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.
All labels observed (3)
| Label | Occurrences |
|---|---|
| SU(N) | 2 |
| 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 ⓘ
linked to:
rotation group SU(2)
SU(2) is isomorphic to the group of unit quaternions ⓘ
linked to:
Quaternions
SU(2) is the double cover of SO(3) ⓘ SU(3) is used in quantum chromodynamics (QCD) ⓘ
linked to:
SU(3)
|
| subsetOf |
SL(n,ℂ)
ⓘ
linked to:
special linear group SL(n,C)
U(n) ⓘ |
| symbol |
SU(n)
ⓘ
linked to:
special unitary group SU(n)
|
| 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
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: special unitary group SU(n) Description of subject: 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.
Referenced by (5)
Full triples — surface form annotated when it differs from this entity's canonical label.
linked to: special unitary group SU(n)
linked to: special unitary group SU(n)
linked to: special unitary group SU(n)
subject linked to:
special unitary group SU(n)
linked to: special unitary group SU(n)