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