$\mathrm{SO}(3)$Rotation group in three dimensions

3 × 3 real orthogonal matrices of determinant 1

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

3

• Lie Groups and Lie AlgebrasSections 6.1–6.3: real skew-symmetric 3 × 3 matrices have dimension 3·2/2=3Open source

Trivial

• Math 210C. Centers of simply connected semisimple groupsPage 4: SO(3) = SU(2)/μ₂; the adjoint A₁ groupOpen source

ℤ₂

• Lie Groups and Lie AlgebrasSection 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³Open 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

Yes

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

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

1

• Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source

Ordinary irreducible complex representations have integer spin j. Half-integer-spin representations belong to the spin cover.

• Lecture 6: Maximal Root, Exponents, Coxeter Numbers, Dual RepresentationsSection 32.5, Example 32.13: sl(2,C) highest-weight modulesOpen 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
• Sporadic isogenies to orthogonal groupsSection 2.1: kernel {±I} of SU(2) → SO(3)Open source

SO(3) is diffeomorphic to ℝP³. Its universal cover is SU(2), with covering kernel ℤ₂.

• Lie Groups and Lie AlgebrasSection 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³Open source

Rotational symmetry, integer-spin multiplets, and the topology of rotations.

• Lecture 6: Maximal Root, Exponents, Coxeter Numbers, Dual RepresentationsSection 32.5, Example 32.13: sl(2,C) highest-weight modulesOpen 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
• Sporadic isogenies to orthogonal groupsSection 2.1: kernel {±I} of SU(2) → SO(3)Open source

group:su-2

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

group:a-4

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

group:s-4

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

group:a-5

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

algebra:su-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

real-form:su-2-r

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

map:su-2-to-so-3

• Sporadic isogenies to orthogonal groupsSection 2.1: SU(2) → SO(3), construction and kernelOpen source

map:u-1-to-so-3

• Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
• Lie Groups and Lie AlgebrasSection 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³Open 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

space:so-3-over-so-2

• Lie Groups and Lie AlgebrasSection 4.17, printed page 31: SU(2)/U(1) and SO(3)/SO(2) as S²; SU(2)=S³ and SO(3)=RP³Open source