special unitary group SU(n)

E593508

Lie group compact Lie group matrix group semisimple Lie group simple Lie group (for n ≥ 2) special unitary group

The special unitary group SU(n) is a fundamental compact Lie group consisting of n×n unitary matrices with determinant 1, central in mathematics and physics, especially in quantum theory and gauge symmetries.

Try in SPARQL Jump to: Surface forms Disambiguation Statements Elicitation Referenced by

All labels observed (2)

Label	Occurrences
SU(N)	2
special unitary group SU(n) canonical	1

How this entity was disambiguated

This entity first appeared as the object of triple T6456475 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.

NED1 Entity disambiguation (via context triple) gpt-5-mini-2025-08-07

Target entity: special unitary group SU(n)
Context triple: [Lie group, hasExample, special unitary group SU(n)]

A. SU(3)
SU(3) is the special unitary group of degree three, a Lie group fundamental to the mathematical description of the strong interaction and the classification of hadrons in particle physics.
B. special orthogonal group SO(n)
The special orthogonal group SO(n) is the group of all n×n real rotation matrices with determinant 1, representing orientation-preserving isometries of n-dimensional Euclidean space that fix the origin.
C. rotation group SU(2)
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.
D. SL(2,C)
SL(2,C) is the complex special linear group of 2×2 matrices with determinant 1, which serves as the double cover and spinor representation group of the proper orthochronous Lorentz group in four-dimensional spacetime.
E. U(1)
U(1) is the group of complex numbers with absolute value 1 under multiplication, commonly representing the symmetry group of electromagnetism and other abelian gauge theories.
F. None of above. chosen
G. Unsure - the case is ambiguous/there is not enough information to decide.

NED2 Entity disambiguation (via description) gpt-5-mini-2025-08-07

Target entity: special unitary group SU(n)
Target entity description: The special unitary group SU(n) is a fundamental compact Lie group consisting of n×n unitary matrices with determinant 1, central in mathematics and physics, especially in quantum theory and gauge symmetries.

A. SU(3)
SU(3) is the special unitary group of degree three, a Lie group fundamental to the mathematical description of the strong interaction and the classification of hadrons in particle physics.
B. special orthogonal group SO(n)
The special orthogonal group SO(n) is the group of all n×n real rotation matrices with determinant 1, representing orientation-preserving isometries of n-dimensional Euclidean space that fix the origin.
C. rotation group SU(2)
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.
D. SL(2,C)
SL(2,C) is the complex special linear group of 2×2 matrices with determinant 1, which serves as the double cover and spinor representation group of the proper orthochronous Lorentz group in four-dimensional spacetime.
E. U(1)
U(1) is the group of complex numbers with absolute value 1 under multiplication, commonly representing the symmetry group of electromagnetism and other abelian gauge theories.
F. None of above. chosen

Statements (48)

Predicate	Object
instanceOf	Lie group ⓘ compact Lie group ⓘ matrix group ⓘ semisimple Lie group ⓘ simple Lie group (for n ≥ 2) ⓘ special unitary group ⓘ
center	cyclic group of order n ⓘ {e^{2πik/n} I_n \| k = 0,…,n−1} ⓘ
definedAs	group of n×n unitary matrices with determinant 1 ⓘ
dimensionOverℝ	n² − 1 ⓘ
DynkinType	A_{n−1} ⓘ
fundamentalGroup	trivial (for n ≥ 2) ⓘ
HaarMeasure	admits a bi-invariant probability Haar measure ⓘ
hasProperty	all finite-dimensional unitary representations are completely reducible ⓘ
is	a classical Lie group ⓘ a compact real form of SL(n,ℂ) ⓘ a compact, connected, simply connected, simple Lie group of type A_{n−1} for n ≥ 2 ⓘ a complex algebraic group ⓘ a real Lie group ⓘ a simple compact Lie group for n ≥ 2 ⓘ abelian for n = 1 ⓘ compact ⓘ non-abelian for n ≥ 2 ⓘ simply connected (for n ≥ 2) ⓘ the intersection U(n) ∩ SL(n,ℂ) ⓘ
LieAlgebra	su(n) ⓘ
LieAlgebraDescription	traceless skew-Hermitian n×n complex matrices ⓘ
LieAlgebraDimension	n² − 1 ⓘ
maximalTorus	diagonal unitary matrices with determinant 1 ⓘ
maximalTorusDimension	n − 1 ⓘ
overField	complex numbers ℂ ⓘ
rank	n − 1 ⓘ
representationTheory	irreducible representations classified by highest weights ⓘ
roleInPhysics	flavor and color symmetries in the Standard Model ⓘ internal symmetry group in gauge theories ⓘ
specialCase	SU(1) is the trivial group ⓘ SU(2) is diffeomorphic to the 3-sphere S³ NERFINISHED ⓘ SU(2) is isomorphic to the group of unit quaternions NERFINISHED ⓘ SU(2) is the double cover of SO(3) ⓘ SU(3) is used in quantum chromodynamics (QCD) NERFINISHED ⓘ
subsetOf	SL(n,ℂ) NERFINISHED ⓘ U(n) ⓘ
symbol	SU(n) NERFINISHED ⓘ
topology	a compact smooth manifold of real dimension n² − 1 ⓘ
usedIn	gauge theory ⓘ particle physics ⓘ quantum field theory ⓘ quantum mechanics ⓘ

How these facts were elicited

Referenced by (3)

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

Lie group → hasExample → special unitary group SU(n) ⓘ

Yang–Mills existence and mass gap problem → typicalGaugeGroup → special unitary group SU(n) ⓘ

this entity surface form: SU(N)

Yang–Mills theory → usesSymmetryGroup → special unitary group SU(n) ⓘ