# Special unitary group

# Special unitary group

In mathematics, the **special unitary group** of degree *n*, denoted SU(*n*), is the Lie group of *n* × *n* unitary matrices with determinant 1.

(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 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 **C***n*.^{[1]} 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.^{[5]}

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 (up to sign), there is a surjective homomorphism from SU(2) to the rotation group SO(3) whose kernel is {+*I*, −*I*}.^{[2]} 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 (though not a complex Lie group). Its dimension as a real manifold is *n*2 − 1. Topologically, it is compact and simply connected.^{[6]} Algebraically, it is a simple Lie group (meaning its Lie algebra is simple; see below).^{[7]}

The center of SU(*n*) is isomorphic to the cyclic group Z*n*, 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 signs being necessary to ensure the determinant is 1).

`TheLie algebraofSU(`

*n*), denoted by, can be identified with the set oftracelessantiHermitian*n*×*n*complex matrices, with the regularcommutatoras Lie bracket.Particle physicistsoften use a different, equivalent representation: the set of tracelessHermitian*n*×*n*complex matrices with Lie bracket given by−*i*times the commutator.Lie algebra

`The Lie algebraofconsists ofskew-Hermitianmatrices with trace zero.`

^{[8]}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 offrom the mathematicians'. With this convention, one can then choose generators*T*_{a}that aretracelessHermitiancomplex*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, the anticommutator and commutator are:

`The factor ofin 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 (*n*2 − 1) -dimensional adjoint representation, the generators are represented by (*n*2 − 1) × (*n*2 − 1) matrices, whose elements are defined by the structure constants themselves:

The group SU(2)

SU(2) is the following group,^{[9]}

where the overline denotes complex conjugation.

There is a 2:1 homomorphism from SU(2) to SO(3).

Diffeomorphism with S3

`If we consideras a pair inwhereand, then the equationbecomes`

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

where M(2, **C**) denotes the set of 2 by 2 complex matrices, is an injective real linear map (by considering **C**2 diffeomorphic to **R**4 and M(2, **C**) diffeomorphic to **R**8). Hence, the restriction of *φ* to the 3-sphere (since modulus is 1), denoted *S*3, is an embedding of the 3-sphere onto a compact submanifold of M(2, **C**), namely *φ*(*S*3) = SU(2).

Therefore, as a manifold, *S*3 is diffeomorphic to SU(2), which shows that *S*3 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 norm of the corresponding quaternion. Clearly any matrix in SU(2) is of this form and, since it has determinant 1, the corresponding quaternion has norm 1. Thus SU(2) is isomorphic to the unit quaternions.^{[10]}

Lie algebra

`TheLie algebraofSU(2)consists ofskew-Hermitianmatrices with trace zero.`

^{[11]}Explicitly, this meansThe Lie algebra is then generated by the following matrices,

which have the form of the general element specified above.

`These satisfy thequaternionrelationshipsandThecommutator bracketis therefore specified by`

`The above generators are related to thePauli matricesbyandThis representation is routinely used inquantum mechanicsto represent thespinoffundamental particlessuch aselectrons. They also serve asunit vectorsfor the description of our 3 spatial dimensions inloop quantum gravity.`

The Lie algebra serves to work out the representations of SU(2).

The group SU(3)

Topology

`The groupSU(3)is a simply-connected, compact Lie group.`

^{[12]}Its topological structure can be understood by noting that SU(3) acts transitively on the unit spherein. Thestabilizerof an arbitrary point in the sphere is isomorphic to SU(2), which topologically is a 3-sphere. It then follows that SU(3) is afiber bundleover the basewith 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 (thelong exact sequence of homotopy groupsfor fiber bundles).^{[13]}`The SU(2)-bundles overare classified by, and asrather than, SU(3) cannot be the trivial bundle, and therefore must be the unique nontrivial (twisted) bundle.`

Representation theory

`The representation theory ofSU(3)is well understood.`

^{[14]}Descriptions of these representations, from the point of view of its complexified Lie algebra, may be found in the articles onLie algebra representationsorthe Clebsch–Gordan coefficients for SU(3).Lie algebra

`The generators,T, of the Lie algebraofSU(3)in the defining (particle physics, Hermitian) representation, are`

where λ, the Gell-Mann matrices, are the SU(3) analog of the Pauli matrices for SU(2):

These *λ*a span all traceless Hermitian matrices H of the Lie algebra, as required. Note that *λ*2, *λ*5, *λ*7 are antisymmetric.

They obey the relations

or, equivalently,

- .

The f are the structure constants of the Lie algebra, given by

- ,,,

while all other *fabc* 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}.^{[3]}

The symmetric coefficients *d* take the values

They vanish if the number of indices from the set {2, 5, 7} is odd.

A generic SU(3) group element generated by a traceless 3×3 Hermitian matrix H, normalized as tr(*H*2) = 2, can be expressed as a *second order* matrix polynomial in H^{[15]}:

where

Lie algebra structure

`As noted above, the Lie algebraofconsists ofskew-Hermitianmatrices with trace zero.`

^{[16]}`Thecomplexificationof the Lie algebrais, the space of allcomplex matrices with trace zero.`

^{[17]}A Cartan subalgebra then consists of the diagonal matrices with trace zero,^{[18]}which we identify with vectors inwhose entries sum to zero. Therootsthen consist of all the*n*(*n*− 1)permutations of(1, −1, 0, ..., 0).A choice of simple roots is

So, SU(*n*) is of rank *n* − 1 and its Dynkin diagram is given by A*n*−1, a chain of *n* − 1 nodes: [[INLINE_IMAGE|//upload.wikimedia.org/wikipedia/commons/8/8b/Dyn-node.png|undefined|Dyn-node.png|h24|w9]][[INLINE_IMAGE|//upload.wikimedia.org/wikipedia/commons/b/b3/Dyn-3.png|undefined|Dyn-3.png|h24|w6]][[INLINE_IMAGE|//upload.wikimedia.org/wikipedia/commons/8/8b/Dyn-node.png|undefined|Dyn-node.png|h24|w9]][[INLINE_IMAGE|//upload.wikimedia.org/wikipedia/commons/b/b3/Dyn-3.png|undefined|Dyn-3.png|h24|w6]][[INLINE_IMAGE|//upload.wikimedia.org/wikipedia/commons/8/8b/Dyn-node.png|undefined|Dyn-node.png|h24|w9]][[INLINE_IMAGE|//upload.wikimedia.org/wikipedia/commons/b/b3/Dyn-3.png|undefined|Dyn-3.png|h24|w6]]...[[INLINE_IMAGE|//upload.wikimedia.org/wikipedia/commons/b/b3/Dyn-3.png|undefined|Dyn-3.png|h24|w6]][[INLINE_IMAGE|//upload.wikimedia.org/wikipedia/commons/8/8b/Dyn-node.png|undefined|Dyn-node.png|h24|w9]].^{[19]} Its Cartan matrix is

Its Weyl group or Coxeter group is the symmetric group *S**n*, the symmetry group of the (*n* − 1)-simplex.

Generalized special unitary group

For a field *F*, the * 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 (

*p*,

*q*). This group is often referred to as the

*. The field*

**p qF***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 (projectively) on complex hyperbolic space of degree two, in the same way that SL(2,9;**Z**) 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 HC2.^{[20]}

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) ,^{[4]} 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.

The group SU(1,1)

`wheredenotes thecomplex conjugateof the complex number`

*u*.`This group is isomorphic toSO(2,1)andSL(2,ℝ)`

^{[21]}where the numbers separated by a comma refer to thesignatureof thequadratic formpreserved by the group. The expressionin the definition ofSU(1,1)is anHermitian formwhich becomes anisotropic quadratic formwhen*u*and*v*are expanded with their real components. An early appearance of this group was as the "unit sphere" ofcoquaternions, introduced byJames Cocklein 1852. Let`ThenAlso i is a square root of −1 (negative of the identity matrix), while j2= k2= identity matrix. Similar to Hamilton's quaternions, here`

*q*=*w*+*x*i +*y*j +*z*k is a coquaternion with conjugate*q** =*w*–*x*i –*y*j –*z*k. The elements i, j, and k have theanticommutativityproperty so that the quadratic form is`Note that the 2-sheethyperboloidcorresponds to theimaginary unitsin the algebra so that any point on this hyperbola can be used as a`

**pole**of a sinusoidal wave according toEuler's formula.The hyperboloid is stable under SU(1,1), illustrating the isomorphism with SO(2,1). 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 pole *p* ≠ ± i. The Poincaré sphere model used since 1892 has been compared to a 2-sheet hyperboloid model.^{[22]}

When an element of SU(1,1) 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 [z.1] in the complex projective line, the action of SU(1,1) is given by

- wheredenotes the similarity ofprojective coordinates.

`Writingcomplex number arithmetic shows`

`WhereThereforeso that their ratio lies in the open disk.`

See also

Unitary group

Projective special unitary group, PSU(

*n*)Generalizations of Pauli matrices

Representation theory of SU(2)

Remarks

## References

*n*) and hence SU(

*n*) in terms of preservation of the standard inner product on ℂ

*n*, see Classical group.

*n*) is the compact real form of Sp(2

*n*,

**C**). It is sometimes denoted USp(

*2n*). The dimension of the Sp(

*n*)-matrices is 2

*n*× 2

*n*.

*Quarks & Leptons: An Introductory Course in Modern Particle Physics*. John Wiley & Sons. ISBN 0-471-88741-2.

*Lie Groups, Lie Algebras, and Representations: An Elementary Introduction*, Graduate Texts in Mathematics,

**222**(2nd ed.), Springer, ISBN 978-3319134666 Proposition 13.11

*Classical Groups for Physicists*, Wiley-Interscience. ISBN 0471965057 .

*Journal of Mathematical Physics*.

**12**(4): 673–681. Bibcode:1971JMP....12..673R. doi:10.1063/1.1665634.; Curtright, T L; Zachos, C K (2015). "Elementary results for the fundamental representation of SU(3)".

*Reports on Mathematical Physics*.

**76**(3): 401–404. arXiv:1508.00868. Bibcode:2015RpMP...76..401C. doi:10.1016/S0034-4877(15)30040-9.