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Lie theory · Map

Spin(4)⟶SO(4)\mathrm{Spin}(4) \longrightarrow \mathrm{SO}(4)Quaternionic spin cover in four Euclidean dimensions

(a,b) ↦ (x ↦ axb⁻¹) on the quaternionic real vector space H, with unit quaternions a,b. Kernel: {(1,1),(−1,−1)} ≅ C₂.

Reference coverageCovering or central quotient
Definition

(a,b) ↦ (x ↦ axb⁻¹) on the quaternionic real vector space H, with unit quaternions a,b.

Definition sources

The group map

Kernel{(1,1),(−1,−1)} ≅ C₂
ImageSO(4)\mathrm{SO}(4)

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

Reference properties

PropertyValueEvidence
Domainspin-4Sources
Codomainso-4Sources
Specified homomorphism(a,b) ↦ (x ↦ axb⁻¹) on the quaternionic real vector space H, with unit quaternions a,b.Sources
Kernel{(1,1),(−1,−1)} ≅ C₂Sources
ImageSO(4)\mathrm{SO}(4)Sources
InjectiveNoSources
SurjectiveYesSources
Covering degree22Sources

Global and representation consequences

The kernel is the diagonal order-two subgroup, not the full C₂ × C₂ center of Spin(4). Hence (jL,jR) descends exactly when jL + jR is an integer. [sources]

Related objects and maps

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,b) ↦ (x ↦ axb⁻¹) on the quaternionic real vector space H, with unit quaternions a,b.

map:spin-4-to-so-4#definition
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Domain

spin-4

map:spin-4-to-so-4#domain
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Codomain

so-4

map:spin-4-to-so-4#codomain
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Specified homomorphism

(a,b) ↦ (x ↦ axb⁻¹) on the quaternionic real vector space H, with unit quaternions a,b.

map:spin-4-to-so-4#formula
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Kernel

{(1,1),(−1,−1)} ≅ C₂

map:spin-4-to-so-4#kernel
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Image

SO(4)

map:spin-4-to-so-4#image
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Injective

No

map:spin-4-to-so-4#injective
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Surjective

Yes

map:spin-4-to-so-4#surjective
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Covering degree

2

map:spin-4-to-so-4#covering-degree
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Global and representation consequences

The kernel is the diagonal order-two subgroup, not the full C₂ × C₂ center of Spin(4). Hence (jL,jR) descends exactly when jL + jR is an integer.

map:spin-4-to-so-4#meaning-1
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Domain group

group:spin-4

map:spin-4-to-so-4#relation-1
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Codomain group

group:so-4

map:spin-4-to-so-4#relation-2
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