In mathematics, the special unitary
group of degree n, denoted SU(n), is the
Lie group of n×n unitary matrices with
determinant 1. The group operation is
that of 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, 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.
Properties
The special unitary group SU(n) is a
real 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.
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 Lie algebra of SU(n), denoted by
su(n), can be identified with the set of
traceless antihermitian n×n complex
matrices, with the regular commutator as
Lie bracket. Particle physicists often
use a different, equivalent
representation: the set of traceless
hermitian n×n complex matrices with Lie
bracket given by −i times the
commutator.
Infinitesimal generators
The Lie algebra su(n) can be generated
by n2 operators , i, j= 1, 2, ..., n,
which satisfy the commutator
relationships
for i, j, k, ℓ = 1, 2, ..., n, where δjk
denotes the Kronecker delta.
Additionally, the operator
satisfies
which implies that the number of
independent generators of the Lie
algebra is n2 − 1 .
= Fundamental representation=
In the defining, or fundamental,
representation of su(n) the generators
Ta are represented by traceless
hermitian matrices complex n×n matrices,
where:
where the f are the structure constants
and are antisymmetric in all indices,
while the d-coefficients are symmetric
in all indices. As a consequence:
We also take
as a normalization convention.
= Adjoint representation=
In the -dimensional adjoint
representation, the generators are
represented by × matrices, whose
elements are defined by the structure
constants themselves:
n = 2
SU(2) is the following group,
where the overline denotes complex
conjugation.
Now, consider the following map,
where M(2, C) denotes the set of 2 by 2
complex matrices. By considering C2
diffeomorphic to R4 and M(2, C)
diffeomorphic to R8, we can see that φ
is an injective real linear map and
hence an embedding. Now, considering the
restriction of φ to the 3-sphere,
denoted S3, we can see that this is an
embedding of the 3-sphere onto a compact
submanifold of M(2, C). However, it is
also clear that φ(S3) = SU(2).
Therefore, as a manifold S3 is
diffeomorphic to SU(2) and so SU(2) is a
compact, connected Lie group.
The Lie algebra of SU(2) is
It is easily verified that matrices of
this form have trace zero and are
antihermitian. The Lie algebra is then
generated by the following matrices,
which are easily seen to have the form
of the general element specified above.
These satisfy u3u2 = −u2u3 = −u1 and
u2u1 = −u1u2 = −u3. The commutator
bracket is therefore specified by
The above generators are related to the
Pauli matrices by u1 = i σ1,u2 = −i σ2
and u3 = i σ3. This representation is
routinely 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 loop quantum gravity.
The Lie algebra serves to work out the
representations of SU(2).
n = 3
The generators of su(3), T, in the
defining representation, are:
where λ the Gell-Mann matrices, are the
SU(3) analog of the Pauli matrices for
SU(2):
These  span all traceless Hermitian
matrices H of the Lie algebra, as
required.
They obey the relations
(or, equivalently, ).
The f are the structure constants of the
Lie algebra, given by:
while all other  not related to these by
permutation are zero.
The symmetric coefficients d take the
values:
As a topological space, SU(3) is a
direct product of a 3-sphere and a
5-sphere, S3⊗ S5.
A generic SU(3) group element generated
by a traceless 3×3 hermitian matrix H,
normalized as tr(H2) = 2 , is given by
where
for elementary representation theory
facts.
Lie algebra structure
The above representation bases
generalize to n > 3, using generalized
Pauli matrices.
If we choose an particular basis, then
the subspace of traceless diagonal n×n
matrices with imaginary entries forms
an-dimensional Cartan subalgebra.
Complexify the Lie algebra, so that any
traceless n×n matrix is now allowed. The
weight eigenvectors are the Cartan
subalgebra itself, as well as the
matrices with only one nonzero entry
which is off diagonal. Even though the
Cartan subalgebra h is only-dimensional,
to simplify calculations, it is often
convenient to introduce an auxiliary
element, the unit matrix which commutes
with everything else for the purpose of
computing weights—and that only. So, we
have a basis where the i-th basis vector
is the matrix with 1 on the i-th
diagonal entry and zero elsewhere.
Weights would then be given by n
coordinates and the sum over all n
coordinates has to be zero.
So, SU(n) is of rank n − 1 and its
Dynkin diagram is given by An−1, a chain
of n − 1 vertices, o−o−o−o---o. Its root
system consists of n(n − 1) roots
spanning a n − 1 Euclidean space. Here,
we use n redundant coordinates instead
of n − 1 to emphasize the symmetries of
the root system.
In other words, we are embedding this n
− 1 dimensional vector space in an
n-dimensional one. Thus, the roots
consists of all the n(n − 1)
permutations of. The construction given
above explains why. A choice of simple
roots is
Its Cartan matrix is
Its Weyl group or Coxeter group is the
symmetric group Sn, the symmetry group
of the-simplex.
Generalized special unitary group
For a field F, the generalized special
unitary group over F, SU(p, q; F), is
the group of all linear transformations
of determinant 1 of a vector space of
rank n = p + q over F which leave
invariant a nondegenerate, Hermitian
form of signature. This group is often
referred to as the special unitary group
of signature p q over F. The field F can
be replaced by a commutative ring, in
which case the vector space is replaced
by a free module.
Specifically, fix a Hermitian matrix A
of signature p q in GL(n, R), then all
satisfy
Often one will see the notation SU(p, q)
without reference to a ring or field; in
this case, the ring or field being
referred to is C and this gives one of
the classical Lie groups. The standard
choice for A when F = C is
However there may be better choices for
A for certain dimensions which exhibit
more behaviour under restriction to
subrings of C.
= Example=
An important example of this type of
group is the Picard modular group SU(2,
1; Z[i]) which acts on complex
hyperbolic space of degree two, in the
same way that SL(2,9;Z) acts on real
hyperbolic space of dimension two. In
2005 Gábor Francsics and Peter Lax
computed an explicit fundamental domain
for the action of this group on HC2.
A further example is SU(1, 1; C), which
is isomorphic to SL(2,R).
Important subgroups
In physics the special unitary group is
used to represent bosonic symmetries. In
theories of symmetry breaking it is
important to be able to find the
subgroups of the special unitary group.
Subgroups of SU(n) that are important in
GUT physics are, for p > 1, n − p > 1 ,
where × denotes the direct product and
U(1), known as the circle group, is the
multiplicative group of all complex
numbers with absolute value 1.
For completeness, there are also the
orthogonal and symplectic subgroups,
Since the rank of SU(n) is n − 1 and of
U(1) is 1, a useful check is that the
sum of the ranks of the subgroups is
less than or equal to the rank of the
original group. SU(n) is a subgroup of
various other Lie groups,
See spin group, and simple Lie groups
for E6, E7, and G2.
There are also the accidental
isomorphisms: SU(4) = Spin(6) , SU(2) =
Spin(3) = Sp(1) , and U(1) = Spin(2) =
SO(2) .
One may finally mention that SU(2) is
the double covering group of SO(3), a
relation that plays an important role in
the theory of rotations of 2-spinors in
non-relativistic quantum mechanics.
See also
Projective special unitary group, PSU(n)
Generalizations of Pauli matrices
Remarks
References
