Lie theory · Group

$\mathrm{SL}(2, \mathbb R)$ Real special linear group in two dimensions

Matrix form, a quotient of the universal cover by 2Z. The real 2 × 2 matrices of determinant 1.

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Reference coverageNoncompact Lie groupSimple algebraGlobal form

Definition

The real 2 × 2 matrices of determinant 1.

Definition sources

Real dimension: $3$
Center: $\mathrm{C}_{2}$
Fundamental group π₁: Z

Reference properties

Property	Value	Evidence
Real dimension	$3$	Sources
Real rank	$1$	Sources
Center	$\mathrm{C}_{2}$	Sources
Fundamental group π₁	Z	Sources
Compact	No	Sources
Connected	Yes	Sources
Abelian	No	Sources
Simple real Lie algebra	Yes	Sources
Global form	Matrix form, a quotient of the universal cover by 2Z	Sources
Real Lie algebra	$\mathfrak{sl}(2, \mathbb R)$	Sources
Maximal compact subgroup	$\mathrm{SO}(2) \cong \mathrm{U}(1)$	Sources

Global structure and topology

Polar decomposition gives the manifold SO(2) × R², so the fundamental group is Z. The center is {±I}. The isomorphism with SU(1,1) is an isomorphism of real matrix Lie groups after a complex change of basis. [sources]

Representation category and descent

Finite-dimensional complex irreducible representations are restrictions of sl(2,C) highest-weight modules with integer highest weight n ≥ 0 and dimension n + 1. The central element −I acts by (−1)ⁿ. These are distinct from infinite-dimensional unitary representations. [sources]

Names and aliases

SU(1,1)

Aliases identify the stated object under this record’s conventions. Type names and broader search terms are kept separate.

Related objects and maps

$\mathfrak{sl}(2, \mathbb R)$
sl(2, R) — real Lie algebra
Real Lie algebra
$\mathfrak{sl}(2, \mathbb R)$
sl(2, R) real form
Real-form context
$\mathrm{SL}(2, \mathbb R) \longrightarrow \mathrm{PSL}(2, \mathbb R)$
Adjoint central quotient of SL(2,R)
Explicit homomorphism
$\widetilde{\mathrm{SL}(2,\mathbb R)} \longrightarrow \mathrm{SL}(2,\mathbb R)$
Universal covering map of SL(2,R)
Explicit homomorphism
$\widetilde{\mathrm{SL}(2,\mathbb R)}$
Simply connected group integrating sl(2,R)
Specified connected covering 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

The real 2 × 2 matrices of determinant 1.

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

group:sl-2-r#definition

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Real dimension

3

Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source
Lecture 6: Maximal Root, Exponents, Coxeter Numbers, Dual RepresentationsSection 32.4, printed pages 175–176: Coxeter and dual Coxeter numbersOpen source

group:sl-2-r#dimension

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Real rank

1

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

group:sl-2-r#real-rank

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Center

C₂

Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source

group:sl-2-r#center

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Fundamental group π₁

Z

Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source
Lecture 17: Classification of Connected Compact and Complex Reductive GroupsSection 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)Open source
Conrad and Landesman · Compact Lie groupsAppendix O, pages 231–232: fundamental groups and covering groups; Appendix V: central quotientsOpen source

Derivation: Cartan decomposition is a deformation retraction onto the stated maximal compact subgroup; compute its fundamental group.

Premises: Maximal compact subgroup: SO(2) ≅ U(1) · π₁(SO(2))=Z, π₁(SO(n))=C₂ for n≥3, π₁(U(2))=Z · For S(U(2)×U(2)), the determinant map has simply connected fiber SU(2)×SU(2), giving π₁=Z

group:sl-2-r#pi1

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Compact

No

Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source

group:sl-2-r#compact

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Connected

Yes

Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source

group:sl-2-r#connected

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Abelian

No

Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source

group:sl-2-r#abelian

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Simple real Lie algebra

Yes

Lecture 15: Classification of Real Forms of Semisimple Lie AlgebrasSections 41.2–41.4: compact, split, and classical real formsOpen source
Conrad and Landesman · Compact Lie groupsLecture 25 and Appendix T: root systems and classical root computationsOpen source

group:sl-2-r#simple-algebra

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Global form

Matrix form, a quotient of the universal cover by 2Z

Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source
Lecture 17: Classification of Connected Compact and Complex Reductive GroupsSection 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)Open source
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

group:sl-2-r#global-form

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Real Lie algebra

sl(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

group:sl-2-r#lie-algebra

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Maximal compact subgroup

SO(2) ≅ U(1)

Lie Groups and Lie AlgebrasSections 6.1–6.3: classical matrix groups and their Lie algebrasOpen source
Lie Groups Beyond an Introduction — Chapter VI: Structure Theory of Semisimple GroupsChapter VI, Sections 3–4: Cartan and Iwasawa decompositions; Theorem 6.46 (Iwasawa), printed pages 317–318; SU(p,q) and SO(p,q) examples on pages 314–316Open source
Lecture 17: Classification of Connected Compact and Complex Reductive GroupsSection 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)Open source

group:sl-2-r#maximal-compact

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Global structure and topology

Polar decomposition gives the manifold SO(2) × R², so the fundamental group is Z. The center is {±I}. The isomorphism with SU(1,1) is an isomorphism of real matrix Lie groups after a complex change of basis.

Lecture 17: Classification of Connected Compact and Complex Reductive GroupsSection 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)Open source
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

group:sl-2-r#topology-1

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Representation category and descent

Finite-dimensional complex irreducible representations are restrictions of sl(2,C) highest-weight modules with integer highest weight n ≥ 0 and dimension n + 1. The central element −I acts by (−1)ⁿ. These are distinct from infinite-dimensional unitary representations.

Conrad and Landesman · Compact Lie groupsLectures 27–28: highest-weight classification and fundamental representationsOpen source
Lie Groups and Lie AlgebrasRemark 36.7, printed page 196: finite-dimensional representations of the universal cover factor through SL(2,R)Open source

group:sl-2-r#representations-1

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Real Lie algebra

algebra:sl-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

group:sl-2-r#relation-1

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Real-form context

real-form:sl-2-r

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

group:sl-2-r#relation-2

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Explicit homomorphism

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

Lecture 17: Classification of Connected Compact and Complex Reductive GroupsSection 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)Open source
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

group:sl-2-r#relation-3

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Explicit homomorphism

map:universal-sl-2-r-to-sl-2-r

Lecture 17: Classification of Connected Compact and Complex Reductive GroupsSection 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)Open source
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

group:sl-2-r#relation-4

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Specified connected covering group

group:universal-cover-sl-2-r

Lecture 17: Classification of Connected Compact and Complex Reductive GroupsSection 43.2: polar decomposition; Example 43.10 for SL(n,R) and SL(n,C)Open source
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

group:sl-2-r#relation-5

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