PSL(2,\mathbb{C})

E898490

Lie group centerless group complex Lie group connected Lie group group linear algebraic group non-compact Lie group semisimple Lie group simple Lie group

PSL(2,ℂ) is the group of Möbius transformations acting as all biholomorphic automorphisms of the Riemann sphere.

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Observed surface forms (2)

Surface form	Occurrences
projective linear group PGL(2)	1
PSL(2,ℂ)	0

Statements (49)

Predicate	Object
instanceOf	Lie group ⓘ centerless group ⓘ complex Lie group ⓘ connected Lie group ⓘ group ⓘ linear algebraic group ⓘ non-compact Lie group ⓘ semisimple Lie group ⓘ simple Lie group ⓘ
acts3TransitivelyOn	Riemann sphere NERFINISHED ⓘ
actsBy	Möbius transformations NERFINISHED ⓘ
actsOn	Riemann sphere NERFINISHED ⓘ extended complex plane ⓘ
actsTransitivelyOn	Riemann sphere NERFINISHED ⓘ
containsSubgroup	PSL(2,ℝ) NERFINISHED ⓘ PSU(2) NERFINISHED ⓘ
definedAs	SL(2,ℂ)/{±I} NERFINISHED ⓘ
hasCenter	trivial group ⓘ
hasDimension	3 complex dimensions ⓘ 6 real dimensions ⓘ
hasElementForm	z ↦ (az + b)/(cz + d) with ad − bc ≠ 0 ⓘ
hasFundamentalGroup	ℤ/2ℤ ⓘ
hasLieAlgebra	sl(2,ℂ) ⓘ
hasMaximalCompactSubgroup	PSU(2) NERFINISHED ⓘ
hasProjectiveRealization	PGL(2,ℂ) GENERATED ⓘ
hasQuotientMapFrom	SL(2,ℂ) ⓘ
hasRank	1 ⓘ
hasRealForm	PSL(2,ℝ) NERFINISHED ⓘ
hasRealPoints	PSL(2,ℝ) NERFINISHED ⓘ
hasTopology	real 6-dimensional manifold ⓘ
hasType	A₁ (complex simple Lie type) ⓘ
hasUniversalCover	SL(2,ℂ) NERFINISHED ⓘ
isAdjointFormOf	SL(2,ℂ) NERFINISHED ⓘ
isAutomorphismGroupOf	Riemann sphere NERFINISHED ⓘ complex projective line ℂℙ¹ NERFINISHED ⓘ
isConnected	true ⓘ
isGroupOf	biholomorphic automorphisms of the Riemann sphere ⓘ
isIsomorphicTo	Isom⁺(ℍ³) ⓘ PGL(2,ℂ) NERFINISHED ⓘ group of orientation-preserving isometries of hyperbolic 3-space ⓘ
isNonAbelian	true ⓘ
isPerfectGroup	true ⓘ
isRealLieGroupOfType	rank 1 non-compact simple ⓘ
isSimplyConnected	false ⓘ
isUsedIn	3-manifold theory ⓘ Kleinian group theory ⓘ complex dynamics ⓘ hyperbolic geometry ⓘ
quotientOf	SL(2,ℂ) ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Riemann sphere → automorphismGroupIsomorphicTo → PSL(2,\mathbb{C}) ⓘ

Clebsch–Aronhold invariants → relatedTo → PSL(2,\mathbb{C}) ⓘ

this entity surface form: projective linear group PGL(2)