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
Famous quotes containing the words special and/or group:
“The very best reason parents are so special . . . is because we are the holders of a priceless gift, a gift we received from countless generations we never knew, a gift that only we now possess and only we can give to our children. That unique gift, of course, is the gift of ourselves. Whatever we can do to give that gift, and to help others receive it, is worth the challenge of all our human endeavor.”
—Fred Rogers (20th century)
“The virtue of dress rehearsals is that they are a free show for a select group of artists and friends of the author, and where for one unique evening the audience is almost expurgated of idiots.”
—Alfred Jarry (1873–1907)