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.
Observed surface forms (2)
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 ⓘ |
Referenced by (3)
Full triples — surface form annotated when it differs from this entity's canonical label.
this entity surface form:
SU(N)
this entity surface form:
SU(N)