su3

SU(3), also written as su(3) to refer to its Lie algebra, denotes the special unitary group of degree 3. It consists of all 3x3 complex matrices U with U†U = I and det U = 1. As a Lie group it is compact, connected, and simply connected, and has dimension 8. The group is the canonical example of a nonabelian compact Lie group of rank 2.

Its Lie algebra su(3) is the set of 3x3 complex traceless anti-Hermitian matrices. An especially common basis is given by the eight Gell-Mann matrices λ1–λ8, which satisfy [λa, λb] = 2i f^{abc} λc and determine the structure constants f^{abc}.

The maximal torus of SU(3) has dimension 2, and the rank is 2; a typical diagonal subgroup is diag(e^{i θ1}, e^{i θ2}, e^{-i(θ1+θ2)}). The center Z(SU(3)) consists of the scalar matrices e^{2π i k/3} I with k = 0,1,2, isomorphic to Z3. The complexification of su(3) is sl(3, C). The root system is type A2.

In representation theory, the defining 3-dimensional representation and its complex conjugate 3̄ are fundamental; the adjoint representation has dimension 8. The group is simple; its Dynkin diagram is A2.

In physics, SU(3) is the gauge group of quantum chromodynamics, where its eight generators correspond to the eight gluons. The Standard Model uses SU(3) color × SU(2)L × U(1)Y as its gauge group. Subgroups such as SU(2) can be embedded in SU(3) in various ways.