The special unitary group of degree n, denoted SU(n), is the group of n × n unitary matrices with determinant 1. (In general, unitary matrices have complex determinants with modulus 1 but arbitrary phase.) The group operation is that of matrix multiplication. The special unitary group is a subgroup of the unitary group U(n), consisting of all n × n unitary matrices, which is itself a subgroup of the general linear group GL(n, C).
The SU(n) groups find wide application in the Standard Model of particle physics, especially SU(2) in the electroweak interaction and SU(3) in QCD.
The simplest case, SU(1), is the trivial group, having only a single element. The group SU(2) is isomorphic to the group of quaternions of norm 1, and is thus diffeomorphic to the 3-sphere. Since unit quaternions can be used to represent rotations in 3-dimensional space (up to sign), we have a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+I, −I}.
Read more about Special Unitary Group: Properties, Generators, n = 2, n = 3, Lie Algebra, Generalized Special Unitary Group, Important Subgroups
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—Joseph Conrad (1857–1924)