Unitary group in two dimensions
(SU(2) × U(1))/diagonal C₂. All complex unitary 2 × 2 matrices, with no determinant restriction.
Reference properties
Global structure and topology
The surjective map (A,z) ↦ zA from SU(2) × U(1) has kernel {(I,1),(−I,−1)}. The determinant map U(2) → U(1) has fiber SU(2), and induces an isomorphism on fundamental groups. [sources]
Representation category and descent
Pulling an irrep back to SU(2) × U(1) gives a highest weight n ≥ 0 and an integer U(1) charge q with n + q even. The parity condition makes (−I,−1) act trivially. Its complex dimension is n + 1. [sources]
Names and aliases
Unitary 2 × 2 group
Related objects and maps
- Real Lie algebra
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
All complex unitary 2 × 2 matrices, with no determinant restriction.
- 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:u-2#definitionReal dimension
4
- 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:u-2#dimensionCompact rank
2
- Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
group:u-2#rankCenter
U(1)
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
group:u-2#centerFundamental group π₁
Z
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
- Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source
group:u-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:u-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:u-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:u-2#abelianSimple real Lie algebra
No
- 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:u-2#simple-algebraGlobal form
(SU(2) × U(1))/diagonal C₂
- Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
- Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source
- Algebraic TopologySection 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375Open source
group:u-2#global-formReal Lie algebra
u(2)
- 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:u-2#lie-algebraGlobal structure and topology
The surjective map (A,z) ↦ zA from SU(2) × U(1) has kernel {(I,1),(−I,−1)}. The determinant map U(2) → U(1) has fiber SU(2), and induces an isomorphism on fundamental groups.
- Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
- Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source
- Algebraic TopologySection 4.2, Theorem 4.41, printed page 376; Hopf-fibration example on page 375Open source
Premises: group:su-2#pi1 · group:u-1#pi1 · Determinant map U(2) → U(1)
group:u-2#topology-1Representation category and descent
Pulling an irrep back to SU(2) × U(1) gives a highest weight n ≥ 0 and an integer U(1) charge q with n + q even. The parity condition makes (−I,−1) act trivially. Its complex dimension is n + 1.
- 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
Premises: SU(2) highest weight n has dimension n+1 and central sign (−1)ⁿ · U(1) character z ↦ zᑫ · U(2) quotient kernel {(I,1),(−I,−1)}
group:u-2#representations-1Real Lie algebra
algebra:u-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:u-2#relation-1