Special unitary group

In abstract algebra, the special unitary group of degree n is the group of n by n unitary matrices with determinant 1 and entries from the field C of complex numbers, with the group operation that of matrix multiplication. It is written as SU(n). This is a subgroup of the unitary group U(n), itself a subgroup of the general linear group Gl(n,C).

The special unitary group SU(n) is a Lie group of dimension n²-1.

The corresponding Lie algebra is denoted by su(n). su(n) is spanned by the traceless antihermitian nxn complex matrices. For example, the following matrices form a basis for su(2) over R:

<math>i{\sigma}_x = \begin{bmatrix}

0 & i \\ i & 0 \end{bmatrix}<math>

<math>i{\sigma}_y = \begin{bmatrix}

0 & \!\!\!1 \\ -1 & 0 \end{bmatrix}<math>

<math>i{\sigma}_z = \begin{bmatrix}

i & 0 \\ 0 & \!\!\!-i \end{bmatrix}<math> (i is the square root of -1.) This representation is often used in quantum mechanics (see Pauli matrices), to represent the spin of fundamental particles such as electrons. They also serve as unit vectors for the description of our 3 spatial dimensions in quantum relativity.

Note that the product of any two different generators is another generator, and that the generators anticommute. Together with the identity matrix (times i),

<math> iI_2 = \begin{bmatrix}

i & 0 \\ 0 & i \end{bmatrix}<math> these are also generators of the Lie algebra u(2).

note: make clearer the fact that under matrix multiplication (which is anticommutative in this case), we generate the Clifford algebra Cl_3, whereas you generate the Lie algebra u(2) with commutator brackets instead.