properties
Global structure and topology
Sp(2) is compact and simply connected, with center {±I}. The isomorphism Spin(5) ≅ Sp(2) is an isomorphism of global groups, not only of Lie algebras. [sources]
Representation category and descent
The defining complex 4 preserves a nondegenerate antisymmetric form and has quaternionic type. The vector 5 is the traceless part of the exterior square of this 4; it factors through SO(5). [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
USp(4) · Spin(5)
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
Quaternionic unitary 2 × 2 matrices. Equivalently, this is the compact subgroup of U(4) preserving a complex symplectic form.
- 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:sp-2#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:sp-2#dimensionCompact rank
2
- Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
group:sp-2#rankCenter
C₂
- 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:sp-2#centerFundamental group π₁
0
- 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:sp-2#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:sp-2#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:sp-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:sp-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:sp-2#simple-algebraGlobal form
Simply connected compact form of B₂ = 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 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
- Math 210C. Centers of simply connected semisimple groupsPages 2–4: centers of simply connected groups by root-system typeOpen source
- Math 210C. Centers of simply connected semisimple groupsPage 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphismsOpen source
group:sp-2#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:sp-2#lie-algebraGlobal structure and topology
Sp(2) is compact and simply connected, with center {±I}. The isomorphism Spin(5) ≅ Sp(2) is an isomorphism of global groups, not only of Lie algebras.
- 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
- Math 210C. Centers of simply connected semisimple groupsPages 2–4: centers of simply connected groups by root-system typeOpen source
- Math 210C. Centers of simply connected semisimple groupsPage 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphismsOpen source
group:sp-2#topology-1Representation category and descent
The defining complex 4 preserves a nondegenerate antisymmetric form and has quaternionic type. The vector 5 is the traceless part of the exterior square of this 4; it factors through SO(5).
- Conrad and Landesman · Compact Lie groupsLectures 27–28: highest-weight classification and fundamental representationsOpen source
- Sporadic isogenies to orthogonal groupsSection 2.6: five-dimensional orthogonal action of compact Sp(2)Open source
group:sp-2#representations-1Symplectic 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-2#notation-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:sp-2#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:sp-2#relation-2Explicit homomorphism
map:sp-2-to-so-5
- 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
- Math 210C. Centers of simply connected semisimple groupsPages 2–4: centers of simply connected groups by root-system typeOpen source
- Math 210C. Centers of simply connected semisimple groupsPage 4: SO(3), SO(4), SO(5), and SO(6) low-rank isomorphismsOpen source
group:sp-2#relation-3