Lie theory · Map

$\mathrm{SL}(2, \mathbb R) \longrightarrow \mathrm{PSL}(2, \mathbb R)$ Adjoint central quotient of SL(2,R)

A ↦ its central coset; equivalently use conjugation on real traceless 2 × 2 matrices with determinant as an invariant quadratic form. Kernel: {±I} ≅ C₂.

Reference coverageCovering or central quotient

Definition

A ↦ its central coset; equivalently use conjugation on real traceless 2 × 2 matrices with determinant as an invariant quadratic form.

Definition sources

The group map

Domain

\mathrm{SL}(2, \mathbb R)

Codomain

\mathrm{PSL}(2, \mathbb R)

Kernel

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

Image

\mathrm{PSL}(2,\mathbb R) \cong \mathrm{SO}_0(2,1)

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

Reference properties

Property	Value	Evidence
Domain	sl-2-r	Sources
Codomain	psl-2-r	Sources
Specified homomorphism	A ↦ its central coset; equivalently use conjugation on real traceless 2 × 2 matrices with determinant as an invariant quadratic form.	Sources
Kernel	$\{\pm I\} \cong \mathrm{C}_{2}$	Sources
Image	$\mathrm{PSL}(2,\mathbb R) \cong \mathrm{SO}_0(2,1)$	Sources
Injective	No	Sources
Surjective	Yes	Sources
Covering degree	$2$	Sources

Global and representation consequences

Both groups have infinite cyclic fundamental group. With compatible circle generators, this double cover induces multiplication by two on π₁; neither group is the universal cover. [sources]

Related objects and maps

$\mathrm{SL}(2, \mathbb R)$
Real special linear group in two dimensions
Domain group
$\mathrm{PSL}(2, \mathbb R)$
Adjoint real group of type A₁
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 ↦ its central coset; equivalently use conjugation on real traceless 2 × 2 matrices with determinant as an invariant quadratic form.

Sporadic isogenies to orthogonal groupsSection 2.2: SL(2,R) covering the identity component of SO(2,1)Open source

map:sl-2-r-to-psl-2-r#definition

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Domain

sl-2-r

Sporadic isogenies to orthogonal groupsSection 2.2: SL(2,R) covering the identity component of SO(2,1)Open source

map:sl-2-r-to-psl-2-r#domain

Link to claim Suggest a correction

Codomain

psl-2-r

Sporadic isogenies to orthogonal groupsSection 2.2: SL(2,R) covering the identity component of SO(2,1)Open source

map:sl-2-r-to-psl-2-r#codomain

Link to claim Suggest a correction

Specified homomorphism

A ↦ its central coset; equivalently use conjugation on real traceless 2 × 2 matrices with determinant as an invariant quadratic form.

Sporadic isogenies to orthogonal groupsSection 2.2: SL(2,R) covering the identity component of SO(2,1)Open source

map:sl-2-r-to-psl-2-r#formula

Link to claim Suggest a correction

Kernel

{±I} ≅ C₂

Sporadic isogenies to orthogonal groupsSection 2.2: SL(2,R) covering the identity component of SO(2,1)Open source

map:sl-2-r-to-psl-2-r#kernel

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Image

PSL(2,R) ≅ SO₀(2,1)

Sporadic isogenies to orthogonal groupsSection 2.2: SL(2,R) covering the identity component of SO(2,1)Open source

map:sl-2-r-to-psl-2-r#image

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Injective

Sporadic isogenies to orthogonal groupsSection 2.2: SL(2,R) covering the identity component of SO(2,1)Open source

map:sl-2-r-to-psl-2-r#injective

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Surjective

Yes

Sporadic isogenies to orthogonal groupsSection 2.2: SL(2,R) covering the identity component of SO(2,1)Open source

map:sl-2-r-to-psl-2-r#surjective

Link to claim Suggest a correction

Covering degree

Sporadic isogenies to orthogonal groupsSection 2.2: SL(2,R) covering the identity component of SO(2,1)Open source

map:sl-2-r-to-psl-2-r#covering-degree

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Global and representation consequences

Both groups have infinite cyclic fundamental group. With compatible circle generators, this double cover induces multiplication by two on π₁; neither group is the universal cover.

Sporadic isogenies to orthogonal groupsSection 2.2: SL(2,R) covering the identity component of SO(2,1)Open 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:sl-2-r-to-psl-2-r#meaning-1

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Domain group

group:sl-2-r

Sporadic isogenies to orthogonal groupsSection 2.2: SL(2,R) covering the identity component of SO(2,1)Open source

map:sl-2-r-to-psl-2-r#relation-1

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Codomain group

group:psl-2-r

Sporadic isogenies to orthogonal groupsSection 2.2: SL(2,R) covering the identity component of SO(2,1)Open source

map:sl-2-r-to-psl-2-r#relation-2

Link to claim Suggest a correction

SL(2,R)⟶PSL(2,R)\mathrm{SL}(2, \mathbb R) \longrightarrow \mathrm{PSL}(2, \mathbb R)SL(2,R)⟶PSL(2,R)Adjoint central quotient of SL(2,R)

The group map

Reference properties

Global and representation consequences

Related objects and maps

Claims and sources

$\mathrm{SL}(2, \mathbb R) \longrightarrow \mathrm{PSL}(2, \mathbb R)$ Adjoint central quotient of SL(2,R)