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Lie theory · Map

SL(2,C)⟶SO0(1,3)\mathrm{SL}(2, \mathbb C) \longrightarrow \mathrm{SO}_0(1,3)Hermitian-matrix Lorentz spin cover

Identify x with X = x⁰I + x¹σ₁ + x²σ₂ + x³σ₃, a Hermitian 2 × 2 matrix. Send X to AXA† for A ∈ SL(2,C). Then det X = (x⁰)² − (x¹)² − (x²)² − (x³)² is preserved. Kernel: {±I} ≅ C₂.

Reference coverageCovering or central quotient
Definition

Identify x with X = x⁰I + x¹σ₁ + x²σ₂ + x³σ₃, a Hermitian 2 × 2 matrix. Send X to AXA† for A ∈ SL(2,C). Then det X = (x⁰)² − (x¹)² − (x²)² − (x³)² is preserved.

Definition sources

The group map

Kernel{±I}≅C2\{\pm I\} \cong \mathrm{C}_{2}
ImageSO0(1,3)\mathrm{SO}_0(1,3)

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

Reference properties

PropertyValueEvidence
Domainsl-2-cSources
Codomainso-1-3-identitySources
Specified homomorphismIdentify x with X = x⁰I + x¹σ₁ + x²σ₂ + x³σ₃, a Hermitian 2 × 2 matrix. Send X to AXA† for A ∈ SL(2,C). Then det X = (x⁰)² − (x¹)² − (x²)² − (x³)² is preserved.Sources
Kernel{±I}≅C2\{\pm I\} \cong \mathrm{C}_{2}Sources
ImageSO0(1,3)\mathrm{SO}_0(1,3)Sources
InjectiveNoSources
SurjectiveYesSources
Covering degree22Sources

Global and representation consequences

The connected source maps onto the identity component of the Lorentz group. It does not cover time reversal or spatial parity components. SL(2,C) is simply connected as a real Lie group. [sources]

Preservation of det X follows from det A = 1. If AXA† = X for every Hermitian X, then A is scalar and equals ±I. The derivative is an isomorphism of six-dimensional real Lie algebras; connectedness identifies the image as SO₀(1,3). [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

Identify x with X = x⁰I + x¹σ₁ + x²σ₂ + x³σ₃, a Hermitian 2 × 2 matrix. Send X to AXA† for A ∈ SL(2,C). Then det X = (x⁰)² − (x¹)² − (x²)² − (x³)² is preserved.

map:sl-2-c-to-lorentz#definition
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Domain

sl-2-c

map:sl-2-c-to-lorentz#domain
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Codomain

so-1-3-identity

map:sl-2-c-to-lorentz#codomain
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Specified homomorphism

Identify x with X = x⁰I + x¹σ₁ + x²σ₂ + x³σ₃, a Hermitian 2 × 2 matrix. Send X to AXA† for A ∈ SL(2,C). Then det X = (x⁰)² − (x¹)² − (x²)² − (x³)² is preserved.

map:sl-2-c-to-lorentz#formula
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Kernel

{±I} ≅ C₂

map:sl-2-c-to-lorentz#kernel
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Image

SO₀(1,3)

map:sl-2-c-to-lorentz#image
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Injective

No

map:sl-2-c-to-lorentz#injective
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Surjective

Yes

map:sl-2-c-to-lorentz#surjective
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Covering degree

2

map:sl-2-c-to-lorentz#covering-degree
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Global and representation consequences

The connected source maps onto the identity component of the Lorentz group. It does not cover time reversal or spatial parity components. SL(2,C) is simply connected as a real Lie group.

map:sl-2-c-to-lorentz#meaning-1
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Global and representation consequences

Preservation of det X follows from det A = 1. If AXA† = X for every Hermitian X, then A is scalar and equals ±I. The derivative is an isomorphism of six-dimensional real Lie algebras; connectedness identifies the image as SO₀(1,3).

Derivation: Check the determinant, central kernel, derivative, and identity component directly from the displayed action.

Premises: det A=1 · Hermitian 2×2 matrices form a real four-dimensional vector space · SL(2,C) is connected

map:sl-2-c-to-lorentz#meaning-2
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Domain group

group:sl-2-c

map:sl-2-c-to-lorentz#relation-1
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Codomain group

group:so-1-3-identity

map:sl-2-c-to-lorentz#relation-2
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