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.