rotation group SU(2)

E443145

Lie algebra Lie group compact Lie group matrix group real 3-dimensional manifold semisimple Lie group simple Lie group simply connected Lie group special unitary group

The rotation group SU(2) is the Lie group of 2×2 unitary matrices with determinant 1 that serves as the double cover of the three-dimensional rotation group SO(3) and underlies the quantum theory of angular momentum and spin.

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Observed surface forms (6)

Surface form	Occurrences
SU(2)	7
su(2)	0
SO(3)	0
SU(2) Lie group	1
Spin^+(3,1)	1
special unitary group SU(2)	1

Statements (58)

Predicate	Object
instanceOf	Lie algebra ⓘ Lie group ⓘ compact Lie group ⓘ matrix group ⓘ real 3-dimensional manifold ⓘ semisimple Lie group ⓘ simple Lie group ⓘ simply connected Lie group ⓘ special unitary group ⓘ
actsOn	spinor states ⓘ
appearsIn	gauge theories ⓘ particle physics ⓘ representation theory of Lie groups ⓘ
centerIsIsomorphicTo	Z/2Z NERFINISHED ⓘ
coveringMapDegree	2 ⓘ
covers	SO(3) NERFINISHED ⓘ
fundamentalRepresentationCalled	spin-1/2 representation ⓘ
hasCartanSubalgebraDimension	1 ⓘ
hasCenter	{±I} ⓘ
hasCoveringMapTo	SO(3) NERFINISHED ⓘ
hasDefinition	group of 2×2 unitary matrices with determinant 1 ⓘ
hasDeterminantCondition	determinant 1 ⓘ
hasDimension	3 ⓘ
hasDynkinType	A1 ⓘ
hasFundamentalGroup	Z/2Z ⓘ trivial group ⓘ
hasFundamentalRepresentationDimension	2 ⓘ
hasHaarMeasure	finite ⓘ
hasIrreducibleRepresentationsLabeledBy	nonnegative half-integers j ⓘ
hasLieAlgebra	su(2) ⓘ
hasMatrixSize	2×2 ⓘ
hasMaximalTorus	U(1) ⓘ
hasRank	1 ⓘ
hasRealForm	compact real form of A1 ⓘ
hasRepresentationDimensionFormula	2j+1 ⓘ
hasRootSystem	type A1 ⓘ
hasUnitarityCondition	U†U = I ⓘ
hasWeylGroup	Z/2Z ⓘ
isCompact	true ⓘ
isCompactRealFormOf	SL(2,C) NERFINISHED ⓘ
isConnected	true ⓘ
isDiffeomorphicTo	S^3 ⓘ
isDoubleCoverOf	SO(3) NERFINISHED ⓘ
isGaugeGroupOf	weak isospin in the Standard Model ⓘ
isIsomorphicTo	Spin(3) NERFINISHED ⓘ so(3) ⓘ unit quaternions ⓘ
isLocallyIsomorphicTo	SO(3) NERFINISHED ⓘ
isNonAbelian	true ⓘ
isSimple	true ⓘ
isSimplyConnected	true ⓘ
isSpinGroupFor	R^3 ⓘ
isSubgroupOf	SL(2,C) NERFINISHED ⓘ
isTopologically	3-sphere S^3 NERFINISHED ⓘ
isUniversalCoverOf	SO(3) NERFINISHED ⓘ
quotientByCenterIs	SO(3) NERFINISHED ⓘ
underliesTheory	quantum angular momentum ⓘ quantum spin ⓘ

Referenced by (11)

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

SL(2,C) → containsSubgroup → rotation group SU(2) ⓘ

this entity surface form: SU(2)

Yang monopole → gaugeGroup → rotation group SU(2) ⓘ

this entity surface form: SU(2)

SL(2,C) → hasMaximalCompactSubgroup → rotation group SU(2) ⓘ

this entity surface form: SU(2)

Wigner–Eckart theorem → involves → rotation group SU(2) ⓘ

SL(2,C) → isIsomorphicTo → rotation group SU(2) ⓘ

this entity surface form: Spin^+(3,1)

Gruppentheorie und Quantenmechanik → relatedConcept → rotation group SU(2) ⓘ

this entity surface form: special unitary group SU(2)

Pauli matrices → relatedConcept → rotation group SU(2) ⓘ

this entity surface form: SU(2)

Clebsch–Gordan coefficients → relatedTo → rotation group SU(2) ⓘ

this entity surface form: SU(2) Lie group

Yang–Mills existence and mass gap problem → typicalGaugeGroup → rotation group SU(2) ⓘ

this entity surface form: SU(2)

rotation group SO(3) → universalCover → rotation group SU(2) ⓘ

subject surface form: SO(3)

this entity surface form: SU(2)

Yang–Mills theory → usesSymmetryGroup → rotation group SU(2) ⓘ

this entity surface form: SU(2)