Lie theory · Map

$\mathrm{SU}(2) \longrightarrow \mathrm{SO}(3)$ Spin cover of three-dimensional rotations

A ↦ (X ↦ AXA⁻¹), acting on the three-dimensional real space su(2) with its invariant Euclidean inner product. Kernel: {±I} ≅ C₂.

Executable coverageCovering or central quotient

Definition

A ↦ (X ↦ AXA⁻¹), acting on the three-dimensional real space su(2) with its invariant Euclidean inner product.

Definition sources

The group map

Domain

\mathrm{SU}(2)

Codomain

\mathrm{SO}(3)

Kernel

\{\pm I\} \cong \mathrm{C}_{2}

Image

\mathrm{SO}(3)

Not injective: keep the kernel when restricting representations. The map is surjective.

Reference properties

Property	Value	Evidence
Domain	su-2	Sources
Codomain	so-3	Sources
Specified homomorphism	A ↦ (X ↦ AXA⁻¹), acting on the three-dimensional real space su(2) with its invariant Euclidean inner product.	Sources
Kernel	$\{\pm I\} \cong \mathrm{C}_{2}$	Sources
Image	$\mathrm{SO}(3)$	Sources
Injective	No	Sources
Surjective	Yes	Sources
Covering degree	$2$	Sources

Global and representation consequences

This is the universal covering map. A spin-j SU(2) representation descends along it exactly when 2j is even. With compatible axis conventions, diag(eⁱθ,e⁻ⁱθ) maps to a rotation through 2θ. [sources]

The torus angle in SU(2) has period 2π, but the image rotation has angle 2θ. Tracking that factor is necessary when comparing the two spherical quotient fibrations. [sources]

Related objects and maps

$\mathrm{SU}(2)$
Special unitary group
Domain group
$\mathrm{SO}(3)$
Rotation group in three dimensions
Codomain 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

A ↦ (X ↦ AXA⁻¹), acting on the three-dimensional real space su(2) with its invariant Euclidean inner product.

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

map:su-2-to-so-3#definition

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Domain

su-2

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

map:su-2-to-so-3#domain

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Codomain

so-3

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

map:su-2-to-so-3#codomain

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Specified homomorphism

A ↦ (X ↦ AXA⁻¹), acting on the three-dimensional real space su(2) with its invariant Euclidean inner product.

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

map:su-2-to-so-3#formula

Link to claim Suggest a correction

Kernel

{±I} ≅ C₂

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

map:su-2-to-so-3#kernel

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Image

SO(3)

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

map:su-2-to-so-3#image

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Injective

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

map:su-2-to-so-3#injective

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Surjective

Yes

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

map:su-2-to-so-3#surjective

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Covering degree

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

map:su-2-to-so-3#covering-degree

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Global and representation consequences

This is the universal covering map. A spin-j SU(2) representation descends along it exactly when 2j is even. With compatible axis conventions, diag(eⁱθ,e⁻ⁱθ) maps to a rotation through 2θ.

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

map:su-2-to-so-3#meaning-1

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Global and representation consequences

The torus angle in SU(2) has period 2π, but the image rotation has angle 2θ. Tracking that factor is necessary when comparing the two spherical quotient fibrations.

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
Sporadic isogenies to orthogonal groupsSection 2.1: adjoint rotation actionOpen source

map:su-2-to-so-3#meaning-2

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Domain group

group:su-2

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

map:su-2-to-so-3#relation-1

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Codomain group

group:so-3

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

map:su-2-to-so-3#relation-2

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SU(2)⟶SO(3)\mathrm{SU}(2) \longrightarrow \mathrm{SO}(3)SU(2)⟶SO(3)Spin cover of three-dimensional rotations

The group map

Reference properties

Global and representation consequences

Related objects and maps

Claims and sources

$\mathrm{SU}(2) \longrightarrow \mathrm{SO}(3)$ Spin cover of three-dimensional rotations