Special unitary group

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

If the field F is the field of real or complex numbers, then the special unitary group SU(n,F) is a Lie group.

A common matrix representation of the generatorss of SU(2) is:

( 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,

these are also the generators of U(2). These 4 matrices then form a complete set on 2x2 matrices.