Special unitary group

In mathematics, the special unitary group of degree n, denoted SU(n), is the Lie group of n × n unitary matrices with determinant 1. In mathematics, the special unitary group of degree n, denoted SU(n), is the Lie group of n × n unitary matrices with determinant 1. (More general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.) The group operation is matrix multiplication. The special unitary group is a subgroup of the unitary group U(n), consisting of all n×n unitary matrices. As a compact classical group, U(n) is the group that preserves the standard inner product on Cn. It is itself a subgroup of the general linear group, SU(n) ⊂ U(n) ⊂ 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 quantum chromodynamics. 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), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+I, −I}. SU(2) is also identical to one of the symmetry groups of spinors, Spin(3), that enables a spinor presentation of rotations. The special unitary group SU(n) is a real Lie group (though not a complex Lie group). Its dimension as a real manifold is n2 − 1. Topologically, it is compact and simply connected. Algebraically, it is a simple Lie group (meaning its Lie algebra is simple; see below). The center of SU(n) is isomorphic to the cyclic group Zn, and is composed of the diagonal matrices ζ I for ζ an nth root of unity and I the n×n identity matrix. Its outer automorphism group, for n ≥ 3, is Z2, while the outer automorphism group of SU(2) is the trivial group. A maximal torus, of rank n − 1, is given by the set of diagonal matrices with determinant 1. The Weyl group is the symmetric group Sn, which is represented by signed permutation matrices (the signs being necessary to ensure the determinant is 1).