Special unitary group

In mathematics, the special unitary group of degree , denoted , is the Lie group of unitary matrices with determinant 1.
The more general unitary matrices may have complex determinants with absolute value 1, rather than real 1 in the special case.
The group operation is matrix multiplication. The special unitary group is a subgroup of the unitary group , consisting of all unitary matrices. As a compact classical group, is the group that preserves the standard inner product on formula_1. It is itself a subgroup of the general linear group, formula_2.
The groups find wide application in the Standard Model of particle physics, especially in the electroweak interaction and in quantum chromodynamics.
The simplest case, , is the trivial group, having only a single element. The group 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 (up to sign), there is a surjective homomorphism from to the rotation group whose kernel is . 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 is a real Lie group (though not a complex Lie group). Its dimension as a real manifold is . Topologically, it is compact and simply connected. Algebraically, it is a simple Lie group (meaning its Lie algebra is simple; see below).
The center of is isomorphic to the cyclic group formula_3, and is composed of the diagonal matrices for an th root of unity and the "n"×"n" identity matrix.
Its outer automorphism group, for , is formula_4, while the outer automorphism group of is the trivial group.
A maximal torus, of rank , is given by the set of diagonal matrices with determinant 1. The Weyl group is the symmetric group , which is represented by signed permutation matrices (the signs being necessary to ensure the determinant is 1).
The Lie algebra of , denoted by formula_5, can be identified with the set of traceless antiHermitian complex matrices, with the regular commutator as a Lie bracket. Particle physicists often use a different, equivalent representation: The set of traceless Hermitian complex matrices with Lie bracket given by times the commutator.
Lie algebra.
The Lie algebra formula_5 of formula_7 consists of formula_8 skew-Hermitian matrices with trace zero. This (real) Lie algebra has dimension formula_9. More information about the structure of this Lie algebra can be found below in the section "Lie algebra structure."
Fundamental representation.
In the physics literature, it is common to identify the Lie algebra with the space of trace-zero "Hermitian" (rather than the skew-Hermitian) matrices. That is to say, the physicists' Lie algebra differs by a factor of formula_10 from the mathematicians'. With this convention, one can then choose generators that are traceless Hermitian complex matrices, where:
where the are the structure constants and are antisymmetric in all indices, while the -coefficients are symmetric in all indices.
As a consequence, the anticommutator and commutator are:
The factor of formula_10 in the commutation relations arises from the physics convention and is not present when using the mathematicians' convention.
We may 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:
The group SU(2).
 is the following group,
where the overline denotes complex conjugation.
Diffeomorphism with S3.
If we consider formula_17 as a pair in formula_18 where formula_19 and formula_20, then the equation formula_21 becomes
formula_22
formula_23
formula_24
formula_23
formula_26
formula_23
formula_28
This is the equation of the 3-sphere S3. This can also be seen using an embedding: the map
where formula_30 denotes the set of 2 by 2 complex matrices, is an injective real linear map (by considering formula_18 diffeomorphic to formula_32 and formula_30 diffeomorphic to formula_34). Hence, the restriction of to the 3-sphere (since modulus is 1), denoted , is an embedding of the 3-sphere onto a compact submanifold of formula_30, namely .
Therefore, as a manifold, is diffeomorphic to , which shows that is simply connected and that can be endowed with the structure of a compact, connected Lie group.
Isomorphism with unit quaternions.
The complex matrix:
can be mapped to the quaternion:
This map is in fact an isomorphism. Additionally, the determinant of the matrix is the square norm of the corresponding quaternion. Clearly any matrix in is of this form and, since it has determinant 1, the corresponding quaternion has norm 1. Thus is isomorphic to the unit quaternions.
Relation to spatial rotations.
Every unit quaternion is naturally associated to a spatial rotation in 3 dimensions, and the product of two quaternions is associated to the composition of the associated rotations. Furthermore, every rotation arises from exactly two unit quaternions in this fashion. In short: there is a 2:1 surjective homomorphism from SU(2) to SO(3); consequently SO(3) is isomorphic to the quotient group SU(2)/{±I}, the manifold underlying SO(3) is obtained by identifying antipodal points of the 3-sphere , and SU(2) is the universal cover of SO(3).
Lie algebra.
The Lie algebra of consists of formula_38 skew-Hermitian matrices with trace zero. Explicitly, this means
The Lie algebra is then generated by the following matrices,
which have the form of the general element specified above.
These satisfy the quaternion relationships formula_41 formula_42 and formula_43 The commutator bracket is therefore specified by
The above generators are related to the Pauli matrices by formula_45 and formula_46 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.
The group SU(3).
formula_47 is an 8-dimensional simple Lie group consisting of all unitary matrices with determinant 1.
Topology.
The group formula_47 is a simply-connected, compact Lie group. Its topological structure can be understood by noting that SU(3) acts transitively on the unit sphere formula_49 in formula_50. The stabilizer of an arbitrary point in the sphere is isomorphic to SU(2), which topologically is a 3-sphere. It then follows that SU(3) is a fiber bundle over the base formula_49 with fiber formula_52. Since the fibers and the base are simply connected, the simple connectedness of SU(3) then follows by means of a standard topological result (the long exact sequence of homotopy groups for fiber bundles).
The formula_53-bundles over formula_49 are classified by formula_55 since any such bundle can be constructed by looking at trivial bundles on the two hemispheres formula_56 and looking at the transition function on their intersection which is homotopy equivalent to formula_57, so
formula_58
Then, all such transition functions are classified by homotopy classes of maps
formula_59
and as formula_60 rather than formula_61, formula_47 cannot be the trivial bundle formula_63, and therefore must be the unique nontrivial (twisted) bundle. This can be shown by looking at the induced long exact sequence on homotopy groups.
Representation theory.
The representation theory of formula_47 is well understood. Descriptions of these representations, from the point of view of its complexified Lie algebra formula_65, may be found in the articles on Lie algebra representations or the Clebsch–Gordan coefficients for SU(3).
Lie algebra.
The generators, , of the Lie algebra formula_66 of formula_47 in the defining (particle physics, Hermitian) representation, are
where , the Gell-Mann matrices, are the analog of the Pauli matrices for :
These span all traceless Hermitian matrices of the Lie algebra, as required. Note that are antisymmetric.
They obey the relations
or, equivalently,
The are the structure constants of the Lie algebra, given by
while all other not related to these by permutation are zero. In general, they vanish unless they contain an odd number of indices from the set {2, 5, 7}.
The symmetric coefficients take the values
They vanish if the number of indices from the set {2, 5, 7} is odd.
A generic group element generated by a traceless 3×3 Hermitian matrix , normalized as , can be expressed as a "second order" matrix polynomial in :
where
Lie algebra structure.
As noted above, the Lie algebra formula_5 of formula_7 consists of formula_78 skew-Hermitian matrices with trace zero.
The complexification of the Lie algebra formula_5 is formula_80, the space of all formula_78 complex matrices with trace zero. A Cartan subalgebra then consists of the diagonal matrices with trace zero, which we identify with vectors in formula_82 whose entries sum to zero. The roots then consist of all the permutations of .
A choice of simple roots is
So, is of rank and its Dynkin diagram is given by , a chain of nodes: ... Its Cartan matrix is
Its Weyl group or Coxeter group is the symmetric group , the symmetry group of the -simplex.
Generalized special unitary group.
For a field , the generalized special unitary group over "F", , is the group of all linear transformations of determinant 1 of a vector space of rank over which leave invariant a nondegenerate, Hermitian form of signature . This group is often referred to as the special unitary group of signature over . The field can be replaced by a commutative ring, in which case the vector space is replaced by a free module.
Specifically, fix a Hermitian matrix of signature in formula_85, then all
satisfy
Often one will see the notation without reference to a ring or field; in this case, the ring or field being referred to is formula_88 and this gives one of the classical Lie groups. The standard choice for when is
However, there may be better choices for for certain dimensions which exhibit more behaviour under restriction to subrings of formula_88.
Example.
An important example of this type of group is the Picard modular group formula_91 which acts (projectively) on complex hyperbolic space of degree two, in the same way that formula_92 acts (projectively) 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 .
A further example is formula_93, which is isomorphic to formula_94.
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 that are important in GUT physics are, for ,
where × denotes the direct product and , 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 is and of 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. 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: , , and .
One may finally mention that is the double covering group of , a relation that plays an important role in the theory of rotations of 2-spinors in non-relativistic quantum mechanics.
The group SU(1,1).
formula_98 where formula_99 denotes the complex conjugate of the complex number .
This group is locally isomorphic to and where the numbers separated by a comma refer to the signature of the quadratic form preserved by the group. The expression formula_100 in the definition of is an Hermitian form which becomes an isotropic quadratic form when and are expanded with their real components. An early appearance of this group was as the "unit sphere" of coquaternions, introduced by James Cockle in 1852. Let
Then formula_102 formula_103 the 2×2 identity matrix, formula_104 and formula_105 and the elements and all anticommute, like regular quaternions. Also formula_10 is still a square root of (negative of the identity matrix), whereas formula_107 are not, unlike the quaternions. For both quaternions and coquaternions, all scalar quantities are treated as implicit multiples of  , called the "unit (co)quaternion", and occasionally explicitly notated as  .
The coquaternion formula_108 with scalar , has conjugate formula_109 similar to Hamilton's quaternions. The quadratic form is formula_110
Note that the 2-sheet hyperboloid formula_111 corresponds to the imaginary units in the algebra so that any point on this hyperboloid can be used as a pole of a sinusoidal wave according to Euler's formula.
The hyperboloid is stable under , illustrating the isomorphism with . The variability of the pole of a wave, as noted in studies of polarization, might view elliptical polarization as an exhibit of the elliptical shape of a wave with The Poincaré sphere model used since 1892 has been compared to a 2-sheet hyperboloid model.
When an element of is interpreted as a Möbius transformation, it leaves the unit disk stable, so this group represents the motions of the Poincaré disk model of hyperbolic plane geometry.
Indeed, for a point in the complex projective line, the action of is given by
since in projective coordinates formula_113
Writing formula_114 complex number arithmetic shows
where formula_116
Therefore, formula_117 so that their ratio lies in the open disk.