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 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.