In mathematics, the special unitary group of degree , denoted , is the Lie group of  unitary matrices with determinant 1.

The matrices of the more general unitary group 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 normal 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 . It is itself a subgroup of the general linear group, .

The  groups find wide application in the Standard Model of particle physics, especially  in the electroweak interaction and  in quantum chromodynamics.
The groups  are important in quantum computing, as they represent the possible quantum logic gate operations in a quantum circuit with  qubits and thus  basis states. (Alternatively, the more general unitary group  can be used, since
multiplying by a global phase factor  does not change the expectation values of a quantum operator.)

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 strictly real Lie group (vs. a more general 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 , and is composed of the diagonal matrices  for  an ‑th root of unity and  the  identity matrix.

Its outer automorphism group for  is  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 of SU(n) 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 , can be identified with the set of traceless anti‑Hermitian  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  of  consists of  skew-Hermitian matrices with trace zero. This (real) Lie algebra has dimension . 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  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 commutator is:

and the corresponding anticommutator is:

The factor of  in the commutation relation arises from the physics convention and is not present when using the mathematicians' convention.

The conventional normalization condition is

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)

Using matrix multiplication for the binary operation,  forms a group,

where the overline denotes complex conjugation.

Diffeomorphism with the 3-sphere S3
If we consider  as a pair in  where  and , then the equation  becomes

This is the equation of the 3-sphere S3. This can also be seen using an embedding: the map

where  denotes the set of 2 by 2 complex matrices, is an injective real linear map (by considering  diffeomorphic to  and  diffeomorphic to ). 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 , 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 group of versors
Quaternions of norm 1 are called versors since they generate the rotation group SO(3):
The  matrix:

can be mapped to the quaternion

This map is in fact a group isomorphism. Additionally, the determinant of the matrix is the squared 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 group of versors.

Relation to spatial rotations 

Every versor is naturally associated to a spatial rotation in 3 dimensions, and the product of versors is associated to the composition of the associated rotations. Furthermore, every rotation arises from exactly two versors 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  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.

This can also be written as  using the Pauli matrices.

These satisfy the quaternion relationships      and   The commutator bracket is therefore specified by

The above generators are related to the Pauli matrices by   and   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. They also correspond to the Pauli X, Y, and Z gates, which are standard generators for the single qubit gates, corresponding to 3d-rotations about the axes of the Bloch sphere.

The Lie algebra serves to work out the representations of .

The group SU(3)

 is an 8-dimensional simple Lie group consisting of all  unitary matrices with determinant 1.

Topology
The group  is a simply-connected, compact Lie group. Its topological structure can be understood by noting that SU(3) acts transitively on the unit sphere  in . 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  with fiber . 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 -bundles over  are classified by  since any such bundle can be constructed by looking at trivial bundles on the two hemispheres  and looking at the transition function on their intersection, which is homotopy equivalent to , so
 

Then, all such transition functions are classified by homotopy classes of maps
 

and as  rather than ,  cannot be the trivial bundle , 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  is well-understood. Descriptions of these representations, from the point of view of its complexified Lie algebra , 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  of  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  of  consists of  skew-Hermitian matrices with trace zero.

The complexification of the Lie algebra  is , the space of all  complex matrices with trace zero. A Cartan subalgebra then consists of the diagonal matrices with trace zero, which we identify with vectors in  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 , 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  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 .

Example
An important example of this type of group is the Picard modular group  which acts (projectively) on complex hyperbolic space of degree two, in the same way that  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 , which is isomorphic to .

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)
 where  denotes the complex conjugate of the complex number .

This group is isomorphic to  and  where the numbers separated by a comma refer to the signature of the quadratic form preserved by the group. The expression  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   the 2×2 identity matrix,  and  and the elements  and  all anticommute, as in quaternions. Also  is still a square root of  (negative of the identity matrix), whereas  are not, unlike in quaternions. For both quaternions and coquaternions, all scalar quantities are treated as implicit multiples of   and notated as  .

The coquaternion  with scalar , has conjugate  similar to Hamilton's quaternions. The quadratic form is 

Note that the 2-sheet hyperboloid  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, and the practice of SU (1,1) interferometry has been introduced.

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 

Writing  complex number arithmetic shows

where 
Therefore,  so that their ratio lies in the open disk.

See also

 Unitary group
 Projective special unitary group, 
 Orthogonal group
 Generalizations of Pauli matrices
 Representation theory of SU(2)

Footnotes

Citations

References
 
 

Lie groups
Mathematical physics